organize begin stochastic problem give description main make place distributed complement proof collect norm norm f fx fy gx recall free delay sequel analyze describe two closely first average nesterov far collect giving dual vector primal mirror dual approximation mirror proximal essentially often recall lipschitz particular imply lipschitz continuous respect convex assumption show optimization function lipschitz smoothly differentiable loss g bound standard assumption sharp choice factor extend mirror update receive stochastic gradient point simple delay uniform delay analysis admit delay long mirror descent simple replace averaging mirror descent follow asynchronous asynchronous vector rule method involve potentially different delay delay smooth arise drastically slightly significant overcome delay smoothness delay perturb since variability result delay essentially second penalty asymptotically negligible set tt delay gradient stepsize mirror sec theorem corollary average convexity addition hold satisfy home corollary theorem asymptotically negligible favorable implication stochastic scenario distribute delay gradient delay abstract away procedure simplex though leave value mirror combine sec tt theorem consequence powerful turn develop application include assign sec scenario simple dataset sample among worker sized subset streaming worker receive stream make simplify worker receive pick replacement base master topology delay protocol compute master time lag distance latter section convergence rate node describe architecture master scheme worker compute master gradient parallel gradient time parallelization alternate delay worker master maintain parameter round update begin computation worker master worker parameter pass update worker worker delay early apply delay cc master worker master node time master gradient compute master communication toward node information store level protocol combine delay delay average worker master work span rooted master node phase leave master parameter neighbor simultaneously parameter depth child fig receive communication iteration leave leaf gradient parent parent gradient leaf average tree take gradient vector round average current gradient pass span description formally delay date vector child master child child child compute random node distribution hierarchy leaf iteration receive gradient parent master root receive delay entire give rise update architecture corollary section achieve asymptotically procedure use synchronization update characteristic network also assume explicitly centralized cyclic protocol update update fx allocate centralized cyclic delay delay compute gradient cyclic assume communication compute sample theorems beneficial master receive locally delay master centralize cyclic local algorithm unit take centralized number average architecture cyclic delay plug count corollary provide architecture unit cyclic figs cyclic compare cyclic locally average algorithm locally always guarantee well quantify improvement path cyclic tree distribute ignore logarithmic cyclic possible modify communication computation compute gradient machine learn natural language reasonable cyclic delay strong convergence guarantee protocol rate delay though focus theoretical present understand aspect real cyclic delay specifically solve regression problem news article economic feature article otherwise optimization delay cyclic method master term worker cyclic delay several worker figure stochastic convert assume define take master master worker compute master receive centralized gradient perform optimality centralize discuss delay enjoy centralized number negligible similar demonstrate linear small investigate network asymptotic communication delay benefit parallelization nevertheless allow delay asynchronous significant improvement section collect technical key choice assumption follow identity hand sum bregman equality straightforward equality lipschitz recall definition negativity replace bind recall strong xt convexity sum proof decrease stepsize delay convexity lipschitz continuity q xt xt non probabilistic xt imply consequence remain second conjugate addition sigma combine xt xt combine early inequality bind essential eq completely prove mirror descent average control involve differently expand condition following leave gradient cardinality chebyshev inequality theorem term proof eq proof theorem term eq non write bound second give expectation happen first condition term bregman delay preserve benefit stochastic relax synchronization specifically resource failure asynchronous penalty delay omit brevity microsoft fellowship support science fellowship program grateful communication like read manuscript collect continuity property proximal operator average dual xt result essentially many context property mirror frequently difference mirror descent minimize mirror q thus q convexity complete update recall lipschitz continuous easy thus schwarz last slightly tight lemma triangle inequality delay indexing know expectation increase h old substituting complete proof essential equality unconstraine upper bind simple side hand side evaluate hand inequality rely application boundedness boundedness iterate gradient without iterate iteration delay berkeley edu department electrical engineering california berkeley delay development master perform parameter worker compute local parallel delay take problem huge internet contribution delay asymptotically achieve overcome communication synchronization requirement architecture asynchronous scale iteration delay additionally statistical stochastic convex computing xt xt analyze receive gradient central asymptotically stochastic delay delay gradient distribute master
sp sp suffice assume moreover r r rp rp rp proof condition exploitation phase attain exploration price price kn fraction offline q plug pricing quantity use pricing limited achieve near bandit style algorithm key designing algorithm price estimate exist respective base see contribution reduction base pricing pricing style setting well well particular use crowdsource conjecture conjecture problematic particular bandit algorithm hard code amount time exploration rise immediate general demand desirable extend possibly respect offline second theorem extend demand demand distribution price one fix follow randomized benchmark fact explore resource pricing iid immediately extend make direction demand time alternatively promise apply algorithm final price small neither likewise price price sale increase price surprising main near regret help question grateful slightly purpose regular distribution least offline benchmark recall agent fix price maximize offline player sp price regular symmetry offer function jensen bind agent never multiplicative strategy compare demand environment sampling yet player correlate exceed include therefore agent draw I excee item happen environment moreover definition correlation q always therefore combine regular price kp pa kp n kp omit kp kp us pass immediate demand approximate offline benchmark several sale denote begin characterization sale moreover log concavity sensitivity support fact plug complete version detail online appear version microsoft designing maximize mechanism offer leave possibly agent maximize mechanism know scenario compare mechanism offline free mechanism whose less offline assumption relax match demand multi armed pricing mab intuition mab set level limit treat price armed bandit another small agent leave price account iid fix distribution bayesian demand tailor one assumption avoid know costly likewise significant demand easily like demand know mechanism call sense specification mechanism spirit demand integral mechanism face mechanism benchmark specific demand paper mechanism offline mechanism depend demand price demand satisfy strong monotone hazard one satisfied price commonly appeal reason first agent needs offer human agent former much easier reveal entire private reveal private third dominant side price mechanism particularly useful demand advance consider item sequentially observe manner influence make item independently demand assume support normalize f whenever agent item item compatible never value observe select henceforth call design pricing strategy compare benchmark assumption demand maximal allow distribution henceforth demand distribution constrain demand fit mab round set arm payoff maximize set correspond mab round exploit mab payoff specifically price behind applicable goal mab converge price wrong maximize treat exploration instead explore arm price schedule payoff much arm suboptimal even index assign history round arm depend history index estimate expect single elegant prior apply section bandit algorithm respective near base pricing limited agent pricing strategy offline pricing trivially follow detail free pricing demand offline minus emphasize pricing know mechanism compatible price focus power well fix offline henceforth fix strategy technical pricing strategy achieve expect benchmark surprisingly demand price minus demand moreover mab sufficiently small improve free demand offline minus constant hazard price item recall price moreover price pricing strategy whose expect offline minus constant condition meet demand hazard property satisfy distribution within offline demand monotone hazard depend achieve pricing vary parameter improve theorem nice trivial arbitrary parameter distribution large depend directly comparable dependence distinction constant fact essentially match latter benchmark expect pricing pricing regret demand demand informally hope theorem uninformative provide another online price meaningful big free pricing demand expect offline minus maximal offline pricing management literature see overview prior price without set study detailed early iid theorem assume depend imply special provide super case continuous specialized equivalent prove upper bound benchmark inferior distinction pricing strategy exploitation demand regret demand parameterize parameter parametrization rely know parametrization current upper upper apply demand improvement demand distribution bind pricing demand dependent pricing round mechanism two benchmark online mechanism unlike logarithmic multiplicative work paper consider opposed agent price design mechanism multiplicative elaborate price mechanism regret al online arrive period designing result offline online multiplicative mab duration reward mab rich literature operation computer science economic proper discussion beyond background relevant work prior free mab payoffs mab lipschitz stochastic payoff e call round arm index pick index arm index bandit arm index essentially available confidence accordingly new price ucb strategy exploitation sample exploitation explore price accord schedule payoff define adopt obvious elegant since expect payoff price word ucb specify standard price sale analysis suboptimal way elegant unfortunately nature adopt trick appropriate proving event rely standard framework round alternative observe maximize payoff mab independent price exploit special mab payoff price arm rely regret neither behind limited informally mab converge highest wrong quickly main conceptual set appeal separate exploitation explore arm continuously observe payoff suboptimal arm choose rarely assign score confidence payoff index depend payoff arm provide reflect limited interpret price ucb payoff strategy obvious elegant rather respectively number define ucb technical analysis sum via elegant limited pricing pricing numerical pick price breaking tie recall fix price approximate round ucb ucb sale price round current sale rate division equal define hold namely least suitable radius want subject quantity observable standard literature use elaborate bad sale see perform well price sale imply proof sale specification nk price set optimal fix price strategy choice parameter regret item think experiment consider pricing continue version realize realize item latter round sale realize realize give execution sale high event describe rest argue loss low probability negligible round via concentration guarantee focus sale respective importantly rather event hold sp second indeed generality th select pricing happen play price union long hold happen pricing price round eventually regret total price round consequence follow immediately sp radius round price price price line bind third sale rearrange bind key instead realize effectively constraint brevity denote sp kp k later respectively set select price plug claim summarize finding far active price active fact active price let price respect tie break else rest note simplify remain take regret improve demand informally formal sale demand easily claim price regular moreover constant demand achieve distribution map sale regularity pick hazard maximizer bind imply regret particularly arm regularity third claim sp np p therefore improve final desirable theorem use pricing strategy pricing set trivial bound arbitrary demand pricing
practice carry approach acknowledgement early stage manuscript grateful anonymous reference part european community reflect universit taylor information technology bayesian analysis sequence tool hoeffding concentration integration hoeffding inequality introduce feedback combine tool although regret bind yet regret potential bayesian tool encounter pac decade st contribution supervise approach lie flexibility bayesian pac optimize resolution pac bound linear classification pac learning pac bayesian long treat suitable almost hoeffde canonical pac handling combine bayesian sequence certain sequentially dependent expectation prominent field reinforcement advantage pac learn recently include suggest state regularization mutual incorporate bellman bayesian justify guarantee batch reinforcement able knowledge informative confirm irrelevant algorithm case pac par situation exploitation batch minimal root number state applicable want bad action reason difficulty pac exploitation observe rest density within evaluate usually situation reason action minimal action pair size bad round resolve weighted strategy commonly bandit usage introduce difficulty influence play influence subsequent variable pac bayesian approach future work sampling grow enable take variance explain gap result bound combine bernstein work survey main present pac bayesian derive bandit conclude variable function lemma precede pac hoeffding tight certain derivation dependent belong distribute special reference depend concentration bernoulli variable type theory ct proof find approximation factorial bernoulli variable empirical average divergence function ct convex convex interval sequentially inequality lemma ready hoeffding verify minimize ia simplify make equal almost equal worse contribution however low lemma tighter relax preferable pac basis take root back physics relate posterior arm distribute distribute irrelevant lemma fix probability great lemma expectation substitution bind obtain simultaneously well tt key ingredient proof ta alternative pac hoeffding inequality theorem martingale ia ta ram hence go bind ta e ta derive obtain great base way furthermore special reward action expect reward eq provide section adapt provide section assertion last substitute back choose get integration conclude technical lemma proof regret tight n unable
select sequence experiment end integer site specific bias denote bias reads observe probability accord view replaced length length error concave infer translate relative per case bias reduce equation slight difference name describe read end appear formulate equivalent directional rna seq confusion pair seq strategy first follow length appropriately site close preferred protocol alternatively site double unclear read analogous modeling error read beyond address issue allele remark possible consider entire compatibility rna seq principle impractical compatibility practical allele infer paper formalism describe utilize probability unknown infer map previously equivalent formulation describe close nice know numerical unique seq address appear biology literature relate property mathematically mean different relative testing equivalent compatibility full certain assumption read typically em read gene three pair initially abundance read assign assign read abundance consider red green read compatibility see subject notational z yx must maximize conditional maximize em initialize expectation pair property every illustration figure em theory derivation multi important case e square equivalent count variance possibly variance multinomial well count suitable computational inference formulation furthermore square approach constrain non heuristic approach except publish bayesian rather infer dirichlet mle use em information theory abundance estimate important seq rna seq differentially term frequently abundance desirable model equivalent virtue condition sum multinomial find analysis observe even biological behave phenomenon refer dispersion alternative multinomial instead binomial poisson thorough differ beyond scope key single uniquely drawback differential gene quantification quantification differential poisson model quantification virtue binomial equivalent quantification different relative abundance generalize read done read quantification although view model optimize seq rna seq advanced year review discuss publish rna seq technology reduce bias sequence technology result throughput development lead rna seq relative abundance introduction eventually single read long rna solve sequence reveal future seq modeling utilize rna sequence require assign short question light remark model relevant practice sequencing technology believe read length read essential accurate quantification differential gene family map read issue read bp long read multi issue effective correction denominator affect abundance protocol expense read quantification fewer note seq uniformity across yet see therefore paper well seq protocol bias model correction datum possible bias affect rna seq protocol sequence establish quantify progress rna seq relative abundance review benchmark cast seq systematic benchmark complexity aspect seq fully connection relative abundance estimation abundance estimate analysis perform reference almost rna seq novel even extensively annotate crucial time abundance accurate ability relative length currently length local estimate abundance available distant reason communication efficiently remain open helpful discussion understand rna seq lead interpretation formulation seq lead seq comment equivalence multinomial rna seq appendix provide preliminary version valuable comment insight equivalence poisson multinomial rna seq model pair end read model extra integer composition note part make convention induce composition composition range element composition equivalence position end align induce weak composition crucial equal factor expression weak eq q number seq mean distribute generalize read replace derivation easy notation name structure experiment primary three l together position alignment length compatibility abundance rate poisson bias weight effective length give thm corollary definition example remark thm conjecture berkeley edu rna seq rapidly become technology application seq quantification accurate measurement relative read review describe approach also explain formulation publish rna seq explain relative abundance crucial model quantification seq sequencing rna seq array include genome annotation comprehensive gene genome seq resolution probe rna seq bioinformatics draw primary quantification topic identification significant compare difficult impossible scope constitute rna seq single quantification hope rna seq ultimately success seq accurate abundance focus problem abundance quantification review begin rna seq quantification rna seq rna seq rna expression refer generate although case consist rna abundance perform translate rna seq amount rna seq allow measurement absolute infer relative able currently common explain outline section model special approximation model individual relative likelihood computation describe rna seq next discuss rna seq examine development yet resort reader rna relative abundance count region length total proxy abundance count understand correspond relative organized rna seq rna seq model another word label publish author model model contain end uniquely read pair end read read uniquely simple modeling bias read read equivalently multi read read formulation pair pair read correspond end pair subsequently uniformity library first model pair empirically random strategy bias bias model content addition modeling effect error read mention ad hoc way review connect dash line read different abundance multiple gene read replace comprise genomic basis genomic number prove projective demonstrate present normalization read uniquely interest abundance identifying replace read count suitably feature restriction read major valuable omit many consist uniquely adjustment equation relevant abundance estimating abundance gene infer frequency pool sequence sequence species rna seq abundance however read show equivalence apply paper publication abundance rna express tag term rna seq assumption est issue comment likelihood describe read read otherwise compatibility eq denominator directly abundance make suggest substantially due addition length derive denominator become apparent inclusion probably denominator length scalar likelihood correct since abundance evaluate qualitatively presence absence denominator result seq denominator read read reason read
inequality inequality immediate production production production sequence consist member I positive integer list latter inequality well eq many certainly ordinal theoretic ordinal formalize part type ordinal smoothly obtain consist color edge color black consist size graph former sequence bad subsequence bad contradiction graph induce subsequence one conjecture follow corollary actually present comparable great element closure quasi category system computer logic type adjoint see induce finitely branch whenever exist finitely branch quasi relation branching rx mx linearization linearization put v fy fy fy order monotone category category category quasi order branching simulation order branching simulation composition z rx category quasi identity rs category resp category category space introduce function originally attempt difficult category section seq object object object former prove part production sequence exactly write assertion monotone define rr lin lin lin rx r rr branching sr x aa transformation transformation resp call lin natural identity lin composite inclusion assertion whenever author thank dr anonymous aid scientific c education science remark teacher type symbol quasi equal mind iff short theory index recursive language learnable monotone inclusion preserve various closure closure monotone direct image type embed category finitely branch simulation subspace space linearity closure branching game target language set symbol short mind topology reverse mathematic finite learnable language learnable system pattern language combinatorial preserve motivate example bring teacher language concatenation observe l follow preserve type game another red x q system finitely branch relation closure preserve image function monotone inclusion regard topological space copy finite discrete topological monotone plus characterization function mx rx coherence stable rx v nothing direct monotone question parallelism trace continuous finitely branch stable closure continuous unbounded finitely order finitely branch set system quasi send branching send coherence question establish monotone continuous organize next part closure language computational section quasi system quasi language computational indexing ordinal section monotone answer category categorical category record sequential category computation duality operator class set relation sequence possibly short quasi bad upper order lattice sequence mean segment say infinite node ordinal supremum immediate extension root number accord tree great tree say system let class module finitely generate sub extended pattern language bound language elementary learnable alphabet alphabet empty definition n u u v closure study algebraic language b I language closure closure language put fact alphabet close positive integer complement nice lemma finitely branch prove preserve alphabet class class subset viewpoint algebraic regard learn system teacher learner teacher hypothesis notion complete lattice element lattice equal lattice whenever finite subset lattice quasi lattice every lattice assertion equivalence obviously assertion lattice ic ic ic iterate process construct elastic closure question preserve closure property sequence advance theoretic topological operation set preserve employ theoretic type argument nonempty set yx arbitrary union intersection generators generator build mean finite sequence boolean formula assignment element generator generator intersection generator hausdorff distinct n ij gx gb say contain set equivalent monotone boolean oracle finitely branching relation define every positivity boolean equivalent ml rx sequence element ii ny ny n n n useful lemma proposition conversely r topology mx finitely branch therefore corollary mind language hold topology positive topology respect topology monotone boolean formula respect positive recall must l monotone simply write monotonicity obvious continuous open preserve range subset inverse image contradiction system fix alphabet belong closure guess closure closure monotone useful derive proof case empty word machine otherwise try oracle partition prove every production sequence empty word production infinite production contradiction done prove counterpart similar
unknown finding solve advanced field however dependence conservative belief evidence element evidence interval avoid taylor algebra initial interval uncertainty ref uncertainty box taylor also review interval polynomial bernstein ref propagate ds close interval support intuition provide extract solve expansion bernstein range bernstein polynomial mathematically separate characterize dynamical uncertain parameter deterministic solved numerically obtain polynomial initial general type order use transform find stochastic condition ie ji expand wiener polynomial variable basis polynomial coefficient polynomial response sec orthogonality property polynomial obtain differential numerically expansion evaluate quadrature nonlinearity basis numerical integration sec integrate however bound ref bernstein compact thank polynomial ds structure ds probability approximate pdf density function two ds white autocorrelation body sec normally distribute find induced structure moment time evolution propagate ode solve central moment structure ds box envelope c use estimate use confidence estimate singleton utility theory function confidence trust cdf cdf obtain variable vary laplace insufficient reasoning pdfs probability million cdf present quantity present approach randomly transformation pdf propagate pdf fig evidence measure indicate much trust lack threshold depend consequence drop two f construct point wise offer use theory action function decision make purpose paper propagate achieve propagate approximation evolution box incorporate author propagate uncertainty polynomial arithmetic moment structure estimate interest probability addition make scalar conjunction cm mm cm author edu lack model uncertainty structure close interval propagation propagate propose uncertainty order uncertainty parametrization approximation probability evolution equation knowledge combine different singleton transformation decision evolution arise deal require propagate segregation create propagate uncertainty evolution evolution govern kolmogorov analytical pdfs literature number technique mixture closure method pdf precisely perfectly practice value amount information available systematic uncertainty great decision system choose representative approximation model evolution equation polynomial propagate uncertainty combine propagation finite box structure uncertainty quantity response classic whenever decision close make propagation problem conclusion future consider uncertain initial vector time variant moment characterize distribution condition characterize structure quantify uncertainty cumulative distribution uncertain interested follow three main looking determine cumulative utility make secondary include order ds structure close represent ds structure x xx cumulative box close induce unique box uniquely body cumulative plausibility function thus cumulative function compute expectation belief finite distribution indicator expectation needed si give construct singleton pdf utility theory applicable way quantify amount take relatively define summarize deal uncertainty variable mean thus confident low
associate unique htbp xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle black yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle node xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle partition easy analyze z z share minimize set partition partition htbp yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle node yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle node edge thick sample retain sub submatrix figure algorithm graph graph retain graph sub un adjacency matrix adjacency see yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift xshift cm rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle thick yshift cm xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle bl correctness adjacency matrix independently appendix expect complexity establish ideally attribute balanced otherwise ny ib iy converge replace eq go size legend south east width height marker xlabel ylabel color mark cd marker value blue dash bind partition restrict define however depend ny nc probability pos east marker node ylabel title blue bar cd explicit red thick color marker thick table scale legend south east height xlabel nod title mark thick bar mark red dash black marker table pos east height pt xlabel ylabel title thick error mark marker thick dash marker thick dash legend pos south east height xlabel color blue bar marker table theory color black marker size value approximation tight assume chernoff sake remark report usually overall advantageous high behavior observation attribute configuration configuration representation represent phenomenon visually pos east width xlabel configuration rank ylabel occurrence marker table color txt color table x color attribute configuration concentrate log select whose configuration occur attribute occur partition attribute occur set attribute say uniform node edge graph graph probability remain require empirically evaluate question graph behave implement available experiment machine ghz processor run three use matrix guarantee valid empirically various xlabel node ylabel black white mark bar prop title xlabel cycle name white color blue mark bar cd mark prop report edge graph grow constant graph confirm observation furthermore strong indeed scale xlabel ylabel node cycle blue error cd prop title xlabel nod cycle list blue thick bar cd prop scalability size various compare vs trial edge sampling graph hour graph million node graph million well least time term number node title pos south xlabel ylabel mark bars cd error cluster mu mark thick explicit mark naive south east blue thick explicit dash mark table x naive naive scalability plot run across range generate therefore grow empirically title pos north east xlabel ylabel mark mark x cluster red dash title pos north node blue thick bar cd explicit mark table mark thick dash cd naive theoretical guarantee algorithm time vary towards run run legend cell anchor east legend pos north east title xlabel ylabel relative name white color blue mark thick mark thick green mark triangle thick color thick table color black mark legend style anchor east pos outer north east title xlabel attribute cycle list thick color green mark table color triangle thick thick table color black mark triangle well configuration diversity hence tendency run since interested plotted factor increase growth reasonably million feasible value title xlabel node ylabel pos mark thick title node legend pos color thick function vary particular fix vary effect section increase exponentially title xlabel ylabel run mark bar mark ds cs xlabel color blue mark thick bar mark cs cs highlight sampling run million node hour currently work rigorously prove guarantee currently investigate high search technique locality lsh apply contain attribute configuration therefore set least partitioning produce first partition without poisson chernoff plug q pos north ylabel coordinate naive legend pos north east xlabel node ylabel coordinate e rectangle cs title legend pos north xlabel coordinate east legend pos north east xlabel nod run coordinate scale legend north east xlabel ylabel title attribute ylabel cycle black mark color blue thick scale title xlabel ylabel cycle list scale title xlabel ylabel run name coordinate scale title xlabel attribute cycle name coordinate scale title xlabel title xlabel ylabel relative run white node attribute node th digit representation problem sample desire prove figure yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle
refinement step accurate approximation probabilistic bring variance sampling parallel monte carlo simulation two possible implement step decide final augment failure krige involve second prove applicability basic reliability example involve mean standard successively influence result carlo reference result agreement monte failure correction also correction increase dimension krige misclassification surrogate krige refinement input instrumental cost induce gradient design search approximate number call number call carlo inspired concern dimension figure longitudinal support fix zero nonlinear elastic surface consider critical load prescribed service hybrid post krige smooth deterministic failure account krige quasi optimal importance algebra meta factor original krige accurate estimate refinement stop krige surrogate build parallel turn application reasonably multiple include reliability author engineering convention national project denote variance sake clarity variance variance also denote coefficient eventually read coefficient range technique variate pdf read note translation fundamental simulation whose key curve easy show eq equivalent slice procedure conditional distribution identify interesting algorithmic improvement pseudo pdfs slice normalize start seed pdf generate uniformly distribute mm mm reliability system prescribe modern engineering resort require run surface expansion krige original substitute cope impossible make substitution krige optimal density function ensure bias estimation meta importance krige active assessment probabilistic dimensional pdf make mathematical estimate simulation failure failure indicator function convenient derive monte nn limit asymptotically estimate decide dramatically failure rare carlo technique intractable world problem expensive black box code frequent event property different alternative force might surrogate amongst surface recent often impossible substitution second reliability taylor expansion mostly hold difficult investigate infer set carlo random attractive fit tail failure probability make monte carlo efficient monte carlo surface probable failure preliminary simulation compute failure estimate monte although much demanding case current practice evaluate probability demand rely hybrid importance krige build adaptively provide krige surrogate build optimal importance failure surrogate compute function krige present quasi introduce adaptive refinement sampling parsimonious describe detail phenomenon interest consideration coherence observation experiment order retrieve large amount relationship input sequel attempt build failure refer define response quantity interest whose spread variance spread source uncertainty model structural reliability family svm krige present make kriging could krige process essence gaussian cast gp correspond basis stationary read define widely class autocorrelation blue point follow gaussian variate moment g read easily surrogate use apply group evaluate krige function author surrogate reliability complete see similar measure krige krige still equal switching shall interpret predictor prescribe negative concept two performance read figure original represent black represent build sign krige prediction state dash fully accurate misclassifie safe accord krige failure domain surely krige I triangle accord center krige probabilistic cumulative center zero whose spread characterize state krige classification propose probabilistic original indicator failure uncertainty sequel matter fact prediction reduce impossible quantify uncertainty motivates approach probabilistic conjunction importance reduction indicator function failure density dominate instrumental density rewrite follow compute instrumental definition nn draw instrumental central quality estimation instrumental instrumental pdf practice probability denominator art build instrumental density quasi instrumental suited problem instance use center standardized inaccurate krige instrumental quantile variate instrumental read illustration instrumental use choose quasi instrumental pdf expression mean failure product augment failure correction factor indicator function krige fully correction identical augment estimator general krige fully probability accounting probability monte carlo sample pdf instrumental pdf central limit theorem normally distribute finally failure read rely reasonably variation read instrumental sample observe constant indeed exact simulation usual multidimensional pdf consist whose asymptotic use technique distribute significance optimality instrumental refine function quasi instrumental counterpart relatively krige approximation g call maximum minimum region sign predict criterion together discuss comparison analytical propose state surface improvement global criterion usually argue reliability region misclassifie propose trade achieve pick well according thus optimization introduction instrumental performance suppose step fill criterion pdfs normalize proceed population criterion technique slice reduce cluster center cluster experimental design krige accuracy refine batch optimal point thus solve
augment sdp extra favorable optimality guarantee greedy branch explore generalize convex approximately adjust guarantee relaxation condition write plus size condition polynomially spherical low scan constant space identify lie candidate search always retrieve component identity identity x optimization since nonnegative matrix decompose present solve along nontrivial rank maximize nonzero loading index loading denote lead term large attain subsequent development spherical lie surface obtain initially follow low rank continue internal maximization rank absolutely behind concept every function absolutely retain expect formation index absolutely sort sort two change point point intersect intersection determine cell construct among candidate check cell retrieve surprisingly exactly intersection count possible combination element cell interval sort point intersection create great value sort region interesting yet check however exploit possible determine index two identify algorithmic intersection pair element intersection sort intersection point value solve problem however ambiguity coordinate resolve visit intersection identical depend point resolve way intersection point combination examine element absolute large ambiguity examine cost main polynomially rank rest constructive proposition identity positive semidefinite maximize constraint begin constructive spherical define set similarly cauchy inequality maximize equivalently rank give element element notice magnitude area retain magnitude expect formation hypercube sort remain sort around change formation region expand sort change determine cell even well set small exhaustive candidate give resemble motivated map set optimal belong number pair comparison magnitude order switching occur v convenience use pair generation di omit v j cells index lead vertex actual sort interior sort may area cell addition show computation lead cell finally associate vertex cell ignore unless determined hypercube hence already examine l guarantee intersection bad correspond cost determine array consequently build recall conclude overall complexity upper principal
straightforward vector produce hyperplane encode inherently quality share carry first reduce symmetric geometrically embed need appear constant clause involve guarantee section formula formula formula moreover clause clause pair formula clause formula assignment satisfie clause clause variable clause variable clause hardness clause appear clause constant clause reduction add global assignment add clause result formula many clause break per impose achieve clause remain connectivity property maximally satisfy assignment would achieve variable clause satisfied assignment clause reduction instance clause construct formula example set clause denote existence construction clause conjunction clause computable clause variable clause indeed theorem completeness fraction clause assignment clause completeness straight assignment clause maximally clause maintain clause reduction change clause clause boundary least assignment contribute least clause satisfied satisfied clause clause bad clause fraction clause assignment theorem question efficient margin try algebraic lead author believe acknowledgment thank optimal solution induce label hyperplane large whose later reason denote give completeness universal unit immediate get guarantee proof lemma spherical origin area estimate correspondingly sphere compute surface area sphere odd low equation fact success occur hyperplane random unit view sphere x vector direction probability sign w occur require semidefinite relaxation simply coordinate sdp achieve sdp gap regardless always yield solution value equally integral question add result answer draw sphere formally coordinate independently deviation separation side linearity cauchy th fact whose entry w rearrange hyperplane entry fact h sufficiently bind close hyperplane change arbitrarily spread sphere normal choose hyperplane trivial bad section claim theorem remark dms paper hyperplane problem machine hyperplane passing maximize separation hyperplane input achieve provide low bound hard margin tight sub separation margin admit present reduction extremely study theory refer thorough reference setup hyperplane maximize subject intuition concept intuition extensive beyond scope ellipsoid time solution hard go beyond scope consider knowledge dealing purely proof possibly constraint xu et coin name maximal indeed behave cluster group assignment produce solve mixed integer use solver encourage xu show well suggest local improve optimality author proof unsupervise side hyperplane appear hard optimize offset claim distant hyperplane yield iterate pair observation reader keep algorithmic claim also begin describe exponential first force look unit sphere vector right hyperplane analysis assume solution unfortunately hardness show allow discard hyperplane separate see discussion ok efficient hyperplane distance optimal singular version prove within preserve get exact optimal sub notation set lie endow inner denote state throughout solution margin refer hyperplane slight abuse usually invariance convenient otherwise base assignment labeling labeling margin margin feasible use pass origin ellipsoid say labeling obtain undirected expansion large absolute adjacency construction infinitely many say enumeration algorithm output hyperplane state feasible classifier vc enumeration label iff one point label degree edge neighbor check separability perform thus labeling exist hyperplane rotation conclude enumeration separable net set hyperplane margin induce enough obtain optimal obtain hyperplane correct labeling optimal labeling I enough hyperplane normal belong deterministic construction net know argument hyperplane dimension attempt margin project dimension preserve apply reduce turn simple pick sphere maximize margin correctness labeling somewhat deferred lemma seem surprising spherical fall spherical however merely induce lemma hyperplane straightforward randomly compute output label hyperplane admit margin solve corollary hardness unless exponential produce optimal approximate one hyperplane fraction point time margin hyperplane whose maximize whose row separate
find moreover possible le recommend separately intercept gives minimize square remark david w robust criterion influential outlier towards perform criterion parsimonious avoid minimize meet challenge small algorithm simulate supplementary robust elastic net high become routine range finance latter array activity help refine disease genetic disease immediate statistical address likelihood fitting inspire relate simultaneously fitting variable generalize elastic nonetheless recover parsimonious attention extend handle outlier datum majority wang discuss penalization model regression response paper hinge loss primarily piecewise hinge illustration apply highlight outlier despite fast computing path investigation surprising outlier outlier influence identify circumstance motivate robust logistic elastic simulated datum regression consist estimate equation extreme alternative approach contribution consider fitting von outlier robust penalize coordinate review logistic regression potentially effect variable penalize loss algorithm fit summary direction adopt follow assume design matrix notation element wise seek predict explain pn expression microarray single nucleotide snp py outlier addition show robust estimator produce panel implication depict penalize logistic regression show path coefficient function penalization add quickly noise variable coefficient relevant covariate combination covariate bottom panel path robust estimator insensitive covariate chance select robust next increase irrelevant add mle second le mass pmf py pmf solution close log minimize impossible minimize unbiased estimate expand summation expectation random draw summation summation mind loss familiar density estimator parametric estimation trading efficiency estimation fact previously divergence mle member divergence parameter explicitly efficient member le represent efficiency le two benefit aforementione robustness median member possess fast solution section seek minimize measure logistic extend sample sensible minimize namely additive constant compactly write divide remarkably loss produce regression close equation intuition le observe value namely predict value far extreme discrepancy free sample predict zero predict tend extreme sigmoid contribute little robustness logistic sigmoid le covariate change accordingly transformation response theory like reader move use line nonparametric solution quadratic minimize minimization easy mm adapt behind mm function instead choose goal mind decrease easy state value real value k always take increase step respect simpler minimize l convex quadratic ii mm material quadratic sharp curvature le supplementary material implication control size solver consequently low speed express kn descent direction compute theorem immediately jj intercept parameter regularization penalty intercept update regularity solve le guarantee follow convergence mm differentiable hand guarantee sufficient convex add ridge meet le net ridge lasso elastic penalty include tend correlate exclude covariate advance lasso could useful perform generate iterate discuss practically regardless solve follow start converge material extension locally experience le issue practice ridge elastic net imply penalize logistic le also group correlate predictor wu coordinate round optimize coordinate hold th round descent th residual threshold nest find section compare follow property first pi f outlier run centroid scenario add ti scenario describe regression le summarize result le mle position outli suffer come spread mle add summary fit supplementary material scenario population pi pi regression le issue address choose amount material tucker optimization scheme choose parameter perform net penalize package robust classifier hybrid machine detail piece linearity path parameterization net supplementary material h table positive scenario heavy contamination le demonstrate sensitivity specificity mle interesting tend drastically close compare three cross validation curve find supplementary first consider regime repository describe consist patient classify belong group normal patient disk patient patient six patient incidence pi slope ss continuous pr pi ss gs disk patient table show correlation attribute disk normal deal expect scenario disk role however mle le show result path mle le similar method show point le examine genome employ study snps current ever ever discovery snps snps rs rs significantly cancer snp mechanic tend correlate rs rs linkage logistic current univariate adjustment take analyst dependency subset restrict miss snp li miss keep retain summarize le rs line dark thick great correlation control path rs behave penalty predictor exclude outlier commonly maximum know attention bias material standard influential cause thresholding mechanism selection predictor response perform method minimize estimate parametric reduce whose coordinate warm logistic regression immediately extend related generalization multinomial straightforward observation otherwise element vector curvature also subroutine perform tensor decomposition common decomposition block alternate lee block minimization regression clear le computationally run le mle occur insight interesting lead predictor algorithm algorithm guarantee result complement characterize speed great importance material supplementary material include detail g criterion estimation variable proof derivation supplement result show along relevant make file datum reason file thank cancer make grant er department david dms national supplementary material curvature strategy quadratic derivative ii surrogate follow observation equality continuity follow derivative argue attain thus ensure existence global minimum global argument eq less sufficient suppose locally notion singleton consist
sample spatial infection report estimate credible period week estimate credible infection less l city statistically different period implication introduction movement observe change incidence movement table amount infection move later none difference statistically plot movement plot incoming figure city movement gp income small community infection reach small introduction infection link distance income mostly distance shorter relative big factor exclude possibility introduction neighbor useful represent phenomenon phenomenon spread disease aspect complicated process necessarily typically inferential traditional approach develop paper address provide flexible inferential fit disease dynamic addition flexibility tractable section employ approach summary construct exist bayesian literature work idea system biology main paper summary normally unknown function utilize model easy summary normally kind consider normal use flexible variance fact utilize transformation statistic process process model function capture complicated relationship parameter multivariate summary uncertainty approach uncertainty fact discrepancy biological account obtain point contribution inferential summary statistical simulation study context pre interesting appear change school enough period separately movement incidence movement identify transmission seem spread infection regardless movement infection might inferential like worth infeasible great around widely expensive realistic likelihood approach relevant characteristic tractable inferential acknowledgement grant foundation edu edu edu probabilistic useful spread infection function expensive may computationally intractable traditional characteristic inspire recent complex motivating focus scientific process model summary via computationally inference result period insight produce poor computation interest disease provide system investigate question biology understand management disease disease epidemic infection account inherent form becoming increasingly allow challenge make furthermore traditional inference may lead poor estimate capture simultaneously address computational challenge inferential dynamic inspire computer computer develop approach disease obtain simulation demonstrate reliable fit motivate infected dynamic receive lot availability rich well issue transmission infection cause persistence population may enter neighbor spatial coupling management infection study disease investigate transmission model demonstrate likelihood expensive minimize likelihood develop infer characterize uncertainty carefully issue arise infer partial allow bayesian uncertainty show challenge problematic reveal example setting reproduce address issue aspect underlie develop simulation choose capture important characteristic dynamic mcmc feature apply investigate scientific change school period structured allow construct amount infect period movement infection seem estimate display infection help transmission generally situation unable scientific forward simulation make approximate abc efficient organized model act motivate inferential challenge describe new alternative traditional method application summarize scientific general disease understand affected model extension infect recover explicit transmission disease division human infect recovered infected individual disease generation city time assumption reflect movement infection movement city take generation length serial transmission positive time epidemic parameter local dynamic henceforth indexing reflect take piece accommodate variability transmission repeat year see move come city computational poisson distribution exploratory fit binomial affect change infection period time age infection balance infect infected unity exploratory fit infection lie full epidemic previously transmission equal assume primarily depend major continuous week attack unity attack infection one could prior dynamic infer infer significantly increase crucially already assume dynamic know call focus description suggest expert inferential tend high pre study capture biological city infection incidence city associate dynamic difficulty integrate high dimensional translate datum pre web algorithm discretization describe simplify assumption detail web note approach require simplify imputation simulate datum location time section datum resemble unconditional likelihood use apparent fix true value ridge value demonstrate ridge move change plot two respectively computational challenge summary statistic simulate parameter gp uncertainty discrepancy approach inference challenge liu begin summary interest proportion let obtain simulation space design word grid element return normal predictive obtain substitute appendix statistic datum connect follow discrepancy discrepancy point discrepancy match model infer consider another parameter note discrepancy reliable fact fit term thought adjustment fact always original use discrepancy informative prior help reduce identifiability inferential give distribution mcmc inferential simulated example pose traditional approach carry describe section provide bayes rely sampling slice continuous grid update analog walk mcmc chain sample core processor process update slice generate length adequate monte summary interest intuition summary informative regard bi make proportion zero course epidemic infection enter neighbor big movement infection case dynamic infection become infection may statistic demand base exploratory analysis conclusion statistic trivial disease link address limited summary important capture statistic bayesian construct non increase summary thereby regard however inferential expensive select ordering summary statistic whether inclusion substantially criteria informative summary technique transform parameter discrepancy term always uniform grid cube equal use permit computationally simulate grid simulation calculated realization highly insensitive model little much resource setting repeat realization cube follow traditional simulate bayes another answer figure confidence region solid gp region contain base gp traditional method traditional bayes set base bayes outline solid shifted truth use gp dash correct contain reproduce estimate gp plot proportion new improve model fit substantially bayes fail capture feature htbp term approach also try wider contain incorrect importance approximate parameter simulate
quadratic decomposition non whiten know row covariance discuss offer proper context whiten naive whitening interpret assume model component important operator discuss finally connection small yield interpretation operator appropriate imply common employ structured field spatial operator structure adjacency mesh connecting represent smoothing embed operator operator smooth variable think weight together structural potential certainly incomplete provide flexibility notice closely spaced series laplacian except boundary second use functional datum move additionally gaussian mat ern class derive function graphical operator possibility diagonal connection employ operator belong class method pca way two us column give difference example show two half smooth svd take another smooth svd outline smooth solution like smoothing half svd half contrast smoothed converse true inverse smoothed operation essentially smooth operation operation share similarity reduction spectral scaling relationship small sum f ensure variable operator relate small concept multi dimensional scaling place proximity provide additional quadratic indicate quadratic operator take unsupervised summary various encode employ interpret orthogonal smooth find reconstruction structural simulation performance operator encode hyperspectral image environmental sensor quadratic operator wide methodology relevant pca pca regularize massive well spatio temporal fmri many spatial location brain inactive extremely automatic smoothing context pca contribution place framework demonstrate general penalty place factor problem approach frame place computing regularize constant penalty bi concave mean versa monotonic converge method penalty scale factor parameter return factor restrict factor avoid employ material prefer former fuse unclear penalty use implicitly factor penalty wish employ penalty smooth adopt matrix speak lagrangian keep mind interpret product induce tractable many alternate maximize step algorithm homogeneous let state one give penalize regression note norm necessarily order function lasso fuse generalize elastic penalty commonly square root similarly minimization algorithm penalty piecewise penalty satisfy avoid problem regularize permit use within algorithm greedy orthogonal sparse pca sparse schmidt update penalty penalty method symmetric norm factor principal interpretability principal factor voxel truly contribute signal hence commonly encourage penalize apply soft simple positive diagonal minimize lasso minimize iterative coordinate coordinate jj row claim obtain find solution compute single warm start iterate lasso fuse norm concave penalty scad mention penalty solve lasso wide range factor sparse framework literature data penalty encourage functional analysis penalty smooth factor factor quadratic fourth penalty minimize penalize group penalize generalized gradient descent note elaborate first solver work solver onto decomposition denote difference row degenerate linear transformation set norm penalize write solution correspondence consider norm op step employ pca quadratic operator necessarily upon structure return smoother yield excellent investigate feature operator respectively receiver roc achieve vary present regularization via accounting spatio bic spatio demonstrate fix r false sparse smooth autoregressive arise matrix arise former size five diagonal element vertex connect vertex variable row covariance column respectively method operator assume unknown structure laplacian replicate dimension perform rank diagonal covariance svd pca functional svd diagonal covariance svd pca functional functional row equally compare compete notice however well un penalize quadratic operator act yield fairly positive positive positive factor row factor sparse decomposition column factor simulate percent false positive method dependency know structure identify substantial demonstrate effectiveness functional classic structure understand spatial autoregressive fix operator pair operator large variance consider neighbor voxel power laplacian five two example study publicly fmri read one yield process voxel dimension reduction unweighted laplacian window ten present three pc pca spatial pc illustrate eight slice pc dot red denote noise correspond experimental period characteristic bold pc incorporate sparsity spatial consistent subject image sentence inferior inferior characteristic stream pathway object identification pc pca advantage reduction variance classical pc initial reduction recognition fmri directly accounting structure variance present incorporate fast massive type factor example regularize powerful present error use measurement grid spaced coordinate environmental spaced structure encode operator close unsupervised covariance behavior measure structured surface particular estimate quadratic via factor separation much future done determine well greatly drive learn optimal quadratic operator class determine fix option explain error always structural actual spatio temporal additionally issue carefully application would develop property structured beyond initial author explore issue well direction closely incorporate way technique frobenius additionally relate pca theorem penalty convergence local need initialization comprise convergence consistency work need broad wide massive common area hyperspectral image sense environmental finance structure recovery exploratory pca regularize pca massive area author reference attention helpful manuscript grateful associate helpful comment suggestion partially support support award svd minimize objective easily verify constraint complete corollary result property equivalent power svd last proposition simplicity problem maximize svd recall amount kk th notice proportion variance result hold analogous kkt necessary follow p c condition put together kkt condition discussion norm occur nan exclude easy satisfy kkt prove solve optimize respect jj j put proposition show algorithm method minimum generalize f respectively op operator linear separable block employ put converge penalize predictor impose orthogonality subsequent factor compute cumulative variance explain k k proportion state result k k k k manner sample assume penalize arrive replace proposition college electrical engineering brain stanford university department statistics stanford massive arise imaging series study technique ignore relationship often poor decomposition svd pca massive dependency generalize relationship decomposition two way penalty sparsity smoothness computational massive set brain reduction keyword sparse singular svd upon form multivariate know poorly enforce sparsity factor consistently recover column relevant dependency two exploratory massive structured set image environmental series pixel voxel hereafter activation location voxel multivariate thus spatial row column apply fmri find spatially contiguous group activation region rarely fmri dependency brain activation interested furthermore component dependency activation remain unclear dependency understand poor examine independently frobenius sum error frobenius norm svd perform poorly strong matrix product loss frobenius permit weight error ii ij good approximation develop decomposition account structural weight element via generalize attention multivariate community gain popularity statistic text publish english review relation mathematical statistical need dimensional flexible mathematical pca account manner computationally analysis data contribution matrix allow dimensional previous review weight applicability imaging engineering section framework generalized section weight quadratic operator discuss interpretation exist methodology quadratic appropriate datum reader intuition aspect future pca pca assumption implication extension singular svd structural massive previously begin write good svd value frobenius generalization weight definite norm nan leave inner take norm svd call notice frobenius motivate respect likelihood frobenius proportional spherical proportional variate norm assume row column covariance check know operator precision account know discuss structured solution state svd decompose let brief regard decomposition similarly eigenvalue decomposition l theorem variate covariance yield identity word value variate row svd pre whiten svd
result noiseless capacity compute shannon size capacity constrain channel local plane shannon capacity plane zero column shannon capacity although remarkably constrain channel yield approximation low capacity order mm extension fig reduce output therefore draw lp lp admissible configuration channel rewrite run fig convergence region belief approximation sequentially basic draw accord proportional involved region belief sample order graph finally kernel vs horizontal dotted show estimate noiseless read size db noisy horizontal dotted noiseless capacity channel average realization information symbol db white symbol snr estimate mutual constraint dimension noiseless estimate finite capacity capacity case constraint additive gaussian information different acknowledgement acknowledge first author thank comment also suggestion presentation physic know free energy introduce basic parameter choice basic plausible since validity array determine slide row array constraint region graph free estimate give operate region beliefs region region energy region region constraint f cf f factor constrained line show result apply capacity shannon capacity bound improve nine digit identical shannon capacity plot versus fig estimate noiseless
rapidly decrease show integral px separately use inequality estimate complete note integration formulate estimate introduce generic context expression also define suppose constant q c h h u r line rely know pseudo x px induction show k since calculation finish distinguish elementary obtain replace bound principal since partial integration twice similarly integration r ix iy complete definition adjoint asymptotic summation pseudo differential operator cf suppose small integer r e application loss generality fix write h follow iv elementary otherwise proof relate note u u satisfy exist constant treat together conclusion dominate expansion proof part r imply r h suppose substitute e assumption uniformly r r ds upper norm r integration integration set h lemma let recall cauchy schwarz see together remark show extend function supremum side motion value brownian two step ff f fp fp f fu fu ff let standard brownian satisfying first weight derivative r equal conclude strictly increase w x fulfil whenever function ng last multiscale rich index suppose ds center standardized decay maximum behave criterion far easy q similarly illustrate multiscale discuss order level value wavelet k remark universit universit unknown important example framework monotonicity simultaneously moderately ill pose fourier density multiscale testing calibration modulus continuity brownian motion theoretical major work x goal distribution estimation deconvolution decade fan selective qualitative study fact rate induce ill band attractive adaptation scale difficulty aim derive simultaneous confidence statement qualitative pseudo differential density assume moment important check convexity concavity monotonicity property moderately ill pose meaning decay polynomial know fan cf pseudo differential combine assumption important error gamma e deconvolution interested increase deconvolution smooth partial ds multiscale follow analytic pair number q negativity decrease tend infinity able simultaneously asymptotic implie allow region prescribe solid kernel low display statement thick line thin figure simulation panel display inspection apparent difficult find monotone display increase decrease plot pick horizontal monotone monotone overlap contradiction statement hold empty plot besides thin uniformity sense confidence statement simultaneously illustrate panel display three reconstruction kernel unimodal kernel surprisingly reconstruction mode completely know want mode plot vary ask mode display density decrease horizontal line monotone increase interval reflect behavior pointwise mode confidence interval merely conclude local local maximum simulation four segment decrease find systematically numerical simulation size statement tool analyze conclusion differential ix l depend class differential fractional identify pf projection subsection implicitly operator monotonicity confidence region shape scale simultaneously generalize need confidence eq h multiscale standard key result distribution free critical compute identify pf multiscale method statement strong statement testing deconvolution cover class ball al paper focus deconvolution deconvolution derive significance view treat sequence tend limit couple multiscale deconvolution multiscale method make attractive choice multiscale interpretation statistic number property mode factor small multiscale differently unlikely detect desirable neither number penalty necessary independent infection interval increase emission detect laplace paper show free approximation shape deconvolution statistical consequence statement performance multiscale numerical expression function stand variation put clear integer norm h shall fairly convergence use structure model observation I uniform q constraint value localize around statement refer distribution free approximation multiscale assumption model process bound generalizing utilize convergence give lemma whenever lebesgue verify sample cf contrary q general cardinality instance make set address carlo start differential operator pseudo fractional integration pseudo differential eq continuous operator paper integration eq define operator identity q differential fractional us mention proof essential integrable formulate symbol however restrict throughout paper axis obtain uniform kind number location root identification specify conclude attain want attain conclude overall sequel empty problem interval analogously whenever interval way solve interval root section collection interval qualitative might interested density random instance transform suppose concavity smooth strictly monotone denote differential therefore confidence problem ball estimate statistically speak observation density iff measurement observe constraint way say operator furthermore arbitrary denote adjoint product formally equality symbol mu formulate symbol number characteristic bound moreover fouri density moderately cf fan real smooth test proceeding consider multiscale notation operator suppose exist side approximate approximate statistic furthermore choice simple ill kf ss r pseudo differential pseudo operator composition compose operator composition heuristic argument theoretical investigate cf section relate restrict confidence result iii confidence link behave coincide density speaking return reject next give h event argue pair large pf h return compare integer old continuous relate confidence statement root location know number trade clear control number root trivial need root separate eventually property simultaneous say zero simplicity root rich neighborhood behave define calculation corollary construction eq length confidence confidence zero therefore become disjoint show number root pick probability observe localization mode log coincide al special density deconvolution comment mode restrictive smoothness contrast rate square strong confidence band comment multiscale theorems calibration multiscale l evy modulus supremum attain uniformly particular attractive construction adaptive multiscale exclude statistic free exclude weak important detect zero outline discussion study deconvolution notice deconvolution inversion operator therefore take deconvolution problem natural variable far thus multiscale statistic particular display repetition concentrate imply boundedness multiscale solid subset complement investigate confidence equal estimate simulation monotone investigate numerically signal average decrease way decrease localization confidence statement instead prominent one good something mean line subset exceed line hold confidence find derivative much achieve statement value come multiscale corresponding belong therefore view superposition statement different qualitative quantitative statement construct band mode whereas qualitative account scale multiscale analyze operator work refine multiscale operator strategy apparent harmonic operator spirit theorem deconvolution model deconvolution viewpoint harmonic multiscale know qualitative construction root require continuous highly impossible statement interval strong band multiscale distribution difficult ill pose include deconvolution become pseudo differential good knowledge formulate space treat subsequent restrict associate limitation curvature handle within allow linearity qualitative integral pseudo fourier handle research acknowledgment support joint research national science partly acknowledge grant comment well associate lead replace eq fairly classical never describe exception brownian bridge gaussian bf f bf coincide brownian uniform define particular empirical brownian universal theorem state supremum behave supremum symmetric hull bridge sequence depend brownian bridge constant refer event cf indicator absolute theorem constant bridge bf x nx let brownian covariance define bf ii standard show surely establish result obviously xt xt h xt q bind application surely almost surely prove c
therefore rule hope good rule decode use regular sentence token length string token vector mt calculate marginal sum possible token generate word string concern computational complexity take sound probability clear next mathematically segmentation show corollary probability structure segmentation unknown give estimation ensemble method method would provide thus serve decode good hypothesis input follow interesting turn distribution certain pr pr hypothesis constant extent method generate segmentation I separate token mean yes lemma source token string translation string number token pair appear prove neither time cover total word let easy picture rest suppose assumption token assumption upper bind assumption proof give also proof appendix token use obtain corollary assumption proof accord small assumption hold pair token similar reasonable conditional ensemble proof monotonic translation length source target side translation try string assign tree node probability input appearance way structure free skip procedure approximate theoretically fact need explanation assumption connect inequality simplification token base give assign oriented parsing size build block tree framework pair represent thus tree probability quite translation justification formalize widely induction nlp cope uncertainty well future field parse field nlp acknowledgment inspire discussion author however belong I appendix lemma proof variable binomial assumption nlp exhaustive advantage translation give exponential work justification nlp view heuristic pattern future language nlp solve induction formalism context dependency substitution etc translation mt learn transfer string tree structure language state aspect show oriented parse bias inconsistent mt almost mt rely method example like parse way heuristic overlap structure train exhaustive cope uncertainty statistical far mt phrase word obvious outperform building need datum valid algorithm investigate empirical large article mathematically sound building show diversity explain many rest article organize formalize show corollary conclude core idea monotonic translation introduce justification monotonic translation monotonic string length monotonic simplified word ignore impact alignment effort incorporation monotonic enough nlp task pair string align word let length string show segmentation token token least one source side
slot policy solve one problem regime trivial tractable index hard chain main bayesian develop treat arm armed bandit problem observation performance problem dynamic spectrum sense secondary select channel maximize reward transmission primary model identical unknown develop channel sense dynamic policy gap obtain model slowly non decrease numerical know reward mab toward unknown treating update observation bellman generalize mab mdp independent fully observable arm activate time arm change offer dependent reward mdp naturally lead programming solution induction incur open present forward arm depend current state optimal reduce exponential arm several researcher come index variant basic classical mab include bandit bandit bandit switch particularly variant classic change optimal play condition find offer check exist analytical optimality regime also approach test constant approximation bandit contribute basic bandit optimal positively channel policy bayesian make identify mab dynamic basic sequential arm associate process performance measure define could obtain know observation know player reward essence thus identify explore sub growth reward first policy characterize single arm parameter markovian logarithmic ucb finite know focus parallel much definition regret regret well sublinear definition imply deviation arbitrarily obtained play know yield pick arm reward arm markov keep activate armed bayesian process priori first describe option markovian say exist finite policy despite finite although player treat multi armed bandit policy sequentially positively stay channel switch visit correspond switch soon channel among visit step channel circular circular circular channel circular circular slot tt odd treat classic armed goal give question arm present next show increase integer ucb policy positive play denote reward constant record non grow dynamic spectrum channel unknown constant relate fact lemma hoeffde trivial chernoff hoeffding allow expectation variable constant eq c na stand generate inequality stand either belief play steady steady throughput belief average policy circular order channel slot order channel j order entry policy channel q theorem eigenvalue generality steady policy different different bind regret c two policy switch constant exposition index possible play ft I equal define play contradict least time similar policy time select time play policy must apply lemma eq difference vice follow readily translate simplified policy however something claim logarithmic policy positively channel whether infinite horizon answer conjecture conjecture offer initial belief initial finite system channel transition different correlation first channel positively transition sequence show simulation quite quickly practically happen great infinity exceed slowly long grow fast though converge quickly also trade bind generally linear ht problem learn treat arm develop spectrum access channel policy strong result bayesian meta state identical arm identify fill finite option structure u edu arm reward chain seek arm
n argument lipschitz derivative cost control cost stability function impose nominal model nominal reason relax oracle robustness learn arbitrarily bad impose nominal subtle maintain oracle polytope polytope strictly large whenever intuitively bad learn nominal reduced oracle update advance scheme denote point input endow feasibility constraint close loop sect control constraint reasoning begin n I n ii n n n mp constraint property n invariance property gx x prove property sect loop system provide b robust trivially varying allow stability system modeling follow prescribe discuss robustness function proceed constraint get maximum continuous continuity generally cf convex reason practice benefit suboptimal hence convex feasibility nonlinear bad behavior formalize stable type controller approximate converge apply approximate bound asymptotically prove key e identically oracle provably bound assume sect feasible exist continuous lyapunov law e satisfy let minimizer unique assume strictly similarly approximate construction proposition imply exist check argument strict convexity satisfied sect certain condition check lyapunov lyapunov check quadratic bound definite lyapunov require linear pa lyapunov converse lyapunov indicate meaning exist second n show begin note positive positive minimize side inequality linear furthermore value equilibrium condition immediately lyapunov situation decrease input answer proof prove theory proceed certain study consequence name reference computer generic continuity boundedness consider take tool identify system natural question example construct oracle secondly law know begin class oracle conclude address law minimizer discuss sufficient ensure minimizer minimizer limiting function often argument trajectory difficult compute generally problem call hill equation type pi set reason situation simplify require give function mx whose intuition correction nominal high nonparametric input without assumption mathematical technique non traditional relate sufficient set lie denote n minimizer constraint converge constraint time relevant result purpose interested theorem let composition ax gx u converge compose dynamic theorem convergence control input trajectory mode activate control ensure key reinforcement nominal sufficiently explore controller ensure se difficult system open se se hold law se regularity correctly follow find se control know u se ergodicity hard verify sample ball center radius inter less consider generic pointwise form decrease make control probability know present prove control nonparametric regression stress meet controller decrease know summarize main law control know n experimentally simulation feature build berkeley air name berkeley energy room use generate energy warm day room able load form adjust name seven office laboratory hybrid control achieve eight energy american home temperature load load adjust action achieve system four steady property kalman correction coefficient make perform successive nonlinear provide robustness amongst display controller overcome phenomenon ground make close ground display mis improve generalization integrate ball nonlinear illustrative purpose engine exhibit type region air engine air engine operate engine possible active model ode describe predict instability flow pressure rise control transfer function n choose take approximate discretization linearization equilibrium r u u linearization unstable pick ensure closed loop choose still linearization small error performance nonlinear lyapunov l use incorporation improve reduce significance setup satisfy feasibility maintain error close loop simulate demonstrate condition check interesting system operating point linear vs perform nonlinear require compute code solver multi toolbox bound deterministic guarantee robustness many type identification tool use require use satisfy robustness simulation improvement translate real amongst future speaking nonparametric work property local globally regularize would acknowledge ram discussion material foundation laboratory agreement nf air office scientific research agreement fa control controller design face reliability practitioner focus however interest growing describe control scheme robustness identification identify rich order optimize design variety tool insight performance reasonable condition framework minimize ensure check input subject furthermore compute face trade stability approximate accurate drive adaptive controller control refine controller design handle input b performance respect c identification tool uncertainty challenge combine purpose deterministic show difficult convergence introduce adaptive predictive base predictive control insight framework tool particular update ensure robust check online difference robustness nonlinear form robustness use learn nominal model focus level define deterministic robustness prove incorporated nonparametric provide convergence know discuss engine estimation filtering note compact transpose distinguish system system state type strictly v lyapunov x mx v difference important transformation tf nr nf pr nf fr qx na nonlinear dynamic modeling lie polytope restrictive determine quantification fitting uncertainty simultaneously perform dynamic measure
design truncation equation fourier specify power spectra differential real net domain order desirable requirement probably frequencie one approach inverting spectrum nonzero coefficient matrix cost learn memory possibility closely approximate spectrum nonzero relate spectra must superposition coherent space central also suggest order take inverse transform differential difference net inspire introduce scale function scale delta function gaussians derivative case forward behaviour generalise elastic net point order cholesky primary train consider orientation origin centroid cell arrange grid stimulus densely arrange map online take long adjust value respective net sequence anneal high value centre axis geometry stop surface note kronecker product critical net axis set order simplify stimulus position order nd net avoid sharp corner hull always intermediate fitness term effort frequency width order periodic net centroid therefore link remain inside hull centroid lie convex hull fitness increase isotropic encourage centroid maxima outside hull net lie hull tendency end net exceed hull end produce corner heavy model would centroids convex hull stimulus preference outside range modulus anneal enough stop training centroid hull exceed keep net inside hull difficult task net investigate map dimension elastic self apply space decompose region diagram diagram obtain forward map figure slice space leave phase diagram contain map reflect contain net tend contain characteristic decrease contour towards orthogonality gradually contour align increase map experimentally match region narrow critical phenomenon map model nearly really phenomenon call present loop smaller attract fast anneal slow phase diagram anneal region contain practically elimination wide another sometimes part cutoff argument forward difference twice square modulus assume net frequency equal map family line slope use logarithmic otherwise evident note slope single correspond thin close curve horizontal geometric relation still differ rigorous explanation phase anneal learn interpret principal along direction freedom annealing become freedom interpretable strength depend small visual dependence connection thing incorporate modify define manually parsimonious define parameter periodic net initial net anneal net bl bl bl c r bl l bl l bl bl bl bl bl bl cholesky map rectangle loop loop elimination net converge training grid net centroid net appear initially continue loop eliminate separate c lb lb lt lb lb lt r phase diagram enough map anneal slow anneal region necessarily contain boundary approximately dotted line region represent curve along start arise curve line mark map curve ex ex ex periodic depend elsewhere ex ex orientation ex ex contour net width orthogonality explanation characteristic periodic although analysis rigorous net effect fitness term arise angle net independent formulation analysis net gradient descent optimisation dynamical analysis type order net periodic self map elastic equilibrium infer consideration eigenfunction operator wave frequency difficult close analog limit space forward difference e transform continuous formation term loop elimination linear relation map require net research elastic define generalise depend define neuron either map centroid related suggest al explicit correspondence model indirect generalise assume binary elastic centroid centroid compare see elastic partly motivated appearance net define limited connection connection proposition far formal represent preference centroid square term add convolution preference play suggest model rather abstract neuron equivalent reproduce orthogonality centre column periodic fitness function type time near zero euler descent fitness net come fitness term elastic trade formation chemical light act however correspondence term operator represent scheme use partial boundary problem regard whether stability whether size tend sparsity coefficient quadratic preferable coefficient desirable memory storage high depend b step solve forward quadratic stability semidefinite whether remain show penalty admit suitable smoothness ill pose nf mathematically concern define second difference constant centroid place centroid annealing system consider latter locality implement spatial work dominate weight wish unbounded cholesky slow operation necessary construct several difficult besides gradient advantage toeplitz toeplitz toeplitz product fast fourier also toeplitz toeplitz extension near structured dimensionality variable density mapping centroid noise dimensionality mode dimensionality reconstruction centroid interpolation centroid traditionally elastic use reduction unseen goal replicate net space perfectly could computer vision graphic density closely som mapping brief som define principled algorithm convergence som reconstruction mapping pass centroid also define prior way derivative net version curvature parametric practically penalty rule momentum seems alternatively define som penalty likely computationally three model suffer curse whose centroid grow manifold locally embed manifold pair term close consider local similar truly construction optimisation method anneal euclidean short elastic try sum distance distance different elastic net lead heuristic opt replace arc locally iterate optimum usually correspondingly generalise likewise complete opt applicable multiple contain city city company home office city city visit length quality splitting start define advance submatrix complex centroid central centroid index get elastic centroid two net disjoint qualitatively justify area dimensionality short approximately proportional depend ff share fix city add centroid extend centroid centroid centroid centroid clearly separate net net perspective globally optimal net matter anneal elastic net e net loop anneal slowly similarity neighbourhood cost space centroid city adjacent equivalent elastic choice accommodate term product pairwise accommodate seems differential general problem difference family elastic family choice spectrum efficiency nonparametric suggest rotation though discrete also particular natural solution work nd equations physics also nd th order case provide constraint operator penalty surprisingly example suggests rigorously consider test sr delta maximally rough increase must make sure square derivative term dominate volume concrete net partly represent partly size complicated difference modulus derivative high discrete triangular height delta net net delta somewhat resolution one centroid small modelling motivation development visual generalise elastic give fitness centroid one sparse cholesky result convolution net approximate differential periodic boundary fourier equivalently however space family repeat order derivative high pass filter filter tend infinity see compete net discrete case qualitatively rich present case rewrite way form alternative subject high context previous elastic net demonstrate equivalence square stimulus look sparse nearest implement connection strength decay distance bring elastic dense connection cholesky annealing explore term phase diagram map interaction fitness term cutoff predict periodic structure strength spectrum map explanation particular geometric thank helpful grant gaussian proper improper semidefinite normalise prior determinant covariance rest take along row improper along convention integral frequency must low impulse delta filter discrete power frequency delta net different piece different besides turn approach integral grid dimensional length w voxel net space delta function real origin coordinate irrelevant wave superposition l concentrate single period compute note cosine term get eq n l expression period valid boundary voxel w fourier difference use forward net physical dimension g piece assume amount modulus forward modulus central decrease know property fourier continuous text proposition section elsewhere integral gr substitution change toeplitz plane wave try wave inner associate mn mn row dft power spectrum matrix note shift correspond invariance row theorem note nn proof frequency differential must identically wave dft proposition obviously matrice tt proposition delta hold delta hold construct divide incorrect gr statement trivially take simplify proposition induction q nm analogously vanish odd function around integral become gr theorem lemma proposition prop conjecture conjecture thm center road dc elastic optimisation fitness term implicitly net operator give generalise elastic net difference derivative map relate prove elastic net generalise elastic net discrete differential area graphic optimisation solution problem subsequently essentially preference square relative success elastic net ease probable attempt beyond specifically complex perhaps approximate cost distance context model whose unknown connection neuron examine relate connectivity suggest consider detail motivation primarily extend model area computer graphic type generalise elastic class generalise elastic third map result net mean original elastic square distance elastic collection centroid dd mm centroid semidefinite prior improper purpose appendix elastic recover possible scale role temperature centroid centroid could shape curvature implicitly centroid centroid semantic typically base scheme statistical training dd posteriori iid ignore independent however anneal alone minimum fitness solution combinatorial problem system latent net iteration index evolution unseen consider hyperparameter validation bayesian g investigate accord criterion differ objective sign multiplication fitness positive latter influence term decrease hereafter however simulate train three cholesky gradient invertible ng training centroid three algorithm cholesky considerably sparse iterate small need elastic net centroid centroid towards centroid confirm simulation short step would obtain related modelling assumption definite since semidefinite depend point fix update guarantee familiar order explicitly compute question ill operation computationally solve g iteration follow compute implement explicitly relaxation trial extra nonzero element quite matrix elastic net iterate equation cholesky find good matrix triangular strictly necessary usually although band structure fill minimum ordering descent certain net net centroid centroid large problem particularly fast anneal term final iteration gauss matrix cholesky generally numerically stage cholesky try cholesky direct gauss iterative try cholesky require cholesky twice high bottleneck net weight take solve cholesky mean cholesky gauss go deeply annealing particularly net centre contrast descent fast anneal considerable cholesky train drastically practically split robustness efficiency cholesky choice allow investigate behaviour range simulation describe practically convenient separate extra copy large case remain define multiply old reduce make take place period fitness elastic mixture associate centroid however non rectangular shape specifically nonzero equivalent implicitly condition type appropriately modify complicated force fitness likewise remove appropriate gradient equation rhs option since operation numerical instability iterative cholesky method involve singular overcome proportion energy become remain centroid modelling approximate primary may patch inactive separate net force net central elastic example centroid periodic elastic net equation nm nm nm nm elastic anneal descent gauss mm definition search incorporate model elastic net concentrate operator ignore dm dm map term fitness represent tensor penalty still restrict secondly note enough symmetric form use lower without generality frobenius specifie centroid net curvature specify centroid one show change centroid line plane etc typically change net via net topology neighbourhood region net represent multiplication periodic b matrix represent generality concentrate particular identical assume invariant consider appropriate arbitrary invariant one q invariance rotation require operator way follow matrix zero choice origin desirable naturally work case positive matrix square matrix sort rotation obtain rotation sign permutation row kk derive transformation since backward difference central elastic sum I element specify boundary simple use modular linear combination periodic since successively toeplitz leave idea elastic net dimension complicate toeplitz analysis simplicity periodic notational remark assume convention central column etc obtain successively ambiguity e assume centroid net dimension coefficient often rather circumstance indexing convention easy later separate tm convolution discrete net ki jk reverse turn basic finite differential difference derivative small curvature consequently element must otherwise would consequence eigenvalue eigenvector centroid centroid rotation fitness invariant newton quadrature operator origin constrain elastic obvious course operator operator contrast readily continuity smoothness geometric accept uniformity term opposite g operator nonsmooth net simple placing parameter mapping net ridge force magnitude represent well net parameterized mapping centroid location coordinate second topology prior spaced essence operator reconstruct net strategy appear repeat along toeplitz slightly delta net modulus net equals construct pass along dimension penalty times modulus eigenvector cosine discrete frequency case bound play describe condition condition net centroid condition differential net zero pattern black sum zero white square analogously term example different net centroid wave wave satisfy condition sample nearly point must wave enough frequency power frequency incur negligible wave wave penalty monotonically net look different truncation error character obvious power spectra happen typically monotonically unimodal view secondary importance net order net different alternatively modulus centroid relevant net plot family question error zero require nonzero order combination order question iterate analyse backward forward difference convolution reflect respective matrix high writing convolution call p p also successively composition derivative associate eigenvalue cosine eigenvector mean convolution one cm p n n repeat irrelevant omit elsewhere backward draw family exactly appear obtain appear except convolution diagram eigenvalue derivative eigenvalue correspond string periodic mode proposition form triangle modulus along otherwise forward curve since nan consistently family practically qualitative explanation behaviour though actual preferred simulation net net idea fitness frequency frequency largest less call cutoff frequency power exceed cutoff predict net expect frequency dominate centre set single decrease gradually decrease fitness energy series minima net way imagine cutoff power horizontal dash line cutoff net centroid centre mass cutoff small cutoff thus behaviour note cutoff proceed energy uniformly decrease cutoff frequency appear develop frequency local one start cutoff high net little confirm annealing understand without curve fitness dominate centroid draw training set fig cutoff cluster differ confirm qualitatively different net cutoff argument effect increase cutoff result cutoff map net cutoff explain plane prefer direction contour approximately circular power cutoff equally result preferred direction fig contour line near become thus occur isotropic result preferred c relatively stress curve cutoff see define central proposition family obtain order divide nm p p frequency low frequency practically fitness generally elastic net result family contain low net part wave value one centroid odd centroid alternate nonzero however structure difference horizontal quadratic pattern map also understand combination consecutive either coefficient overlap term avoid map net show individually fig b visit give discuss differential operator desirable centroid eigenvalue dft delta penalize integration opposite though integral smoothing c c etc net domain look curve look forward central
incorrect however occur neighbor number denote frequency figure frequency neighbor list note neighborhood histogram neighborhood useful determine likelihood give frequent element likely neighbor big neighbor additional ordering form list let symmetry neighboring neighbor neighbor vote receive suppose however identify indeed think average bad fold cross validation procedure see net value rather compute pick empirically vector pick entry node interpret rank elastic neighborhood estimate order weighted neighbor additional correlation neighborhood paper paper ise penalize penalty recovery recall model addition rate confirm maximum edge wise neighborhood produce way node structure lemma example david attract considerable sense biology language protein estimate gaussian ising extend work include penalty numerical introduce leverage introduction sparsity neighborhood graphical simplify let vertex simple concentration underlie neighborhood random neighborhood set importantly variable give nr nr employ pseudo measure compare penalty base enjoy property domain treat variable generalize model reconstruct full edge graph allow al structure gaussian field model neighborhood regime max improve examine build upon work expand scope graphical multinomial discrete lasso drawback moreover highly correlate variable incorporate elastic net retain introduce union estimate idea neighborhood pair node necessarily neighborhood pair design usual neighborhood undirected encode clique basic form comprise interaction node max write normalization function mrf sx inverse covariance correlation node zero neighborhood reveal neighborhood expectation represent mrf distribution describe full distribution form term eq hessian fisher much correlation entry extend expansion describe parameterized indicator represent particular variety relationship despite factorization simplify parameterization denote agreement ise x binary note transformation discrete indicator classical multinomial regression ns l equation selection approach take condition penalty relative term preserve sparsity elastic net original package multinomial library al evaluate mrf distribution enough force pe positive experimentally start increment value sampling computationally expensive long temperature overcome wang augment iff formulation node vice versa value otherwise generate sample update substantially vertex chain explore outcome rapidly wang regression net setup observe rate number neighborhood large bad matter small neighborhood counterpart plot vertex degree even sample scale test star star star connect center add neighbor star maximum number impact fix recover significantly hard density maximum star density dependence degree additionally equally sized plot error rate top edge see plot bit bring error rate bottom time sample clique star edge maximum degree recover exceed figure density increase sample simulation achieve consider group vertice community connect belong community complex group rest application community individual page modular however available achieve rate increase perform regularization produce ht figure elastic mrf ising degree run range trial ise clarity note represent neighborhood neighborhood neighborhood multinomial penalty benefit neighborhood sample value penalty chance penalty allow select exhibit effectively average parameter provide essence figure dimensional curve consistently provide precision recover ise model introduce drop
unique main color colored green solution hyperplane strictly separate red rewrite solution system substitution variable substitution lp common hyperplane separate hyperplane separate random separate remain hyperplane draw leave distribution let subset unchanged permutation equally vary uniform come first vary step experimental x success z axis curve correspond three label success vary vary space rate z axis label set success transition suggest conjecture entry absolutely distribution lp recover also tb conjecture system give solution integer entry sufficient recover lp relaxation minimize exchangeable well geometry knowledge compressive empirically distribution lp recovery succeeds always fail conjecture binary entropy interested grid wish retrieve connectivity three power series hour send customer single phase principle measurement phase determine customer empirical collect time customer imply underlie system reduce problem binary checking difficulty et one uniqueness lp norm lp recover compute generate instance almost study condition entry minimize recover sufficient satisfied recovery albeit lp relaxation definition constraint relaxation lie lp lie polytope face uniqueness tight entry lie polytope
eq follow uniquely identifiable string identifiable already distribution string long uniquely determine extra work employ date binary value stationarity stochastic row resp column translate column summarize whose function rise matrix translate zero probability string equation string collect insight length sum string avoid string infinitely crucial write ideal comment point independent outline string computation confirm example q ideal characterization variety apply markov model due necessary remain whether computation also base early follow determine submatrix setting refer string length necessity definition express plug close let alphabet determine parametrization state generic parametrization hide return routine compute parametrization subsequent note invertible apply work invertible eigenvalue lemma choose b bm x recall decide incorrectly form incorrectly algorithm proposition relation membership thereby p n exclude iteration infer parametrization respective process finally eq e unique column thank algebraic great stay berkeley special present algebraic berkeley author would thank participant thank david private stay berkeley thm notation observation thm thm remark thm thm generic decide infer subset answer process markov process root algebraic statistic algebraic set algebra allow algorithmic elementary statistic concern finite alphabet refer process hide infer representative list contribution exhaustive list estimation practical practical argue explain contribution form extra center around exception cone stationary solution algorithmic date string exercise hide introduce turn attention identification string due hide course value root algebraic draw particular concept variable uniquely determine rise hide answer yes yes stem process density infinite state read algebraic unique course ideal characterization process arbitrary base argument hide relationship note review summarize result turn note correspond note algebraic treat convenient require stationarity characterization binary novel basic definition algebraic serve treat algebraic setting formal definition markov definition statistical algebraic model hence binary markov formulation alphabet string letter concatenation letter stochastic process generate string technical convenience confusion write throughout none exceed string length alphabet polynomial complex affine zero iff imply correspondence irreducible counterpart complex value boolean stress algebraic list auxiliary virtual string short elimination necessary fact algebraic family algebraic statistical call process take value variety string length generate stationary rise unless process stationary often remain unclear difficult determine computation stationarity mention stationarity stationarity geometric implication among stationarity technical among remark practical process stationary evident particular establish gold example speech classification hmms gene certainly early identification linearly model finite markov runtime say iff sum whose parametrization immediate parametrization process process parametrization admits provide condition notation rank integer choice parametrization state tuple emission matrix hide state write refer write hide relaxed follow consider proceed initially observe probability symbol subsequently addition use correspondingly eq technical computation reveal reflect forward backward observable state make obvious hide admit dimensional parametrization definition process process act state rank db b invertible write process parametrization rank admit would algebraic family algebraic model write zero reflect every parametrization parametrization obtain algebraic complex require row sum make eq row imply process analogy hide hide string write variety associate n h reflect state translate invariant hide work one thereby obvious immediately equivalent elementary argument variety insight rise process statement q imply statement equivalent rise straightforward generalization previous let q quasi projective standard applying choose example process exist invertible transform dimension invertible lemma existence transform summary invertible form variety cardinality generic reflect permutation state yield observe stationarity value argument prove identifiability also value identifiability
regularizer asymmetric estimate maximize joint similarity thus strictly psd wherein general zero structure show theoretically ground truth uniform many real application estimator report demonstrate variance report gaussian fix draw th integer run draw th distribution sample ti single average stein also compare randomized fold validate stein validate choose minimax stein low leave minimax stein validate mean q cv comparable span propose minimax datum set axis random draw percent mean risk stein reduce black blue green line draw risk vs single mean estimator average draw percent estimator risk stein overlap identical blue green line h percent vs mean half x axis mean apart average stein task estimator provide gain dominate apart outperform stein estimator performance indicate benefit level early bad opposite minimax optimize cross validation albeit cross green blue dotted summary variance wide margin stein achievable oracle variance calculation pairwise separate separate issue use always minimax eq exclude cross validation performance pairwise well indicate estimate improve th real application use application final student student application project constitute single experiment student treat task never student cross pool student pool class risk average across student dataset percent vs cv denote standard deviation cv low percent stein percent validate version bad counterpart minimax estimate similarity rare impossible student result never bad task gain stein performance surprisingly student score good task estimate wrong draw choose task sale impact business amount customer spend time period customer order customer amount customer amount compare task ground compare fold choice statistically two sided test observation stein statistically outperform estimate counterpart cross non estimator customer task stein minimax minimax cv cv mt kde averages recall task kde assume estimate density query un average average estimate mt kde kde laboratory adversarial open primary create rich region attack security relate risk event goal km market event attribute group seven treat kernel kernel bandwidth minimax also expert school assess seven period similarity force unknown separately kde grid grid leave validation assess kde mt kde kde mt sort seven reciprocal ideally worse kde datum big inferior preference perhaps multiple fashion improve test stein cluster kernel denoise acknowledgment research event dataset helpful united award united states office admit rewrite matrix laplacian notation asymmetric derivative optimal exist prove lt tb tt clear every disk plane real eigenvalue invertible negative clear entry true part eigenvalue already prove wise negative sum third sum zero entry wise derivation similarity minimize limitation q parameter g version replace plug simplify eq q minimax derivation estimator minimize estimator prior minimize risk prior prior find need follow minimax solution minimizer favorable constraint least favorable constraint bad put mass constrain make write clearly weight minimax guess bayes indeed expression risk page tb u risk minimize minimize proof proposition formula write inspection clear strictly general al cs task jointly task result task minimax estimate outperform estimator stein motivate leverage evidence stein stein square mean surprising perhaps often provably nice effective practice particular amount stein true estimate sale error report need apply average section task th task th mean pairwise solution variance problem task iid task key pair generality minimize loss jointly together multi regularizer separate minimization produce normalization estimate empirical dominate formulation regularization set experiment restrict error formulation many generalize trivially rather scalar notational similarity often convert task appropriate choice case minimize estimator background stein learn manifold body stein strategy simultaneously draw stein th dominate stein replace unbiased stein towards regularization shrinkage shrinkage implicit assume stein estimator become q work r decrease degree see number stein multiple vector task large stein give admissible theoretically estimate eigenvalue trace covariance matrix replace effective result stein separate effective maximum stein estimation use effective address explicitly mean special task fit main regression task task regularizer minimize empirical matrix norm row alternate feature closely relate constraint estimation estimate notion task provide task matrix joint joint alternating degenerate simplicity enable similarity general form et regularize least square supervise ny leave laplacian negativity strict positivity investigate list collaborative recommendation task et diffusion test graph svm application task also inspection estimate combination task average combination analyze sample respectively suppose variance iid finite variance without loss fix simplify mse single low separation variance sample approach infinity mean helpful differentiable specify minimize obtain non specify task ideally task minimize derivative effect varied error similarity use use analogous bayesian prior precede design minimize method minimax estimator linear result recall generalize result try risk minimize approach fold tractable freedom considerably try minimize g analytic task estimate solve
hierarchical likelihood identically goal new showing mixture great relevance posterior computation user far specify prior prior fundamental augmentation gamma associate turn upon evy fisher extensively likelihood mixture normal result parsimonious compare involve example random utility logit logistic table odd use logistic model closely extent new ratio logistic analysis chapter issue poisson table paper contingency include focus factor multi cccc center total total present concern logit way begin representation computation illustrate propose jeffreys prior remark generalization defer trial design control behind trial observe denote success control product advance write odd easily analytically traditionally use approximation hasting intractable mixture bivariate logit likelihood lead describe conditionally gamma name closely relate distribution fact moment special mixture inverse gaussians interesting straightforwardly gamma main normal latent v n arise treatment center patient let correspond single success correspond log odd ratio favor independent assume bivariate apply v estimating employ em mode begin pre iteration I estimate converge log odd ratio odd marginally augmentation normal give justify maximize conditionally maximize trivially use essentially weight least interesting comparison method run plug step compute apply clear whereby current obvious hybrid either I converge estimate log odd ratio sample relevant equation far draw gibbs density still incorporate normal wishart improper applying theory wishart figure part sum gamma work make important sampler examine robustness inference generation gamma merely simulate single modern computer far less consuming appear environment rapidly become norm task core available pc initial v draw eq sampling odd assess avoid odd ratio center vertical central credible dot posterior ratio treatment center observe draw posterior indicate wishart describe choose match identity hyperparameter marginal correlation pre specify result figure compare likelihood pool treatment effect center produce support efficacy quite uncertainty incorporate many pose tail tail logistic widely inference west particularly case logit sensitive default informative log lie informative particularly model use right tail approximate agree reasoning lead modification framework augmentation directly match tail extra compare student distributional odd odd definition distributional many logit prominent example boltzmann logistic currently comment affect em algorithm parsimonious updating quite situation arise inference precisely exponentially lead moreover strictly generate ever appeal bayes evy relate brownian motion offer advantage simulate step situation applicable insight gamma random generate inverse gaussian detail relate independent moment namely arise observation rearrange kk exponential density result euler write moment generate moment generate gamma proof result straightforwardly use expression recall write normal conditionally specific turning form arrive result straightforwardly distributional evaluate reduce explore property odd ratio odd odd ratio lead distributional author odd ratio ratio augmentation likelihood recall representation mean gaussians augmentation odd mixture theorem inverse moment generate e gamma also gamma direct avoid calculation simple representation parameter augmentation scheme integral via identity easily exponentially version distribution fall euler use generate require derivative integral consider inference logit suitable constant conditional note multi consider size odd transformation transform gamma gamma distribution normal gaussian usual bayesian calculate define sampler therefore gibbs sampler augmentation conjunction identity map odd consist step extend way default mcmc regression possibility design also penaltie conjugate likelihood interpret moment gamma gamma log z I lead exact sampler repeat issue use fisher define jeffreys jeffreys logit tail prior west make penalty matrix z pseudo avoid influential row motivation place exchangeable logit help lead prior likelihood match exactly pseudo logit code design b pre stack iterate step update agree drop covariate imply posterior use multiple odd ratio exchangeable logit parameter meta number odd ratio family functional form likelihood penalty vector odd ratio give ii parameter hyper active total set variable moment normal stack covariance combine conditionally posterior style algorithm iterate estimate em inverse wishart hyperparameter estimate repeat point emphasis weight though weight appear second new consider place categorical multi usual lead shrinkage sometimes require propose alternative log odd norm version hierarchical van scheme conditional augmentation model study normal wishart tr likelihood b family natural b observe wishart add
derivative upon loss term view market lot trading strategy loss even view coin flip chance stay coin section present help course rather practice compute mathematically result product skip large direct exposure expect side replace add explain give example bring market agree everything apart variance market kind plausible mathematically reader immediately unbounde bend justify leverage see give propagation leverage belief leverage avoid substitute realize payoff swap swap theoretically well familiar scenario succeed turn express extreme fall finite cover look exposure outside range practical traditional calibrate sophisticated express consider designing course question finally motivation process vice bank thank former bank present author herein author necessarily view email com bayesian probably logical optimizing turn separate presentation summary happen remove case capital eqs depend whether thing normalization end return probability one payoff careful numerical asymptotic remove capital risk grow e mathematically odd ref risk free paradigm denote normalization coincide capital risk free ordinary accept research market really exist market available belief payoff drop leave proceed sense market mm mm around traditional modeling extend propose quantitative framework creating meet relatively simple last world financial development quantitative method remarkably little innovation derivative business compare product recognize payoff financial challenge solution quantitative innovation serious quantitative innovation purpose introduce elementary argument rational sake clarity generality principle back trading derivative demand solution reason natural fair serious reason belief much economic treat fact belief convert payoff imply belief payoff structure check behind position agree obvious rational view physical ability product result depth one raw consider represent market stock everything know summarize depend whether knowledge powerful probability modern knowledge mechanic certainly describe finance belief material impact reality heart introduce distinct imply realize role outline market represent familiar derive distribution price entirely realize reality market predict realize realize variance auxiliary definition surprising language intuition market might trade indeed imagine market variable put opinion separate express view probability question redundant growth objective support subsequent section general present imagine expert market confident high return formal near return people would risk analysis simple limit imagine absolutely certain lie distribution belief fall within pay variation payment view digital call spread believe future outcome summarize probability illustrate nontrivial question maximize utility formulate allocate mathematical connect think win well understand understand return proportion choose subject constraint maximize expect return realize simplicity start odd market imply th spread rewrite derivation allocate oppose logical make mathematic appendix remove logic reader speak use price market probabilistic market return probabilistic interpretation odd people odd true even bid offer imply imply pricing exist belief probability payoff bayes tell update research regardless actual q spread end consider underlie market payoff run imply rearrange eq fundamental payoff rational market market logical market perform assume bayes theorem explicit posterior leave recognize case logic equally growth interesting sake brevity focus couple practical paper shape growth eqs payoff function payoff match importantly payoff even actual growth section payoff expect rational imagine justify view must adequate nothing pure justify relatively growth imagine variable market imply still agree illustrate growth mention choose payoff close follow risk optimal payoff fall analogue introduce happen ignore beyond answer boundary realize position never intend matter many scenario potential side relevant tuned boundary product really example ensure market naturally market classic payoff current pressure simplicity imply european digital option belief must far consensus justify leave reader become
qx real risk ii convenience remark map objective depend mcp risk thus concept simplify selector mathematical finance way extend therein definition application subtle scope similar implicitly depend merely risk maps markovian since mcp assumption homogeneity see policy construct replace expectation determine know introduce correspondence preference convexity concavity example economic represent uncertainty outcome first risk preference minimize preferred intuitively convex categorization objective maximize categorization convexity suggest see confirm reasonable preference situation safe state policy conservative high uncertain even apply within mixed preference risk map complementary obviously risk since real value measurable map everywhere power mixed preference map exist map property exist several map economic mathematical finance literature merely space bound restrict definition mostly mathematical finance verify coherent neutral name also field optimal sensitive control preference everywhere convex everywhere everywhere equivalent besides tradeoff expansion e minimize avoid contrary variance prefer intuitively coincide categorization concavity transition exactly probability contain kernel cost adapt apparent verify intuition scenario consider dynamic state also notable regularity unbounded deviation whole chain preference map mean variance tradeoff idea generalize measure analogously additive additive integral homogeneous everywhere behavioral economic interpret behavior risk integral model analogous classical eq discount follow control sensitive markov rest convenience objective notation solve cf discount rest shall operator simply accordingly tv define selector action deterministic fx cx assumption selector economic discount reflect valuable outcome effect mathematical property multiply widely economic finance discount map homogeneous optimal optimize well define discount multiply discount immediate discount see homogeneous equivalent discount homogeneous merely discount indeed discount objective merely risk map convex merely contrary later objective rather define show limit exist nonnegative constant w x obtain proposition f iterate proposition assumption v selector thus proposition u f banach discount show selector hence due let arbitrary f ii exist extend mcp framework prove existence measurable ii x aa aa become condition lyapunov widely reference therein optimization unbounded deterministic comparison map study risk mcp literature map countable therein discuss sup merely risk case follow connect note value change hence set obtain weak condition state existence sufficient sup condition kb assumption condition cover assumption average fu w eq xu xu yu repeat switch xu yu optimal selector selector ii jx jx lemma iterating apply mcp yet white value measurable transition kernel mcp dy ba weight indeed next k hand mcp hold proposition hold acknowledgment careful constructive suggestion des tu author b author support remark measure risk context know risk measure finance behavioral economic weight infinite horizon sensitive discount solve optimization discount propose conventional lyapunov generalize chain existence solution bellman risk stability lyapunov stability control total core mcp framework description mechanism switching action immediate action however policy mcp usually incorporate replace immediate utility whereas require sophisticated commonly cause environment employ probability finance decade therein temporal horizon infinite sensitive apply coherent base suppose rational limit make risk economic measure one make overcome limitation mention behavioral economic maintain existence infinite horizon albeit map stage discount depict discount minimize objective select two objective notice discount objective objective risk neutral expectation similar obtain treatment context horizon risk consider finance behavioral economic space prove type objective discount optimize discount discount conventional coherent measure generalize lyapunov literature markov ensure existence solution organize follow context borel generalize finance behavioral economics markovian temporal call accordingly risk control parameter subsection construct simple one discussion example subsection infinite horizon sensitive objective discount satisfied introduce framework control follow concept state borel space xy markov ax x component borel space feasible action x k ac r random capital variable letter e g denote choose markov rx x say ii resp coherent resp concave slight abuse terminology maintain monotonicity translation invariance risk map investigate risk module fx f gx l map let real value iii sublinear r v consequence remain prove iii r measure counterpart construction real map I sublinear ii coherent rv l xu v lx xu xu xu value map module axiom risk upper apply control growth iteration specify consider sign u norm rv w w due homogeneity key role investigate originally study ergodicity lemma reader v reverse give vx vx q risk important role study property mainly trick real value measure whenever r pp remark probability ergodicity state generalize map subgradient
must find solution estimate counterpart kolmogorov nonparametric record arise size obtain statistic order record perform poorly suggest basic statistic distance region evidence record arise simplify w ny df df n df df distribution record come ml distribution function equation depend desire result calculate record nan hypothesis exceed present simulate critical provide record order level early reduce alternative hypothesis leave open testing obtain level estimation restriction eq degree freedom sample go degree minute company al follow al record complete mle assume model calculate table let approach accept testing statistic support et give basis example air conditioning time ccccc approximate therefore support assumption simulate record arise ccccc mle assume kolmogorov goodness fit test record goodness suggest record first accept procedure arise exponential accept exponentially reject reject parametric result arise sequence sample obtain sequentially record sample induce statistic power conduct proposition ac ir department statistics school sciences box reduce cost running check inference chapter p record testing case summarie substantial determine within consequently check kolmogorov von goodness fit record data mm von exponential goodness reliability primarily fail course record continue service fail case long long long situation terminate failure item test observation item fail obtain call censor variety censor example censor encounter data shape exponential appear frequent life short portion parent limit parametric mind exponential record obtain estimator point inference area goodness fit base record publish direction check inherent bound variation consequently check motivated checking record record
include trial track dynamic therein mab receive provide tradeoff exploitation holding potential future arm address bandit achievable sublinear play arm propose fine ideal play reward know achieve reward sublinear regret reward arbitrarily great logarithmic achieve regret growth reward distribution simple policy know distribution bound assume support develop two calculate past play arm tradeoff exploitation reflect arm intuitive logarithmic need index confidence sufficient ensure develop mab deterministic sequencing exploitation classic separate objective exploration exploration sequence exploitation sequence player arm fashion play properly mean exploitation reflect see sequence exploration spend bad nevertheless need ensure ensure exploitation caused incorrectly identify exploration reward tail logarithmic use sequence cardinality heavy tailed reward achieve reward th moment tail classic policy ml approach knowledge reward classic support range apply know knowledge difference arm demand distribution achieve regret order sublinear heavy tail high classic ml adjust hardness term observation variation decentralize mab player observation mab often arise minimum spanning dominate weight decentralize player observation player affect player arm necessarily refer player whether involve receive reflect immediate communication multiple unknown successfully channel search collect target location agent way player agent deterministic however exploitation carry reliable ideal centralized scheduling grow mab arbitrary rank necessarily solely markov model combinatorial arm involve study extend classic policy mab setting ucb policy order heavy reward basic ucb estimator chernoff hoeffding result however high observation differently achieve heavy tailed mab tail heavy tailed also offer variation discuss option sublinear heavy sublinear knowledge reward decentralize player regardless observe reward ability system logarithmic classic mab division fair construct decentralize mab address set reward liu general interference light tailed reward arm address I propose decentralized regret reward focus light bandit play arm draw player observation arm permutation policy sublinear average reward performance tail heavy tail divided exploitation player play exploitation play choose past sample sublinear htbp notation denote include sample tradeoff exploration exploitation balanced exploration cardinality exploration set minimum reward loss exploitation order cardinality reward light tailed tail gaussian light tailed sub gaussian extended chernoff hoeffding mean subsection truncate regret tail carefully cardinality sequence variation regret memory significantly ucb store sample truncate time instant idea satisfy truncate lemma result include exploitation play truncate arm policy argument substitute consider r logarithmic regret certain address certain model cardinality exploration sequence calculate mean extend variation include objective decentralize incomplete mab dynamic arm arm objective arise multiple player constraint cost arm ml example ucb arm large tend due combine bind arm complete arm rank ucb arm however directly extend rank sequence consequence general simply choose arm specifically assume define regret cost choose theorem sample mean exploitation theorems key play arm exploration sample non small ensure properly alternative incur whenever play similarly theorem satisfy player arm large sample truncate exploitation sequence relaxed need due distinguish arm specify select rank vary objective extension class mab incomplete arm arm arm time player involve focus good cause take unknown dependency solely exchange player involve equivalently receive reflect arm local arm policy history concatenation total reward compare centralized player mean play policy similar sublinear sum reward reliable obtains solely utilize decentralize construct sequence player play fashion eliminate player calculate either fair sharing across player randomness reward limit carefully learn fair sharing player need address fair sharing round fair share sharing achieve average reward player exploration exploitation decentralize policy regret decentralize policy determine good player happen least incorrectly arm consequence analyze exploitation consider proof classical mab formulation arm one arm span dominate weight path dependency arm ignore exist naive often grow exponentially combinatorial mab polynomially exponentially maintain properly detailed rather mab formulation assume model dynamic practical scheduling continue evolve player continue evolve grow target continue mab arm continue evolve change markovian transition arbitrary unknown extend mab centralize equivalently decentralized distribute player logarithmic call weak detailed derivation address fundamental tradeoff exploitation mab separate objective clearly handle furthermore exploitation variation decentralize player incomplete
interaction participant define framework occurrence direct count change arise occurrence event kind occur specified generation implement specify generate actor termination condition semantic state responsible change termination achieve closure event fulfil consequence new state event therefore constitute trace representation process formal translate equivalent knowledge language follow convention calculus action language component atom atom atom say support create ab ic atom intuitively atom know body call head constraint indicate satisfy body program theory conjunction semantic set assignment atom program minimal consistent solution mapping part model component framework independent component deal generation predicate responsible single observe atom identify describe time establish state event event event occurrence fact translate expression translation formal augment specify length trace interested occurrence event specify incomplete trace complete trace set matching trace detail answer linear model course complete trace detail inductive programming concern prior observation contain fact trivial accurately language head write schema indicate output example bias compatible mode iff schema head replace replace atom implicit body rule hypothesis head body well form rule compatible tuple set property mode inductive approach incremental system support synthesis rule case theory monotonic inductive logic programming exist minimal tr system bias similar define transform program differently different transformation task call program program theory theory task distance ii let main availability system system intend behaviour incorrect require trace specify event denote expect output case relate instance avoid effect trace time expect body body head rectangle draw minimum height cm cm height body minimum width height cm body minimum cm body cm right cm translation must split form trace case framework tuple default static relation mode schema crucial increase computation conversely choice problematic iterative representation describe language perform event indicate answer acceptable conceptually e use satisfied set step ultimately choose rich enough demonstrate description agent find digital object block large entity file agent copy copy agent restriction request share generate agent terminate sequence event generate trace share initial case propose addition leave rule yet intend additional request set feedback refine specifie occur answer improve set include rule body combine change original specification change since include leave intel gb ram b correct add variable intend learn case detail describe integrated program reasoning execute derive available constraint construct provide perform transformation describe conceptual step reader phase example meta hypothesis rule exception case body add head body learn mode newly predicate pre try I mode execute describe phase generate informally exception rule condition body condition plus exception empty body exception condition pre phase syntactic transformation preserve solver use monotonic transformation recently particularly suited reason default observational causal dependency monotonic hypothesis essential none exist mention learn within semantic permit provide solver contain ultimately section transform logic programming notation obtain replace type replace omit top head definition predicate true whenever ib ib atom label argument argument use third limitation completeness true translate inductive match requirement case specification requirement converge course case practice system accurate solution case input instant block b b try try exception rule level lr level lr l b sec motivation behind upon correct incorrect behaviour practical experience able verification seem broadly framework al system check case logical capture completeness modality capture new assimilation time agent framework form indicate insufficient participant unable frequently course agent prefer evolution framework either put forward management whether norm purpose develop additionally explicit likewise actually choose purpose presentation line much discussion address effectively mechanism something yet software engineering undesirable behaviour background specification perspective logic program support tailor particular design requirement author propose complete differently employ stem framework high nevertheless human identify specification framework process formal program achieve mean inductive working representation inform instance behaviour actual coincide propose modification correct trace process foundation properly connect system aim criterion suggestion provide currently investigate select likely minimal e
ij equation unit equally iteratively proceed maximum ml mle hessian straightforward depend easy hill involve estimate intensity quantile estimate term mcmc approximation value provide htp estimate cc dyadic intensity obtain glm cycles jk ki ji grey bar place feature demonstrate efficacy real united directional flows united number people state datum allow consider require transformation away continuous parameter flexibility function thick tail network independent specification glm previous suggest population significant dyadic change cauchy glm restriction provide degree freedom significant statistically effect cluster effect flow offset find leave warm substantial increase population year evidence away state cauchy far depict level glm theoretically would exhibit anti anti spike place likely anti reciprocal modeling framework greatly scope technology expect tool span science across network production knowledge relational phenomenon ability structural network graph limit introduce exponential capable network greatly expand analyze examine network vary binary unbounded important graph capture generative yet limitation network binary tie exclude gene various transaction network inference thus subset specify broad gibbs sampler strength limitation apparent specification adjacency direct parameter network permutation give much contain statistic capture covariate challenge model apparent specification come specify evaluate represent proper constant distribution support infinite every ij specification capture sum product subgraph particularly unit reciprocal process likely value vertex star edge transform support multivariate transformation xy j derivative diagonal way inverse parameterized covariate specify elegant feature distribute case gaussian covariate reduce allow ratio upon interpret dependence modeling via discrete edge interpret
partly analytical etc yield valuable insight significance statement along associated great non may limit limit purpose allow derive detection would n possible bin may varied constitute trial statistic would hypothesis include composite hypothesis framework consider maximization likelihood give analytically amplitude detection series maximize also maximum statistic hypothesis independently variable cumulative cdf independent essentially signal independently distribute random cdf eq cdf parameter amplitude pd phase likelihood show particular posterior consider amplitude amplitude dominate bin main kind integration simpler comparable illustrative case instead amplitude pa observable since amplitude prior simply frequentist limit particular since also upper event happen quantile background probability large amplitude already upper imply fact intend frequentist bound suppose fall maximization integration parameter primary origin maximization behaviour affect little crucial amplitude contain limit behave ratio snr apparent amplitude amplitude frequentist confidence amplitude value simply quantile true amplitude chance frequentist limit happen zero trial elsewhere snr differently constraint amplitude actually conservative realistic wave much something improper frequentist limit usually via numerical bootstrapping frequentist desire would probably monte carlo integral get marginal amplitude quantile frequentist snr translate constraint snr amplitude present amplitude consideration parameter generally frequentist suggest bayesian complicated frequency restrict fourier sec construction may case yield detect frequentist require specification alarm connect also commonly upper limit
account x z substituting yield condition corrupt group maximum optimum procedure outline overlap need lasso derive overlap group group draw display function child scale wavelet length haar wavelet decompose image bind recover compressed sense pay group overlap tight subgaussian penalty result toy agree acknowledgement comment correctness proposition definition plus em minus width compressive signal measurement extremely exhibit pattern pattern predefine group I measurement reduce measurement need universal group group hold overlap configuration many field recover high relatively possible signal indicate one measurement exactly recover signal knowledge sparsity arrange pathway highly correlate coefficient model belong tree child property spectrum display e pathway tree knowledge help recover theoretic variety ensemble author need lie union version recover derive iid need signal sparsity analyze emphasize assume gaussian subgaussian constant gaussian case highlight main keep nature short generic group variable overlap partition ambient distinct theoretical asymptotic group similarly condition lasso author derive lasso author derive group provide group vector complement distinction make overlap complement union union set complexity compressive framework measurement gaussian focus contain restrict singular operator measurement first grow coefficient close per consideration g etc disjoint remarkably bound regard somewhat group challenge support use group sparsity loose upper group overlap specific case make tight derive overlap group derive take group require recovery correspond rest signal sparse multidimensional coefficient set I lie assume subscript convex hull wish solve paper recovery signal atomic govern formalize notion atomic group signal block define eq atomic simple representation hence look minimize atomic subject equation q standard norm atomic aware correspond atomic lasso norm group atomic group substitute give atomic overlapping case norm respect tangent cone guarantee recovery main recovery unit sphere width width certain specialize mean gaussian program optimum provide empty cone gaussian euclidean jensen obtain tangent cone cone versa thus cone atomic sphere cone quantity disjoint overlap recover group iid lemma variable freedom ii monotonicity merely optimize application particular magnitude duality suffice g follow random construct bind distance support normal cone
insight connection multiple mcmc monte carlo integral mcmc markov stationary metropolis hasting mh mh mode try chain independent identically I enable sampler make jump portion molecular chapter chapter candidate pdf accord directly target fix scheme pdfs produce identically candidate extension technique analytic weight acceptance probability find two correlated formulate rule detailed balance condition sampler exploration improve reduce proposal pdfs apply introduce procedure candidate generate early construct improve follow recall metropolis novel rigorous novel detailed balance numerical advantage relationship classical mh draw movement approach correlate constant correlate different distribution eq brevity notation vector choose pdf sequentially since z rewrite consist pdf calculate normalize accord step kernel balance need weight j draw weight normalize accord remain set repeat step weight weight criterion improve computational target possibility choice follow pdf remark recently clearly great flexibility construction weight statistical cost study need unclear classical propose eqs different acceptance probability reason eqs eqs step coincide reference different fix variable recall q multiply member notation write repeat development detailed balance pdf show generic pdf draw technique pdfs step weight variable pdf kind weight point pdfs weight compare run multi point candidate calculate acceptance movement compare use use statistically long moreover observe candidate become step j analytic weight coincide depict solid normalize figure average weight use depict dash dotted triangle rate result always correlation htb solid normalize histogram b acceptance try dot line triangle solid line circle estimate scheme correlate function analytic arbitrarily balance draw approach novel flexible correlate different instance procedure pdfs different pdfs tune proposal candidate successive proposal pdfs improve technique weight unlike point metropolis weight improve computational avoid check existence selection broader observe depend proposal important independently pdf use namely proposal separately theoretical study good
dc u r copula popular page gamma rule respectively score associate dc z estimator dimension parameter function system version idea inversion solve obtain previous inversion estimator parameter iterate inversion parameter system four omit estimator consider straight asymptotic normality fulfil dc r u u u dc u tend solution converge proof correspond assumption verify copula satisfy q u dc u dc u monotonicity dc u dc u q assumption natural parametric copula second check copula family k u dl dl u dc assumption iterate readily obviously integrable denote matrix view q family give us context evaluate estimator simulation study iterate evaluation bias th outline repeat different size repeat copula kk select parameter copula choice assign moderate dependence select copula specify reach choose true moderate dependence couple numerical corresponding summarize proceed follow copulas estimator software simulation reasonable result notably strong dependence family size reasonable family method look subsection family respectively size inversion md inversion md criterion clear table well far concerned hand case inversion md rmse rmse become reasonable observed md estimate hour notably execution term minute md system equation study inversion contaminate copula mixture refer proceed contamination model copula inversion md estimator parameter computing bias summarize contamination rmse inversion well bias rmse sensitive indeed contamination equal contamination md sensitive among md formula bivariate copula introduce moreover simulation minimum md focusing bias estimation perform reasonably rmse worth quite md one author grateful improve mm statement mm bivariate bivariate asymptotic normality carry compare rank perform well bias concern root sensitive outlier cite copula secondary g copula dependence multivariate copulas author copula construct copulas area simplicity restrict two df copula copula df j u g generalize quantile arise topic copula include maximum margin present method front produce result univariate interpretation possess feature moment extension heavy tail mention method relative sensitivity propose copula compare approximate apply multivariate copula univariate bivariate copula example performance give first begin classical moment yy statistic moment analogy moment skewness respectively functional representation quantile p shift polynomial sequel transformation orthogonality term summary fit brief moment water science especially moment point relationship definition moment show wide moment redundant moment drop moment suffice statistical definition moment medical application see pearson exceed g type iterate less equal iterate maximal family minimal iterate discuss give copula iterate iterate equal bivariate tool copula moment iterate copula three bivariate iterate copula frank table twice differentiable generator example generator one parameter frank instance copula continuous concave copula generator generator c generator page bivariate copula moment give formula bivariate moment obtain corresponding system bivariate copula moment counterpart see semi parametric
grow linearly increase pattern also look cell relationship strategy naive develop maximize usage locality demonstrate benefit obtain naive partition strategy partition schema schema compare less intermediate reduce intermediate usage locality surprising state capacity software experiment graph speedup rate low speedup law machine reason practical upper fashion conduct experiment check pointing balance speedup low job large whole rest resource correctness implementation parallel implementation setting text range finally wikipedia corpus page wikipedia english website comparison test default optimization separable implementation accuracy besides linear promise enjoy soft experiment train subset instance start kernel multiplicative table major suffer svm attractive also grow example acceptable acceptable adopt scale interesting column dropping cause terminate guess believe heart binary cm p training kernel full capability scale factorization symmetric performance two consider multiplication interest also sparsity pattern find list dominate reason large partition cost reason dramatically drop accordingly general fit computation although computational current may local machine environment cm first dataset adjacent university website web edu corpus adopt calculation sort descent plot notice since small edu reduce logarithm verify see fit side remain suggest converge evaluate execute wikipedia within hour provide plotted figure certain notice large shift regard interpretation leave chart list sort although wikipedia release popularity rank measure life experience list implementation also entry drop list top sorted mark c wikipedia entry united united english negative successfully paradigm scale problem extract discuss concern parallel setting configuration schema sparsity various scale main distance multiplicative update individually experiment successfully multiplication encourage speedup reduce schema discover crucial speedup performance show speedup computation relate svm achieve comparable matrix wikipedia entry hour encourage work suggest sophisticated involve instant communication online version development collection algorithms time fortunately prototype design version experimental widely reveal potential may far improve schema discuss prototype http advanced centre http www uk rgb song liu multiplicative suitable paradigm implement version large scale multiplication characteristic focus fundamental algorithm promise speedup multiplication design considerably efficient maintain core produce encouraging programming keyword multiplicative deal algorithm require learning paradigm remarkable representation question whether paradigm fashion wide major similarity dominant position attempt variety algorithm available paradigm solution several neighbor early stage good report individual implementation graphic unit locality hash lsh google news broad effort establish generic solve et algorithm introduce inspire realize implementation greatly model interest et propose methodology web non multiplicative lee two multiplicative machine include negative factorization support machine svm multiplicative paradigm organized follow setting three adopt extract common parallelization conclusion summarize implement machine engine google service page repository news google suggestion direct evidence google google number idea effort novel way factorization series multiplication multiplication strategy stage balance maximize parallelization huge permutation resource handle extremely acceptable plan algorithm et investigate processing pattern take lead iterative advantage limitation immediately research fashion illustrate community query property theory summation form fit sufficient query aggregate core responsible examine study generally notable exception illustrate graphic processor fail pc minimal may medium sized iterative overhead cluster implementation fail handle programming use report typical reconstruct view nmf divide observation nmf nmf lee soft k I uses please refer iy recursive quickly get single represent eigenvector chain effective direct markov chain principal eigenvector eigenvalue stationary method employ multiplication generic extract dot involve dot product vector multiplication require multiplication calculate measure distance handle euclidean distance sparsity communication stage e piece intermediate group computational call locality maximize multiply summation partitioning determine piece go locate input naive p however cause severe hash hash parameter uniformity unfortunately locality violate three parameter secondary stage factor affect stage aggregated note schema matrix result summation dense partition large profile matrix partition intermediate parallel utilize aggregate partial I implementation straightforward machine implementation computation communication random piece long row cache regard small among allow load two general essence nmf decompose multiplication nmf similarity multiplication consider unique profile utilize four multiplication generate dense sensible schema across short plan largely partition similarly schema list matrix multiply generate splitting across prefer column unity feasible conduct row entire row go calculate file format retrieve column computing read single sequentially store final output actually computation kernel matrix represent kernel calculate schema dimension account challenge storage result intermediate grow partition also prefer conduct see
tail covariate interested year affect univariate location tail trend mix estimation parametric parametric know parametric introduce fitting use author focus estimator result extend estimate propose regular twice continuously similarly author obtain base move use value weight log select advantage regularity covariate multidimensional require define normality highlight discuss present hill address minimum practical arise illustration associate metric conditional eq fix q precisely want focus design non random us ball radius tend go response belong design concentrate go belong z n tt order estimator situation covariate latter increase negative function tail discuss universal asymptotic normality weight asymptotic normality sequel fix first slowly vary function say vary detailed cumulative constant satisfying uniform condition regularity distribution slowly normality tail normality hill close call index estimator necessarily introduce approximation continuous satisfying extreme value analysis simplify rescale rewrite main establishes exponential rescaled type pareto variable covariate approximation hill sum exponential random refer near use study main normality define multiplicative weighting similarly proportional multiplicative force index impose negligible standard estimator evaluation theorem particular slowly hold arbitrarily slowly condition lemma problem usefulness two extend classical estimator covariate second choice give overcome restrictive first introduce tw tw w ts tt corollary estimator knowledge estimation order well estimator plug subsection arbitrary choose estimator replace second direct w sign misspecification figure extreme second consequence misspecification efficiency unconditional case view easily remainder paragraph devote partition define concern variance plane reason represent propose cubic centre united available http www uk daily build year distance hill estimator eq heuristic functional idea pair estimate approximately select smoothing contains rescale validate distance goodness locate interval tail day month extreme flow rest sake sequel sufficient constant verify well slowly account exist x entail conclusion sufficient integrable origin since absolutely monotone verify origin covariate rectangular ball lattice suppose approximated jt jt jt assume uv I z tt uv enough thus tt uv see theorem q I tf h introduce theorem sufficient integrable focus finite conclude theorem n k w w k conclusion theorem assume thus hold straightforwardly choose arbitrarily verify conclusion normality
ratio fall balanced use require sequence balanced set mix normalize nest sampling set self need ahead accord disadvantage property deterministic integration upon nested fall ahead present combine include nest posterior ise present discussion ad ask operate set away center take move poisson allow approximation show introduce describe balanced simultaneously member family set discuss area exploration four bb bb aa describe way describe py slowly fall attention condition vast mcmc critical sample reader reference therein turning ability execute line well demand beyond modify simulating draw key let identically exponential mean function call let reasoning py dominate observe natural uniform mean say run homogeneous describe number point run furthermore point process run use ise physics inverse call way either hx add auxiliary lebesgue center mind work yx hx go z graph add change continuously find normalize constant known estimate nb z connection finish job suppose time call mean give approximately interval build similarly easy find give prior lastly output acceptance follow bind tail special start difficulty refine estimate final estimate within ii phase create estimate success distance failure right hand chance phase expect need complete running collecting run poisson call ei simply sum iid suppose simultaneously doubly intractable arise bayesian process form independent generalize necessary far chernoff markov probability upper bounded poisson taylor series expansion simplify bind yield yield tail yield must term let prove taylor consider draw condition ise wherein adjacent similar eq integral hand run straightforward perspective come approximation valid technique answer present method lattice return suffice right node node mean circumstance instance fall give constant make solely
uk college example corollary proposition brain human claim make static multiscale modular letter relationship substantially random variation correlation decrease topology increasingly edge letter synthetic phenomena module potentially multiscale unweighted modularity weight sign adjacency c topological lattice module keep throughout relationship edge module regular lattice graph module modular different color simulation unweighted edge show tend topological unweighted characterize regular time type module graph decrease add entirely nest modular brain cast temporal organization reduce window discussion letter
objective location polytope doubly stochastic gradient backpropagation range gradient demonstrate information retrieval straightforward document supervise define goodness rank build algorithm capable query label lead aspect vary ordering function rank rich order feasible query therefore produce permutation term label piecewise gradient training learn without become infeasible supervised rank doubly relaxation permutation ranking preserve doubly interpretation propagate doubly lead choice pre normalization approach integrate powerful suited development network rank set size item label query rank permutation document aim learn permutation rank document study group ndcg indicate ndcg persistence maximize aggregate subject difficulty change order piecewise expectation objective ranking gain objective sum e kronecker delta call objective previously one remarkable aspect objective capture entry expectation doubly matrix nonnegative special doubly every doubly stochastic permutation think incorporate uncertainty appropriate gain row properly position probability expect leverage interpretation distribution doubly interpret entry ndcg permutation describe counterpart permutation matrix degenerate differentiable objective remain produce single ordering doubly natural maximize maximization bipartite problem solve time cubic bottleneck short bipartite document pass imply rank sort order permutation procedure suit rank ndcg heavily influence rank focus computation sensible second way act row training objective efficiently backward discriminative training backpropagation proceed provide computing layer input normalization gradient column index amount differentiable optimize unconstrained square piece family example probability computed define edge define construction cumulative row mass appear great preference rank reasonable probit logit cdf fast compute matrix document alternative sorting recover imply one discuss everywhere differentiable derivative tie practical primary optimize value vector possibility network sophisticate seven benchmark five fold distinct split number large query smooth anneal initialize early stop ndcg prediction mle select reduce small query turn derive document document perform normalization five short cut seven baseline figure ndcg graph art substantial build rapidly expand early employ surrogate gain aforementioned evaluation measure include gain gain ranking rank entail sort rank use ranking score view optimize ranking concentrate mass peak select variety also method approximate scale recently normalization minimization permutation directly incorporate normalization balance time post rank unlike optimize objective half balance optimize conceptual expectation certain rank objective objective doubly stochastic polytope appropriate possible projection remarkably call
estimator graphical lasso graphical inference treatment reason difficulty sample fully bayes decision theoretic involve autoregressive section propose prior imply computation shrinkage estimation covariance matrix symmetric unimodal write scale strategy prior traditional mention scale distribution early usage include sensitive regression error seek shrinkage prior flexible wide shrinkage potential highlight salient feature augmentation obtaining draw shrinkage none permutation invariant estimation carry solely metropolis hasting involve move metropolis high experiment point prior autoregressive model framework encourage integration assessment consequently paper organize outline shrinkage covariance construct augmentation conduct simulation multivariate discuss vector multivariate normal wish unimodal mode symmetric control motivation often real individual incorporate shrink clearly flexible posterior precision main unimodal density unimodal form q shrinkage variable express v may scale gaussian distribution scale uniform example popular prior applicable popular construct exponent gamma extensively scale power shrinkage ht px pareto prior shrinkage distribution shrinkage property infinite spike heavy tail precisely shrinkage normal standard half distribution close denote scale representation provide simple way distribution fy fy augment former involve sampling truncate often break gibbs shrinkage uniform latent parameter cumulative advantage cumulative distribution density provide use cumulative whenever inverse shrinkage introduce outline density inverse cdf student double pareto logarithmic cdf extend shrinkage parameter direct approach propose direct cholesky metropolis gibbs step sample first sampler systematically matrix note ij jj truncate truncate normal detail strategy generate method step complete class feature sampling constrain undirected set useful information involve multivariate modification ig start scenario involve normalize choose substitution integral hand scheme single shrinkage element towards idea rate entry hyper conditional sampler cumulative section model prior towards normalize constant ij fix represent prior hierarchy deal constant therein approach utility compare three frequentist wishart zero b model yield risk standard loss stein loss tr pl lasso validation shrinkage specification reversible jump fit wishart default total scale mixture shrinkage power double iteration rapid good autocorrelation lag mixture wishart one dataset core ghz run mixture respectively lasso compare loss thus indicate relative relative indicate method report shrinkage outperform less sparse mass prior model encourage sparsity power double pareto prior illustrate indeed ep ep wishart generalize double logarithmic constitute powerful proximity cx ip ci popular autoregressive distribution row constrain maximum ensure location set satisfying determine prior estimation uncertainty conjugate wishart flexibility model assume wishart provide author recommend model proximity prior high mass close encourage equal theory autoregressive place prior leave extension far flexibility incorporate knowledge example similarity easily diagonal functional form parameter gibbs choose plausible range association specify range recommend light recommendation prefer increasingly favor close association smooth across region concern study cancer analyze count type record state plus year collect national health count miss cancer surveillance number tumor follow spatial intercept tumor mean associate hyper place cauchy expect robust shrinkage simulation let ji neighbor heavy tail assess sample experiment divide count bin bin follow mean sample discard burn cancer report measure square mean wishart wishart maintain variance overall credible sensitivity plot validation display surprise seem influence appear tight influence implementation hour validation take day runtime core ghz flexible htbp var non shrinkage double logarithmic property construct mixture recently shrinkage receive proceed via prior model regression coefficient author independently propose base kernel bridge bayesian posterior suggest introduce sampler implement simulate normal simulating exponential posterior exponential analog bridge challenge posterior mixture normal logarithmic prior advantage many specifically independently identically standard normal parameter conjugate mahalanobis report error logarithmic iii outperform performance exponential logarithmic prior median report bootstrap error cb iii ep ep three recommend generalize ep exponential mixture unified framework shrinkage matrix wide uniform insight advance
financial time series arise proper inverse development indicate time besides evy pareto paper analyze brownian inverse difference property analyze mainly moreover asymptotic gamma large exhibit motion motion inverse increase evy transform evy exponent drift evy moreover characterize dirac delta differential define memory kernel laplace transform classical anomalous brownian length period distribute describe period besides stable case namely gamma review function stability skewness distribution skew assume simplicity property sum stable tail stable distribution c moment finite attractive physics anomalous phenomenon physic define exponential gamma infinitely tail gamma mean tb stable stable l evy laplace q form moment brownian stable stable evy increment evy laplace operator proportional derivative fractional see process case calculate function generalize memory formula simple consequence calculate stable derivation increment evy laplace also namely evy transform consequence moreover inverse gamma calculate eq consider summarize c stable stable ts c tt te du process plot panel observe period tb stable ga stable equal stable focus one popular characteristic record trajectory particle analyze motion scale matter stable stable gamma scale square ensemble obtain curve gamma whole obtain law mean trajectory process plot stable ga stable law plot gray line tb ts ga motion property analyze process point asymptotic squared consider show case small appendix gamma define gamma
condition probability theorem obtain decrease curve shift right problem hard certain sense error versus rescale sample curve align different panel b show plot rescale axis predict stack establishe appealing choose large condition although theorem guarantee minimizer nonconvex program minima impossible local minima global optimum nonetheless program reasonable geometrically project gradient constrain update stepsize assume statistical consistency optimum constrain universal gradient descent bound let denote objective optimum gradient update contraction claim deterministic probabilistic condition enter corollary involve surrogate noisy extension descent high state impose apply nonconvex note nonconvex program addition minimizer considerable understand upper iterate polynomial compute focus involve verify right essentially term optimum similar guarantee composite descent apply lagrangian project objective decrease geometrically experimentally prediction simulation show apply additive panel panel random apply projected method project gradient iterate statistical trace trace statistical geometric rate straight regardless start iterate exhibit converge statistical geometrically point theorem require meet statement corollary universal positive constant corollary triplet scaling mean instance draw scale pm theorem signal covariance ratio measure note covariate agree know corollary compare derive tolerance sub particular notation arise must noise use sophisticated chapter replicate form n corollary note sample example sub quantity conditioning fraction establish uniformity bind matrix curve align suitably rescale perturbation figure rescale curve miss rescale autoregressive additive perturbation drive point excellent agreement scaling run appear corollary additive value run project gradient q plot rescale roughly b rescale error rescale miss curve roughly constant showing point trial noise run plot rescale finally type arrange linear except two set rescale rescaled come randomly first entry probability condition number rescale generate sample appropriate corrupt miss error run chain additive plot missing panel rescale rescaled prediction qualitatively similar star technical corollaries supplementary pt minimize indeed regularize imply equivalent remainder derive condition combine low left side side triangle piece conclude hand sparsity piece conclude inequality inequality combine stronger regularize returning obtain q piece rearrange claim side final use combine giving prove claim project use state require proof show argument hinge restrict smoothness apart exploit update appear paper tolerance condition hold final assumption similarly combine turn lagrangian version correspond within contraction take previous remain verify put piece compound formulate sparse corrupt may project guarantee statistical draw estimation exist type dependency covariate perform corrupted replicate point interesting nonconvex broadly discussion grateful associate anonymous improvement section notation remark part foundation fellowship grant dms air force office scientific although standard involve draw involve noisy miss high novel datum inherently nonconvex difficult establish guarantee practical approach nonconvex program optimum project minimizer provide miss dependent condition project converge geometric global minimizer standard prediction partially observe voting behavior vote behavior survey suffer miss question tend noisy partially variety deal well involve book reference become noisy consequently converge minimum guarantee optimum study sparse develop despite still error optimum show scale perfectly fully observe miss corrupt g well reference therein mean require relevant allow miss converge close covariate noise selector multiplicative estimator past dimensional model remainder description estimator descent section devote main include optimization corollary case noisy miss section confirm theoretically spectral hadamard background description motivate class discuss descent link vector via observe covariate include independent say vector entry observe hadamard multiplicative noise however interested probability random consider fix accord vector autoregressive predictor grow exceed impossible unless endowed structure instance sparsity also infinity class examine simple constrain program long radius unique program know matrix certainly try estimate nonetheless provide suggest principle quantity consider regularize user note equivalent lagrangian duality objective set shorthand row calculation concentration inequality sequel well program involve contrast matrix loss appear unbounded make eq generally impossible presence minima remarkably issue project descent program close optimum illustrate serve unbiased noise pair noiseless semidefinite regime cause equal consider miss random extension entry use diagonal cause consequence semidefinite convex multiplicative generalization application row I noise arise reader discussion q calculation past observe contain various closeness focus closely understand condition type matrix satisfie eigenvalue condition support bound magnitude conversely since convenient analyze fully row n choice early also probability previously discuss generally negative finally analogous upper restrict eigenvalue formalize upper restrict tolerance high project et loss rsc smoothness random scaling choice minima possibly program polynomial procedure optima simple either program quadratic function application method generate recursion stepsize rp procedure update variant project regularization also onto detail semidefinite converge global objective generic iterate family program iterate actually close statement
utilize stick break study break beyond dp identically distribute observation convergence infinitely compactly continuous derivative optimal require intervention derive old well old stop short deriving derivation require class development assign useful recall two dp distribution partition stick break interpretation first approach characterize measure measurable characterization say independent base measure determine admit lebesgue density strictly positive constant variate lebesgue equip borel qx p identically iid establishes calibration induce infinitely density rate compactly twice continuously differentiable result section probability compactly continuously n tool condition slightly two number kp px px qx qx kp ball cover hellinger positive hellinger call sequence existence quantitative lie stick break dp mass precise fix ml dp b b let net let simplex h ds h h small equal h assertion multiplication hellinger root break z hz h describe paragraph md md formula assertion subset adapt nearly super finitely differentiable function arbitrary super n n n l hellinger sm prove second fourth bound bound possibly n c old n hellinger correspond old derivative care establish class ba cn n immediately proposition include first tackle density big challenge smooth proof minor need handle leave gap intend ordinary compactly ba cn n density z du kk du n u mm dp fu fu db u diameter u gamma super along simple development n n n straightforward extension display distribution point propagate extension discrete n p support p supremum continuously differentiable let closely need dy x integral form remainder dy
correlation perform strongly suggest fast range presence hardness dimensional frequency parameter frequency model quantify change response bm natural characterization essential bm generally short response correlation frequency also experiment letter infer keep recursively must overfitte enough important conventional expansion pseudo intend implementation function boltzmann ising model eq bm ip expand later spin loss impose come increase allow calculate large maximal size hardness obtain individual frequency expansion ise grid noise spin allow entropy entropy length short entropy common interaction sign depend absolute separately entropy smaller discard low neighbor cluster select reach summation partial entropy vs perfect sampling label dot configuration b deviation entropy tail interaction differ affect entropy absolute value correlation first extensive overfitting make secondly entropy universal coincide system behave perfect accurately cluster entropy path systematic enumeration pass pl fail interaction critical size error decrease rao fig system cross core processor ghz performance seem interaction recognize pl vs grid obtain grid infer grid dependence boundary condition vs b spin spin understand calculated bm fluctuation configuration equilibrium expression poor correspond criterion justify fluctuation comparable bootstrap find fig merely bottom leave vs unity underlying size analyze second activity bm cluster reach fig c error infer correlation fig interaction find select cell small cpu computer expand alone entropy generally achieve fluctuation fig coincide good expansion system rather dense severe version include base ij even bm problem exhibit correlation due accurately ise range decay interaction consider even direct widely closure change dr r correlation redundant redundancy origin locality property cluster entropy useful biology advanced nj paris france l paris procedure correlation build successfully recover
intrinsic notion improvement tail replace slightly weak tail negligible tail inequality inequality sum inequality moment tail use shorthand concave due result give inductive first inequality jensen inductive combine inequality n inequality corollary estimate proof technique first subgaussian subgaussian q require bernstein bind surely almost eq q eq q tail dependence dimension moment trace roughly form apply full tail improvement applicability inequality subsection disadvantage quantity replace theorem bind early tail case point elaborate far rademacher equal x remove factor side perhaps importantly deviation whereas former latter provide tail roughly put optimality share inequality exponential moment relate nearly moment technique count contribution beyond entry worst denote small eigenvalue subgaussian constant tail state previous work tail infinite subgaussian finite example give error moment matrix I copy relevant high x rather spectral generally spectral error appropriate spectrum show dimension inequality sometimes strong vector ix dd potentially dimension n tail tail cover appendix comparison scenario compare rapidly decay time square factor factor roughly example latter much weak subsection independent concentration expectation put main factor give approximate fix column computation prohibitive alternative say q allow failure give sample outer product identity inequalitie b union b bind grateful useful pointing subtle early comment reference tail small large subgaussian result explicit originally vector surely application lemma simply completeness subgaussian subgaussian certain combination relate tail tail particular almost chernoff bounding lemma therefore
novel name normalization assessment embedding assess similarity correspond low dimensional embedding caused eliminate aspect embed still geometric structure manifold preserve global assessment evaluate skeleton manifold preserve local assessment evaluate preserved tool selection embedding optimal specific include validate effectiveness literature describe independent embed depict conclude assessment review presentation notation table throughout paper euclidean lie embedding lie datum local index near neighbor low near neighbor identity frobenius norm motivation embed quality assessment global work review pm enable quantitative translation match finally assessment result show pm provide quality point pm neighborhood although modify pm scale neighborhood address coordinate low pm follow neighbor work overlap representative lc meta serve diagnostic tool assessment sum consist index neighbor accord euclidean quantify embed embed exploit develop agreement ar shares ar assessment france rand index reduction lee propose name co ranking criterion aforementione rank neighborhood cast unified framework visualize extend dependency high assessment pairwise preserve pairwise long keep degree inspection residual variance diagnostic evaluate standard approximated geodesic distance value embed nevertheless normalize embed geodesic distance preserve method manifold ground truth within euclidean geodesic ground truth set assessment case hold manifold short construct ranking branch length overall assessment normalization aspect assessment assessment measure scale coordinate subsection demonstrate normalization motivation description subsection state dot black origin blue neighbor origin embedding randomly normalize embed mark dot origin mark circle dot nearest origin mark red square meanwhile near marked circle see origin lot overlap meanwhile clearly observe distortion overlap cause normalization topological preserve neighborhood structure manifold neighborhood view subspace ambient normalization rational choice measure assess coordinate formally index motion rotation translation assume stand goal find optimal match solve follow constrain matrix square transformation formally q normalization item introduce eliminate scale optimization admit alternatively orthogonal manifold riemannian embed multiplication convenience presentation ia square form yield nn rewrite form let objective proposition trace expand derivative strict convex substitute solution fp db jj mb jj since make follow substitute fp ib jj jj ia jj jj let rewrite tm j tm tm column stand hadamard product transform descent problem function gradient follow calculus dimensional stacking column derive w elsewhere tp iw w mh mh mi tp matrix calculus formula length method vanish iteration update q qr triangular high decomposition greatly possible project densely optimally manifold method step step otherwise decomposition assess embedding need evaluation manifold embed preserve geometric neighbor embed assessment address normalization local subsection low characterize neighborhood preserve assessment index geometric preserve topology embed word treat representative pairwise geodesic preserve geometric motivated assessment motion scale computation global step respectively neighbor treat length euclidean graph short distance count short path finally denote define geodesic obtain optimally preserve position equal one correspond define global corresponding need experience state replace yet complicated method subsection implement suppose namely well locality vice versa say well topology vice versa compute specific combination assessment score test benchmark use select method embedding compare commonly assessment unified assessment indicate embed method description embed surface embed surface notation description measure residual measure assessment eq assessment truth manifold le number near embedding give shape le recover geometric structure training c embedding method state bar plot learn embedding bar correspond bar visualize fig bar embedding embedding learn affect normalize still report reasonable design give overall reasonable method local neighborhood le good completely match inspection demonstrate effectively bar matching match intrinsic degree learn name bar learn character share intrinsic convex rectangular geodesic connect training sample neighbor learn bar plot approach distortion global shape bar fig report quality assessment match illustrate embedding evaluation geodesic short approximate distance intrinsic degree freedom various method bar low bar evaluation training embedding bar observe except manifold global isotropic report evaluation still fail assess correctly successfully match isotropic name learn embedding experiment code intrinsic freedom graph construct learn neighbor bar see give orientation successfully extract intrinsic despite inspection bar fig indicate experiment lie geodesic short geometric manifold value intrinsic freedom manifold subsection application important near first section select manifold
time sum obtain choose action sp kullback horizon arms reward max q recommend arm confidence newton strictly tie maximizer choose random idea suggest section main paper asymptotic state problem optimal regret maximal ucb choose arm bernoulli ucb asymptotically kl ucb generalize tend kl asymptotically chernoff obtain ucb ucb maximize choose choice sub logarithmic proposition concern kl ucb kullback omit ucb study difference compute arm soon exploration instead case quite large simulation suggest ucb reward ucb large principle rely bernoulli makes present bernoulli ucb choose appropriate arm belong density respect reference partition twice differentiable poisson gamma easily check upper replace line kl choose reward state apply remain object generalization possibly different arm technical topological discussion conclude observe require upper policy price slight loss theorem canonical exponential precisely observe cx b concave point poorly concentrated remain correctly almost begin arm estimate play good mistake important bandit design event decay fast skewed slowly vanish arm reward suboptimal time logarithmic compare five tune suboptimal represent plot six panel clearly highlight effect mention simulation reliable typically ucb instant note show ucb ucb ucb proposition arm scenario hence kl quite actual ucb criterion whether play criterion ucb ucb estimate compare arm correspond upper consequence arm play ucb share common effect ucb generally preferable elimination instead outperform conjecture ucb term ucb line remain preferable final comment tune perform though slightly bad kl ucb right panel fact control ucb ucb differ ucb simply asymptotic bernstein inequality draw grow exploration significantly difficult inspire reward nine arm different reward regret scale tune reach par kl significantly significant cp kl binomial confidence easily verify quantity satisfy property pearson ucb differ ucb arm compute interval leibler easily adapt proof show bound cp ucb kl often actually decision term ucb achieve marginally guarantee cp arbitrary truncate reward ucb ucb clearly ucb account reward value distribution ucb horizon yet ucb tune ucb variant kl ucb kl ucb prescribe easily slightly performance excellent consider positive arm generality denote bt kl ucb confidence expectation last lemma prove appendix positive provide two lemma ta dr dr self present improve ucb make policy simple even binary f common field hold proceed make trick slice geometrically slice trivial nn nn kt x state deviation moment ts france analysis kl ucb algorithm online horizon bandit distinct bound reward kl ucb uniformly bind reward reach furthermore simple optimal unbounded exponential study compare ucb main ucb tune kl ucb remarkably stable ucb rely arm simple agent slot arm try maximize choice draw opponent subject bandit choose receive conditionally arm choice reward rule observation measure reward horizon reward would accumulate arm exploration exploitation rate play sufficiently since work several variant solution see reference therein distinguish family distribution belong family bind policy determine optimal policy multi draw optimal maintain work refine recently ucb term parametric bandit online call kl ucb horizon tune recently together learn ucb denote kullback consequence upper arm fact divergence specific case reward rescale
hardware fuzzy circuit consist three part create work procedure part since currently work fuzzy consequently layer layer specify gray dash purpose rule describe work fuzzy consider become strength input big degree reason circuit confidence treat circuit value confidence degree concept neuron create fuzzy number system represent concept pixel interpret aforementioned concept intensity degree hand concept first become specify activation concept fuzzy universe one big terminal represent concept big complement terminal represent small confidence validity application detection pixel represent minus input dark fuzzy concept purpose simple circuit fig vertical act universe universe weight membership universe fact specify membership function circuit weight fuzzy big small universe shape circuit depict side specification shape membership support similar evident function implement work procedure simple column locate maximum input row create membership define universe variable low generate universe fuzzy function x x fuzzy circuit correspond number fuzzy column circuit second layer specify call create role circuit compare column perform dot fuzzy layer weight form column middle therefore fuzzy pre create output correspond weight negative say column implement input implement output variable maximum equivalently similar implement rule output fuzzy column implement variable big complement configuration two part rule distinct fuzzy strength output value compare way recognize concept simultaneously input apply rule input want determine strength activation fuzzy actually product function hide vector finally artificial neuron fuzzy create consider difference neuron kind activation another operation fuzzy part conjunction simple part happen evaluation rule evaluation result rule tie reveal similarity rule basis last circuit layer consequence evaluation per visible system method summation triangular fuzzy primary fuzzy fuzzy intend system simple herein note sum operation kind degree determine degree concept assign neuron neuron represent output neuron concept big assign increase big output aggregation neuron big neuron output unit variable simply consider final unlike fuzzy system circuit fig single output properly get fuzzy number circuit want gate briefly image locate intensity word intensity intensity edge otherwise consequently pixel application logical neighbor intensity indicator existence pixel one detect kind gate work input pixel intensity output function intensity output increase output proportional build consecutive strength higher weak system generate gate fuzzy horizontal vertical fuzzy loss generality circuit fuzzy merge pair consecutive fuzzy fuzzy manner construct extract vertical use circuit vertical extract degree repeat horizontal extract circuit fig analog extract consecutive pixel applicability hardware fuzzy system fuzzy structure pre membership numerical specification output require present easy modify specification shape force probable smoothed smoothing extract present fig merge single final simply fig circuit extract intensity degree directly concern view detection e smoothing fig edge counterpart area edge visible analog therefore operate fuzzy surface different versus behavior fuzzy function clearly area similar intensity zero difference begin I activation hide circuit fuzzy inference method fuzzy surface describe fuzzy function next propose extract image extract structure apply illustrate stability white gradient algorithm noisy result information neighbor hardware fuzzy express fuzzy fuzzy implement newly I fuzzy extract edge simulation show edge detector noisy concentrated detection obvious structure implementation fuzzy rule inference fuzzy system lack hardware tried overcome fuzzy inference fuzzy achieve introduce fuzzy realize detector analog edge detector advantage edge image integrate circuit successfully however limit extend addition small device powerful device computer beyond limit scale equivalent circuit currently work digital gate array construct digital logic perform make former separation inefficient later lead consume computing establish flexibility also limited precision arithmetic operation carry limited publication scientific progress compute system front digital fast counterpart almost logic recently capabilitie use fuzzy logic digital addition fuzzy logic concept analog completely analog fuzzy advantage way implement fuzzy introduce fuzzy analog fuzzy edge fuzzy fuzzy construct fuzzy consecutive extract simulation meaningful edge benefit fuzzy detector hardware implementation analog advantage note system use even hardware artificial construct binary network demonstrate devote fuzzy fuzzy inference system fuzzy edge eventually section passive field artificial intelligence become clear passive many analog neural building analog circuit hardware soft computing implement digital circuit mathematical passive neuron connection build output neuron connection create logical say neural fig implement traditional circuit capability weight fuzzy relation activation layer fuzzy probable similarity logic usually interpret work another figure connection layer weight usage denote locate matrix connection locate layer neuron represent linguistic concept neuron neuron active logic concept mean output time neuron consequently consider work fig network membership connection connection matter neural network fig
rate finitely rate always converge finite subset candidate recommend support determined reflect mixture relatively modify approximation finite location mixture focus admissible substantially decrease combinatorial solve reasoning calculate consideration pick good score maximize profile essentially marginal integrate reasonable sort along mix support fix discrete choice henceforth dirichlet measure linearity without explicitly evaluate via sequential imputation function sense define reasonable properly account preference two support contain justified assume restriction closely support grid advantage nonparametric become parametric fact possible estimator drawback however maximize combinatorial satisfactory computational second bayesian fraction value nonparametric know density borel present show process computation mix absolutely continuous measure almost surely depend shall counting application continuous large two density kullback divergence respectively mix weak closure km show conservative obtain near see base likelihood density marginal recall f dirichlet hierarchical mixture prior single act marginal bayes work maximize restrict moderately set support cardinality despite give anneal combinatorial problem anneal move current new tend simulated decrease temperature take follow give acceptable discussion subset associate determine suffice anneal set generate define characterize else visit herein choice success annealing move move likely flip prevent get temperature make shall exactly component encourage number assign great mass q equivalently sample uniform next I likelihood reduce dependence marginal likelihood predictive averaging computationally herein stable keep specifically corresponding example distance consecutive run choice option drive force behind simulate subroutine marginal likelihood suppose subset recursion convergence rate present focus case focus galaxy set consist velocity galaxy consideration reasonable second version liu imputation likelihood study mixture seven consideration respective list table support method take point complicated take aic scad model estimate single tend example bic ccccc bic bic aic bic aic bic aic bic bic proportion estimate size take procedure able lattice higher become costly kernel outline construct space collection quite advantageous introduce simple mixture could reduce size space important location interval respectively location fine grid mixture unimodal cover mixture student fix degree vertical accommodate restriction simulated location adjust indicator characterize enter pair admissible subset optimization restrict subset reduction structure section change chance respectively maintain encourage accomplish encourage entry select likely algorithm mixture anneal flip coin move accept move bad candidate help simulate annealing get location scale galaxy five overall good elsewhere five need six computation panel mix simulation component make relatively detect distinct component vary size location anneal optimize collection admissible subset subset second comparison include estimate seem increase large quite competitive method good across rw rw rw true take present novel hybrid stochastic annealing algorithm finite introduce unknown bounding approximation simulated annealing maximize likelihood candidate benefit rate approximate work example competitive exist experience method fast spread support bound bound bound need since explicit mix simulated annealing proposal implicit anonymous incorporate approximate candidate consider grid fine nearby concern want unknown algorithm replace profile version experience expensive computationally acknowledgment grateful lead go liu suggestion work complete author university recursion anneal efficient grid anneal method employ maximize likelihood estimate novel annealing compare predictive recursion complicate describe mixture difficult specify mixing assume observation density eq
function respectively I qx te put give bind partial centering iteration tolerance p ij x I v I p ij q qx ij c il conjunction center tolerance k ij q p q qx ij p variational low example conjunction initialization initialization observe initialization work cluster four replicate array analyze al ng version approximately ng four go adjust index partition four category illustrate mixing unlike go weight length cluster high among five axis axis profile covariate gene attempt ng et category addition gene observe category consistently separate remain gene category probability fit axis probability substitute pz represent belong mixture belong go impact center five component marginal attain take average second average efficiency clear hierarchical center hill tool journal k genomic analysis systematically perturb ng ng effect
x qr growth theorem notation restriction observation available method section order order reduce dedicated variable order perform procedure variable way order sort value variable technique improve perform make asymptotically appearance variable number select high give penalty order frequency decrease false discovery rate require indeed see relevant place order successively eq stop accept set relevant generalization since multiple sake understanding deal procedure order testing test successively nan accept section recall k ts v q introduce space first upper multiple I ks kt kk u simulate choose accordance follow ensure procedure next fix estimate differ part indeed lie order essential chance occur restrictive appear family stochastically order define derive let set eq reject explicit model orthonormal cardinality let accord procedure k p already define tn k tv unknown nan hypothesis introduce relie define kk tv kt let accordance reject second procedure ensure next give condition notation follow theorem tm tm tm nx consider state b I powerful step important procedure combine perform right bound constant depend rewrite theorem remark context procedure show call almost estimate consider begin section section applicable modification concern account denote dimension successively collection alternative x k apply comment result table selection procedure present implement project six method procedure order denote order table section fdr method comparison several framework orthonormal latter fdr variable compare adjust fdr fdr calculate variable onto introduction fdr biology order selection calculation avoid schmidt orthonormal le kx l l j lot redundant calculation gram schmidt case order unknown family chi degree freedom define simulation family unknown concern simulation coordinate orthonormal orthonormal principal example mention consider reflect well condition mean column percentage actually label variable replication record select average mse square calculate cross threshold pe also objective distinguish remain assume reach frequency frequency stay decrease assumption wrong procedure display order variable accordance far powerful order unknown lastly context give excellent dimensional table result surprising choice powerful almost concern test orthonormal orthonormal compare fdr weak fdr indeed nearly procedure outperform combination induce decrease become satisfactory l correct replication record record correct average replication zero l l fdr mse truth c mse replication record replication average l fdr c correct truth truth mse replication column truth time record select replication mse non sparse sample hypothesis procedure prove powerful signal show outperform especially h k k h obtain eq conclude construction k condition u bind quantile u n condition q x log upper u inequality hold positive number n lead j k orthonormal j k upper z explicit let gaussian order v u expression upper part k denote proof condition verify quantile n give event n f n ta n g k g u tm mb ba n nx combining k k k n tc c mc two lie theorem remark estimate linear new order variable procedure inspire propose new procedure powerful condition give discovery fdr recent provide biology throughput still major challenge fit also purpose discover variable lead high linear probably penalization set lot consistency lasso penalization induce pls pls kind aic information penalize portion high develop powerful high selector recent show selector exhibit behavior criterion model selection procedure
work ensure step triple correctly orient v describe application orientation path l four vertex two successive conditionally dependent hx ix jx jx kx x hx j applied lx search length consecutive dependent structure conditional relationship satisfy orient one relationship triple create depend path input orientation rule open possibility path yield skeleton skeleton order infer skeleton need determine trade slow obtain modify instead consider subset output identical output output equivalence however detail suppose independence ccc dag skeleton skeleton skeleton denote initial skeleton final skeleton skeleton absence skeleton x skeleton find pt step possible result mechanism triple orient consider check triple check would contradict information output correspond interpret dag latent output output page read dag b namely reason independence skeleton sep appear output encode consider adapt add pointing modification conditionally belong conditional read class interpret specify graphical underlie identical theorem output infer edge output induce extend jx jx induce connection induce see page x shorthand skeleton vertex three induce induce induce moreover induce induce path give relative pt ix must adjacent occur member j ii dag condition c c role formulation illustrate adjacency matrix entry replace variable nonzero entry strength mutually random variable assess half parent child restrict particularly scenario oracle version output identical oracle sample version version processor ghz gb ram r first version simulation average observe integer assess case number additional compare give difference additional almost difference next performance version compare simulation value dag run parameter replicate replicate perform slightly large replicate mark replicate replicate mark replicate see text dag average dag set size run computationally graph eight hour termination occur times nine latter nine lead algorithm compute number extra mark b remain first sep time sep vertice consider combination run sample version simulate possible replicate new use max drastically consider subset cc still next modification setting run second replicate text show running replicate scale version run correspondence time fast graph fast second test adjacency vertex large see really fast independence exist small interpret output weak mean identical algorithm similar output cause structure find version rare consistency considerably modification show use assess impact use building causal observational start reason open investigate completeness investigate edge mark maximally section remark e nsf grant ai direct dag arbitrarily selection explicitly infer independence causal setting computationally infeasible fast output informative independence causal setting package consider acyclic system background outline system satisfy causal causal acyclic effect cause possibly indirect cause direct path dag read dag dag equivalence markov give shorthand equivalence dag alone markov equivalence dag uniquely complete partially independence relationship exactly imply suppose causal independence consist dag idea prominent example pc sound complete I maximally informative causal implement r asymptotically want dag case dag calculus intervention calculus represent dag conceptually causal idea form basis observational datum generate causal dag stand intervention calculus algorithm validate expression practice variable record speak whether measure sample speak see several problem inference pc incorrect relationship observe dag pc algorithm would lead cause neither direct box latent circle represent observe latent dags marginalization conditioning follow dag conditioning dag variable example x imply entail exactly solve introduce observed graph dag latent transform observed page describe infinitely dags latent selection relationship mark selection variable possible underlie observe separation separation see definition describe relationship partial selection relationship cause selection variable dag true imply mark reflect independence arise dag cause arise cause markov information among alone pc algorithm originally orient induce path sound presence et introduce extra orientation output interpret despite intensive modify less cut typically tail sound size situation latent causal conditional independence variable much order hand compare define prove consistency setting sparsity need strong exist supplementary document section simulation error similar modification package implementation graphical compose edge six bi mark graph direct direct bi presence mark graph one present say pair graph vertex vertex call sp ne vertex say path path together vertex definition vertex x form cycle three adjacent adjacent path three ii vertex adjacent adjacent mark star path pair vertex graph maximally subgraph call graph almost undirecte parent dag entail relationship via call se separation say middle structure mx x joint density conditional qx p qx implication conditional infer separation separate dag separate ix selection selection variable e disjoint correspond conditional relationship page path definition moreover ix encode independence variable selection separate separate importantly preserve relationship dag remainder event distribution conditional relationship want causal relationship underlie dag represent purpose new first make distinction partition c mm mm g iii absence two exist presence vertex pt edge j jx vertex gp iv condition x strong several consequence dag skeleton skeleton correspond equivalence class dag example worth may skeleton graph also discusse propose modification speed remain sound discuss several algorithm give skeleton final skeleton structure mark conditionally ix jx j I mean j x conditional call pt k jx x h pt sep vertex since sep skeleton pc starting test adjacency vertex responsible final skeleton triple mm kx kx x information sep thus algorithm every element x j arrange edge remove independence complete orientation necessarily iv orient algorithm replace circle tail orientation computational effort two set testing subset set infeasible contain sep set play important defining must sep first modification sep decrease edge type mm j x jx hx hx kx ix ix ix line replace modified modification possibility use conservative structure fewer especially algorithm v bi edge set orientation work follow result subset separate separate separate separate
paper pareto front locate deep weighting dissimilarity efficiently linearly theorem pareto validate theoretical advantage real acknowledgment labeling thank suggest computing nf present proof theorem measurable conditioning py complete assumption x hence fy dy dy substituting integral special simplify recall note large complete event q interior hull condition independent let pe b pe q eq calculation change proof theorems two correspond theorem deal asymptotic suggest hold even though experimental rigorous two denote absolute coordinate result experiment suggest grow figure show curve regression give half power growth dot align experimental criterion induce domain test current pareto front denote near connect near dissimilarity detect anomaly specifically anomalous nominal sample small sample near anomalous anomalous may much like hand choose sample nominal mean nominal near neighbor connect near neighbor e isolate could anomalous undesirable keeping try problem nominal deep retain connectivity require connect implicitly consist proximity work return situation design construct dissimilarity connect choose independent probably choose jointly could perform choose dissimilarity poorly example problematic previously paper area heuristic include trajectory low trajectory nominal detect criterion speed trajectory anomalous anomalous anomalous anomalous perform quite auc advance example identify pattern exhibit anomalous often anomaly detection anomaly single euclidean dissimilarity capture anomalous case test anomaly advance algorithm need execute choice combination parametric anomaly pareto anomaly criterion propose criterion provably criteria anomaly variety diverse detection detection method parametric typically dissimilaritie high multiple dissimilarity measure anomaly detect trajectory multiple criterion use anomaly order anomaly combine dissimilarity advance difficult dissimilarity choose furthermore change execute anomaly detection concept pareto detect anomaly choose typical define multiple compare item item say pareto well equal criterion pareto necessarily hence anomaly form create corresponding dissimilarity dissimilarity pareto pareto front depth dominate pareto front depth remove front front continue remain front depth see figure nominal anomalous locate pareto front front discriminate nominal grid exponentially number criterion assumption multi criterion realization pareto front precise use combination theoretical validate several art anomaly synthetic set tp illustrative training black dot pareto green pareto induce concentrate around triangle concentrate follow front relate pareto anomaly lead detection algorithm experiment several utilize optimality previously propose overview typically formulate problem pareto quite difficult pareto optimality introduce rank create datum pareto correspond dissimilarity sample pareto another feature view case view improve view disagreement investigate criterion could severe disagreement performance another differ multi area learn anomaly detection survey distance th neighbor score distance neighbor nearest adaptive anomaly span bipartite anomaly nn anomaly scheme define aforementioned setting execute unlike anomaly section utilize optimality area economic science social sciences pareto pareto front criterion item function criterion non negative weight combination minimizer linear involve find item another item criterion write pareto dominate combination include item combination pareto front pareto front pareto dominate item member front convenience pareto front seminal much pareto front relevant I pareto belong cardinality general pareto framework feasible vector common construct criterion combination easy identify pareto hull pareto good many pareto pareto selection weight linear compare paper pareto point identify anomaly anomalous reject much anomaly pareto linear front two domain induce open density vanish outside large density strictly asymptotic z pareto point normal convexity identify pareto optimal geometry panel theorem enough pareto neighborhood attention randomness even convex front method pareto let tp pareto front induce geometry randomness pareto black experimentally close likely pareto pareto nominal sample anomaly anomaly significantly criterion associate measure dissimilaritie dissimilarity compute ki ix convenience strictly dominate another pareto front dominate second pareto strictly section front correspond locate pareto dissimilarity combination criterion nominal basic criterion pareto require front connection anomaly test near neighbor choice jx jk nearest say front near distance anomaly mention multiple criterion property know pareto pareto linear pareto weight independent dependent support appendix pareto observation near anomalous anomaly mean depth easily use phase calculate pairwise dissimilarity create pareto front testing dissimilarity near create training sample anomaly show provide heuristic practice require anomaly detection mention
hide vector set information perform activity mutually exclusive across note hidden vector translate require across vector connection variant illustration dot connect layer constraint valid activation single j conditional sampling efficiently show softmax vector pair train use cd generative gibbs initialize vector probability divergence compute copy useful feature set relevant desirable robust situation maintain connection target accomplish copy energy dependency constraint activity least call illustration constraint example activation j exception condition tractable generative perform layer hide cd update implicitly pool soft input definition softmax pooling disadvantage begin essentially activation unit actually hard operation definition apply gradient class approximate likelihood classify input learn classification problem drug categorization year propose parallel diverse expectation extension citation knn support mi decision loss expressed accordingly present rather maximum generalize classifier without function product radial base family function gaussian convenient kernel kernel instance kernel max mi tend quadratic compute kernel set use probabilistic convolutional rbm propose see pooling rbms mail rbms set baseline describe size feature idea baseline indicate gray bold student classification accuracy good variant never statistically worse importantly outperform maxout confirm usefulness level oppose input purely discriminative scale pooling however categorization piece mail page page handwritten letter form white mail thousand day process couple day document management page process handwritten produce feature correspond presence word page vocabulary recognize word resolution gray scale regular box handwritten compute statistic predefine page detector bank kind paper output recognition high noisy extent mail expect page label label assignment page natural enough imply label one page individually develop vs experiment collect dataset mail mail page piece mail validation different important reject piece mail uncertain piece mail process human whole threshold goodness multiclass micro b cc observe emphasize large fast boltzmann could investigate achieve also success mail categorization experiment confirm usefulness pooling oppose deep neural architecture acknowledgement derivation simplify derivation assume size necessary sum use softmax go e softmax softmax softmax f recover unit free energy softmax h provide free assume size use mutual use inequality hence c softmax c softmax c softmax general function cm ia sa paris france ia com department computer science edu classification occur problem mail page web site propose generalization restrict machine rbm explore relationship rbm experiment dataset rbms variant incoming mail classification vast machine develop vector process one setting consist find incoming piece mail document represent example simple approach would away mail useful pre computer another sequence relevant dynamic take e informative speech content popular classify multiple belong originally motivated drug disease drug shape label bind implicit recognize appropriate partial observation enough restrict boltzmann multiclass set information present learning extension deal competitive result dataset mail rbm binary joint input target correspond label take belong position
assume discrepancy median log within fit highly lead ignore discrepancy verification uncertainty perturbation control explain discrepancy acceptable inherent uncertainty input conventional sampler abc rejection mechanism adjustment inference may view high combine adjustment marginal distribution posterior dimensional principle abc replace univariate precision particularly summary parameter dependence initial sample poor however accurate approximation extreme increase marginally approximation roughly constitute obviously poorly joint point anonymous linear summary marginal allow method moderate beyond abc likely abc acknowledgement support thank discussion adjustment contribution water reservoir reproduce mixture normal file contain entropy txt contain reproduce file library analysis fan mm linear computation abc complex connect demonstrate regression moment approximate adjust bayes adjustment role exploratory problem low abc first rough abc separately dimensional provide joint illustrate material article regression bayes linear tool analysis think fundamental primitive quantity first article method model write discuss computationally intractable generate smoothing evaluation regression framework transform independent error coefficient weight give estimate estimate interest writing adjust approximately model hold globally extension generalise model feed forward neural perspective regression adjustment adopt adopt regression quality occur reliably obtain complex obvious strategy summary possible quality largely dependent curse argument overhead increase imply date moderate article primary contribution first abc summary adjustment motivated contribution propose regression adjustment approach method obtain abc univariate estimation improve estimate explore literature extend applicability beyond abc practice structure introduce adjustment describe adjustment abc regression adjustment simulation study real present conclude suppose summary statistic construct vector covariance adjust expectation posterior coincide obtain specification full first key interpretation bayes abc full outer side distribution relatively bayes hold globally locally unweighted hold appropriate summary ordinary ss ns abc population fix sequence infinity ss es justify adjust demonstrate adjusted method interpretation regardless globally bayes connection surprising monte prior square present purpose interpretation helpful exploratory adjustment high anonymous shrinkage implement bayes adjustment continue kernel weight first second adjustment replace matrix diagonal possible take square element adjustment outside bayes framework however nonlinear appropriate expansion give certain regression adjustment suggest transformation bayes broadly applicable adjustment fitting choose set statistic construct truncate bayes aspect paper summary sampler smc rejection importance strategy increase dimension force summary cause sampler abc suffer hand often linear difficult validate may preferable low adjustment essence rough approximate abc marginal easier full reduce statistic parameter estimate base margin joint marginal rough estimate order statistic adjustment maintain multivariate original reasonable expect fit sample without exclude marginally informative abc distribution sample space replace quantile precisely write margin sample generate save suggest sample summary statistic something leave work point marginal compatible ordinary approach joint statistic quantile spirit procedure theoretical improvement parametric marginal joint beneficial normal copula mixture normal motivation come determined projection arise joint distribution characteristic adjust adjustment relationship locate negative slope summary statistic line marginal posterior summary estimate use rejection rapidly curse restriction fixed accept within increase simulation return sample curse beyond density estimate poor adjustment sample rejection clearly adjustment improve estimate quality approximation increase albeit sample follow adjustment contour improvement regression component appear density marginal dimensional margin adjust figure improvement location place adjustment transform message estimate figure abc true kullback pd draw compare computational conclusion rejection sampler adjustment rapidly summary regard gain though marginal adjustment marginally lower marginally beyond sampling adjust posterior largely represent uniform able degree improvement correlation lose beyond benefit adjustment use lower accurate abc discrimination incidence consist zero one incidence represent random observe lattice field rs symmetric function elliptical thresholding geometric set element fairly fraction probability expect one west lag respectively approximately probability describe lie sensitive reason prior variance field package summary similarly number pair matter estimate n v n summary adjustment perform r nonlinear method incidence solid denote individually marginal margin component line margin adjustment solid estimate individually estimate accurately verify rejection predictive wrong interesting insight gain figure scatter versus informative statistic order appear change function adjustment help overcome versus incidence method computer aim uncertainty high discrepancy approximate treatment assessment view regard model input modelling subset type input g sampling ignore involved involve assessment due exercise see different aspect computer external function uncertainty abc simulation accord precision handling round abc water water design management water three area store input produce add split fractional area flow store water stream total fix beneficial vary dimensional great input denote run
reduce disk record string generator seed simulation code original string whether accept proposal acceptance calculation involve ratio proposal probability degree freedom care number must stream particular evaluating acceptance discard stream platform corruption computer processor architecture source eliminate code point platform valid original run nontrivial mh spherical particle particle spherical two surface sum potential particle center site intra particle particle center coordinate site file source file material run runtime version execute highlight straightforward degree particle obviously inefficient store update value change degree format transition simultaneously particle format potential generally method code would enable reconstruction without compare storage scheme information sample sequence sample change energy string sample proposal calculation iterate take enable analysis histogram angle intra particle pairwise accumulate string evaluation efficiently change original computation particle bit string record energy evaluation iteration degree htbp operation second record bit string string enable long spend proposal compare ratio fortunately satisfied sample determine degree freedom component implementation minimum achievable consumption execution record sequence reject analysis expensive instance physical heat capacity would additional bit much alone far reduce acceptance string corruption single incorrectly subsequent solution encode error storage consumption low check persistent device would constitute redundant non freedom sample equality hash apply bit string retain accept reject expand technique replica exchange metropolis simulate efficient analysis require system theoretic perspective broad algorithms foundation national medical corresponding author edu prototype carlo transitions state accept reject random gets record practical mh implementation discard disk writing contain acceptance rejection desire also nontrivial impose restriction simulation compatible mh carlo representation kf monte enable importance sample complicated metropolis hasting prototype proposal applicability ratio easily broad usage since scale resource capacity accumulation discard advance mh algorithms experience illustrative simulation equilibrium interact atom
distribution coupling binomial negative binomial motivate beta mark regard process evy mass measure mark beta produce kb binomial set count atom count bernoulli draw like distribution replacement beta bernoulli process atom since count generalization version rather simply select already evy measure express define include show ibp beta bernoulli beta ibp customer walk try previous popularity previous customer additionally customer try place ibp take unlike ibp posterior occur conjugate binomial compute atom produce poisson q finite hence go simplification consider restrictive desirable construct evy non infinite limit evy measure q conclude approach bp k r also stick representation infinite count hide encoding encode sample analysis useful assign latent derive inference exploited compare characteristic eq cf characterize observed section count factor distribution equivalently gamma count gamma naturally beta gamma hierarchical structure formulate hierarchical strength group exploit binomial beta gamma poisson eq x x relationship r k c pr k k newton metropolis hasting mh stepsize rejection gr strictly p r strictly gr strictly unique hierarchical way connect result gamma support require restrictive mention investigate binomial process examine subset table include factorization allocation lda topic generally follow dirichlet distribution characteristic infer infer nmf lda let pa substitute em algorithm non nmf kl kl gamma poisson gap summarize special impose case change ik x ip ik lda variational equation brevity impose prior essentially distinction beta prior framework conditioning ki topic ibp compound prior actually give construction however arise sense ss uci research toolbox restrict vocabulary occur document corpus include journal unique comparison keep remove corpus detail corpus holding hold pi five testing partition discard similar nmf gap inference lda closely compare inference prior prior gap thus prior toolbox setting iteration corpus ghz pc show infer variance factor count draw active binomial close close factor case corpora tb infer factor assign result mcmc dominant topic example dominant characterize effect stop dominant topic keep diverse prominent prominent example capture adjust negative binomial infer factor stop produce readily interpretable remove show dominant easily interpretable infer iteration top corpus increase sign upper automatically fig last test count topic result setting factor automatically infer influence size accuracy generally support specialized factor loading binomial beta poisson modeling count augment evy convenient efficient poisson binomial reveal factorization demonstrate factor whose variance via capture common corpus acknowledgement thm thm proposition multi process gamma gamma poisson approximation beta evy convenient computation augmentation marginalization encourage document count setting medical care specie model corpora poisson binomial univariate repeat measure count extension incorporate variable principal factor analysis discover low incorporate variable tend prohibitive restrictive count
point algorithm scale impractical large moment scale old thank admm estimation make auxiliary develop exact choose soft upon may discard decomposition value parameter plug know slightly efficient run linearly default good consider great arbitrary application unnecessary uncertainty slowly simple fix scale beyond variable ideal concern sparse expect poorly see issue note convergence characteristic date find open use appear efficacy regularize discriminant overall plot lambda covariate diagonal diagonal also group nonzero upper use vary simulation unbiased misclassification l l lda signal snr adaptively third entry misclassification adaptively sparsity account outperform roc unbiased run realization enough perform though lda label choose train one classification peak middle end cv match nearly indicate solution converge concern reach compare lda choose simple maximize st regularize drop minimize bias nonetheless appear motivated estimating sparse efficacy plan admm rewrite affect motivate move finally arrive ease notation unfortunately necessarily property could calculate converge original unfortunately calculate minimize fix admm employ calculate first true initialize iterate q q instead step step update begin must take q c ignoring solve soft eq eigenvalue require reason hundred linear common distribute within pool covariance separate rarely insufficient covariance regularize model adaptively adaptively pool paper propose fit moderate problem efficacy simulate penalize discriminant interaction class observation covariate classify far independently normally mean come classify propose simple discriminant classify observation find high class classify class discriminant analysis mle constrain rarely covariance matrix decrease increase discriminant note pool find tradeoff convex combination generally partial wise equal covariate interact identically group give natural constrain ie eq likelihood unfortunately require relaxation sparsity among sparsity graphical recently joint partial many reader convex later pool discriminant wherein assignment forward bayes simplify term logistic identically nonzero nonzero forward model method sparse interaction recent interaction basic model continuous response entry often simple q show nonzero term estimate differ estimate boundary discriminant lda partition estimate back decision quadratic rather expected decision hybrid number pooling fill idea
simpler stop establish satisfy regularity mu strategy state preserve polynomial assess although mat ern old brownian motion old view characterize sequential resort make fine sound corollary lemma thm axiom thm thm thm thm thm remark thm problem thm principle pt ex france mail fr fr let formally set mu mu consist deterministic arbitrary sequence amount denote class sequentially class bound behave global justify proposition combine bad may whether correspond question view path theoretic random knowledge make quantity interest evaluate widely explore optimization computer assess article brief concern article part assumption deal let assume krige observation projection onto evaluation n mu n solve equation estimation q mat ern assumption equivalent eq domain empty assess quality mu mean mmse mean q screening optimality parametric design mention case adaptive even process view know mmse induction select mmse prove claim case linear criterion bad error ball denote ball adaptive mu n equal nx krige isometry concern mmse strategy gaussian satisfying lipschitz interior condition rate decay criterion prove non
probability direct crucial capture small irrespective probability drug worker utilize relationship member target facilitate chain expect participant disease typically conduct stage survey member attempt biased sampling study typically multiplier estimate population elaborate difference since briefly strict capture service almost decade past little justification interest survey complete survey hide mean attempt exactly present ht paragraph design restrictive adjusted analyze estimator show adjust desirable converge stage various empirical focus mostly survey also additional set regard design consideration contrast discuss way could apply easily circumstance unknown twice individual appear unfortunately positively correlate stage mark sample consequence study stage comprise record associate likely infect take hoc analogous assumption capture individual capture individual rearrange covariate uncorrelated particular know trivial therefore correlate use sample proportional degree good homogeneous heterogeneous stage perform walk result walk person capture thus capture exclude possibility consideration plug jensen equality upper bind simplify give q conclude homogeneous e naive low opposite true relatively yielding figure strategy prove numerator denominator sum like concentrated close expectation find concentrate close prove capture matter capture individual observable concerned remark capture twice thus stage demonstrate heterogeneity capture sampling stage heterogeneity individual capture sample calculated value true decrease naive decrease relatively across range similar irrespective simulate seed follow construct repetition focus repetition population mean bar plot vs combine yield population adjust similar test select population note accord encourage weight naive repetition general bar deviation plot black bar notice survey capture capture take record age discuss highly privacy record record age stage uniformly sample stage repetition group plot age group variability probably e qualitative survey reveal head average age yield relatively behavior attribute describe one might size g variance group head naive project simulate evaluate time complement make university source construct network census project assess adjust evaluate time obtain agreement finding ref bias large adjusted estimator true record conduct group record check white appear rest tag look explanation convenience seed group close size record slight consideration record ref population c c black common application sampling individual proportional concern direct regard implicit application incorrectly report possibly second note population assess deviation generally problematic sample nice capture preference like population weighting yield marked individual correction give age datum idea patient study population reference hand estimating exclude incidence list suggest sec even first stage non random individual privacy record thus suggest adapt use diverse utilize sec chain individual sample manner detail x exp nc stage generality mutual denote sum k easily apart sum cb centre modelling department population school white ac ny cb es study new drive capture size
ed dd ib ib b lb iv ix j z learner tree ensemble full pure full tree ensemble tree training subset attribute forest decrease forest total attribute drawback ensemble member classify agree case sample member sufficient prediction high via evaluation evaluating need voting stop probability evaluation voting classify ensemble initially step randomly remove vote run vote accumulate member criterion safe vote safe member draw safe ensemble member agnostic base vote vote process vote irrelevant disagreement I many way ensemble voting ensemble member chebyshev vote observe binary categorization random vote member algorithm invoke binomial infer around observe unobserved vote agree ensemble fall outside side terminate q critical usual finite correction account model stop simulation describe require upper power computing side far treat treat vote stop show error al stop early inference vote frequency model distribution dirichlet vote proportion update compute yet member current exceed specify class approximation exist class ensemble fit memory memory ensemble fit memory rule classify send read sub evaluate map test receive one test reach decision file copy read base read job vote vote output second every input explore efficacy vote behavior stop threshold method number vote need ensemble threshold practical ensemble threshold compute care avoid numerical avoid table mm factor loop produce complexity threshold take difference ensemble ms ms vote generate vote otherwise vote rule prediction evaluate label accuracy repeat million million report stop increase error compare five ensemble evaluation five approach gaussian tail g population require vote importantly benefit vary value evaluate show ensemble perform web web page compress language categorization predict web page english feature proportion character character use extract feature randomly divide full hour create file gb contain nearly randomly extract block gb us reference count broken species location count record describe environment much effort observer spend scale hundred american base environmental american united complex vary region make task testing count convert create intend appendix detail testing block example gb storage include file implementation nod one intel ghz block speed without accuracy serial train range ensemble train across ensemble member serial range min serial total ensemble version evaluate ensemble specify always vote require uncertain figure train block varied varied likewise gain parameter except ensemble block show bag node construction randomize e set evaluation ensemble datum use full single figure decrease average number figure size provide speed entire speed top row relative less ensemble need ensemble drop less bottom evaluating finally evaluation scale stop vast majority require require machine unchanged parallel way disjoint partition find voting tree partition enough produce boost tree accurate partition learn partition simple vote job selection choose voting aggregation partition et al work empirically create merge comparable classification serial fast distribute bag size benefit easy datum memory failure incorporate ensemble large ensemble particularly important wu train tree ensemble evaluation evaluate boost train disjoint combine fast experiment boost ensemble distribute overhead creates independently create massive tree accurate ensemble ensemble overfitte uniform evaluation train sub appear bag unlike unclear cost incur yield accuracy expect building build benefit node split local controller choose split communication exception construct tree pass construct involve fall collect sufficient controller final building include overhead task job job et al build framework support easy ensemble specialized specify test avoid file system scale gb remove unnecessary ensemble pruning study dynamically time decide evaluation stop differ reliably fix order voting ensemble evaluation pruning base decide meta train reliable different accuracy lead fast pass build block appropriate scale fit compare subsample serial ensemble combine ensemble efficiently evaluation cost option easy requirement prune could remove could network ultimately small accuracy domain future contrast tree pass source version consume direct comparison imagine trade acknowledgment thank david particularly grateful david suggest learn task national laboratory national nuclear security contract perform preprocesse step record cover rare group count type respectively attribute attribute thousand spurious mistake miss attribute token outside handle record categorical multiple tree handle long short build forest ensemble distribute block bag gb time subsample serial propose dynamically member reduce evaluation cost x tree ensemble evaluation integration technology daily life massive volume practically computer memory massive website transaction throughput biological sensor massive process streaming sized subset c datum computer analyse parallel often stream benefit make process large consume processor construct massive volume datum evenly partition construct ensemble form ensemble vote classification complexity resource scheduling use require approach minimize disk overhead short prove novel local employ importance instead usual bagging train ensemble bag forest feature example local ensemble ensemble thousand large point classify use ensemble confident easy implement approach follow divide massive single decision use approach bagging implement fast compute bayesian approach bag ensemble build partition phase job point evaluate large build ensemble variant bag difficult however vote equal weight trivially merge ensemble single easy vast majority
realization transition expression countable j j due assignment arm independently great choose state arm break favor arm center optimal thing probability matrix subtract construction break tie arm number center belief finite repeat none belief exist hold hold hold symmetric illustrate next consider arm center since optimal would arm center belief p reach never arm play infinitely long arm play center never reach finite reality never belief integer hold action optimal exist transition conjecture reason player arm affect arm evolve arbitrarily reward arm claim hold probability reward distribution word arm transition subset facilitate iid problem reward hence bandit assume arm state slot arm say reward k reward arm reward arm k kx correspond constant belief kk kx equivalent assumption iid set arm symmetric symmetric appear subset assumption sufficiently dependent regret player know choose assume start know player set proceed separately admissible compactly represent rhs degree denote arm transition transition extension clear admissible policy individually exploitation c count dependence drop bound play time arm least time take place exploration l chernoff hoeffding estimate transition true constant transition estimate sufficiently close great real q continuous action e transition solution true exist depend player appendix k appendix appendix bind hold l least least away true minimum threshold let constant arm strong hold hold lemma depend know exploration ensure exploitation logarithmic know theorem true transition probability type regret instance free open regret transition probability reward expression probability prove continuity equation system exploration player achieve know step solve finite mdp form estimate state finite transition original mdp except play computed fp sublinear regret fa show fa fp inefficient practically state space comment place pls transition c arm choose term reward similar keep use whenever arm explore player choose high one separately choose parameter unless constant fa policy fa transition know use algorithm average reward show fa perform equally well information state assumption state never fa assumption violate runtime fa comment elaborate please although regret bound perform policy total identical suboptimal decrease make equally fa see prove c almost fa reward arm policy high reward know two arm sufficient optimality horizon discount guarantee scope fa potentially c c c table reward fa propose sublinear learn poor fa fa arm step fa total fa exploration exploitation continue final sublinear use multiply fixed see computation visit fa step guarantee fa online matlab policy exploitation computationally practically policy policy avg run fa constant hold fa choose impact exploitation reward decision step poorly much great exploration step gain exploitation step em c bandit regret horizon player solve often costly practice belief discretization solution player structure greedy policy effort performance hold continuity property w substitute time approximation action begin measurable transition kernel q indicate variation countable correspond chain ergodicity ergodicity uniformly chain perturb kernel c c chernoff hoeffding bind frequently reward range let relate belief player true upper difference product size unit pairwise number set consider induction clearly observe player select arm observe next arm happen estimate u v l induction eq hence radius hold b compact guarantee implie exist give liu paper problem optimal policy bandit play select player maximize horizon arm transition time dynamic policy contrast commonly study respect single action policy achieve mdps partially variation performance online exploitation discrete rule user reward observe current state arm control selection system however arm player exploitation state design optimal lie balance optimal horizon discount stationary policy contraction stationary optimal transition probability knowledge arm arise many sequential channel channel wireless initially track movement design fast minimum horizon reward give statistic model causal system commonly policy static policy arm determine need regret show arm player know logarithmic regret finite near would iid bernoulli prove iid logarithmic knowledge attempt extend decision process mdps process sub parallel aggregation prove regret problem reward arm horizon reward regret logarithmic regret specific see remainder paper countable finite bind strong algorithm increase variant aim assumption respectively extensive prior armed start seminal arm iid policy asymptotically parameterize logarithmic order refer order optimal logarithmic consider index easier optimal et policy parametrize arm parameterize iid bandits states exist achieve regret prove bandit problem upper strong algorithm numerically index iid logarithmic regret process uniformly asymptotically extend iid bandits markovian markovian whereby arm activate whereby arm change markovian version bandit know discount bandit different arm arm drive optimal policy policy always play arm weak greatly arm reward arm observation arm evolve static aforementioned difficulty optimal bandit know criterion design bandit typically compare static always play arm expect reward logarithmic arm extend decentralize merge continuous path liu al sequencing explore exploit block length geometrically version iid problem work design bandit identical arm whereby finite player consider feedback bandit study computationally dynamic outside mdps possible strong policy logarithmic regret estimate reward kullback kl divergence show optimal term instead divergence regret find probability another logarithmic reduce computation mdp reward literature apply regret contrast mdp learn logarithmic regret bandit regret take estimate transition learn play contrast action transition sublinear take pair infinitely index arm prove mutually markovian arm index whose evolve unknown transition arm simplicity without generality true either perfectly play reward perfectly observe receive uniquely space space arm probability arm transition ergodic exist notation let represent unit vector th element denote natural number integer product vector element whose transpose denote arm get select arm reward reward horizon player select state need arm probability reduce exploit acquire high carefully player player optimally unknown vector variable represent state choose step select simplicity assume label player observe action evolution random represent action observation matrix policy policie program player probability player know state system initially take take sublinear regret sublinear convergence rate mention early probability matrix learn approach section problem often belief simplex observe select arm belief arm choose eq reward optimality expect assumption existence markov chain probability space follow true consider function finite bound condition existence solution prove prove integer state therefore hold boundedness belief state adopt belief form arm select write state state observe state note countable subset player exploit continuity player method thus distinction state space statistic calculate subsequently belief subsequently analyze upon completion information possible state player select arm step select matrix knowledge state denote state set probability state information ic iw ft w w arbitrarily u arm ti j average exploitation phase phase player play phase accord transition player exploitation state arm player clearly q non negative player explore player exploit time player keep play player depend state probability exploitation player issue q advantage action gain index player action action
epidemic manner capture must result benefit extensively potential impact evaluation inform decision comprehensive transmission traditionally formulate deterministic equation partial increase computer performance prominent particularly low possibility infect ability epidemic ordinary ode prefer develop type impact discuss incorporate incidence specific study methodology ordinary equation epidemic consequence serious inclusion currently availability incidence pattern behaviour furthermore phenomena transmission clear area efficacy clinical transmission duration duration currently test duration nature acquire relationship contact population demonstrate uncertainty calibration ode epidemic statistically outcome contribution individual uncertainty population monte carlo sampling gain essential perform review review extend statistically challenging relevance interpretation calibration response incorporate tune typically result follow rate stationarity chain example widely hasting markov proposal tuning performance mcmc typically chain stationarity regard theoretically derive act complicated discussion ode epidemic ode ode parameter space provide proposal fact incorporate stage ode therefore construction avoid computationally expensive tuning process hence incorporation mechanism carlo demonstrate improvement automate avoid mechanism approach classified mechanism distinguish generate name combination state sequence depend history markov chain kernel ergodic stationary distribution recent propose ergodicity prove design adaptation satisfy law posterior ensure law large number reader refer detail successfully metropolis formulate transmission treat incorporate whether agree available calibrate model key modelling framework paradigm couple ode automate methodology base incorporate response predictive adaptive assess collect source specific dynamic view distribute stage disease present intend describe move recover move epidemic assumption discuss assumption model people five age group absence people range year survey trial model may grow furthermore activity cause infection serve contribute age number activity age e transition vice versa model population people belong least group group never leave activity level old activity drop active risk group individual age gender choose opposite gender age mix treatment become aware status lose upon state correspond use response conversely general long infection infection support whereby infection lead complicated formulation ordinary capital people activity group denote derivative kronecker delta gender I take since comprise year member move age maintain people term obtain divide individual year reach g evenly add distribution mechanism enter leave propagate age period leave discussion transmission infected subtracting leave go either g role infection linearity system specific behaviour usually subsection construction discuss construction development member infection transmission feature developed age group community certain formation process refer mix find medical literature parameter throughout calibration ode epidemic behaviour survey serve reliable source information number participant survey survey survey likely irrespective experimental design population survey understand result take assume specify model formulation individual gender group opposite gender group age gender age opposite gender challenge take calibration reasonably practitioner worker social worker alternative involve simplify uncertainty specification suitable calibration mixing importantly present transmission use approach degree specify infection equation transmission gender become infect infected person gender easily incorporate several extent reflect formation pattern individual thereby make extent old prefer tendency old gender opposite gender last become whenever gender meet demand individual gender degree activity group uncertain specify matrix unknown parameter low complexity enable scenario epidemic ode addition epidemic specification ode purely conditional trajectory ode conditionally specify differential literature purely epidemic form ode detail forward obtain ode evolution ode gender age activity time component conditional ode forward ode population note fine study subset system observation several solver notice choice solver efficiency depend parametrization parametrization one system repeat generic computationally inefficient write solver intel differential equation library solver distinction explicit implicit degree system solver employ solver explicitly modification stage implicitly fourth jacobian interface remainder model two matlab solvers solver matlab solver ode system point speed considerably minimal obtain discuss option jacobian specify model epidemic extension mix matrix make choice work choice basic describe assign report specification completely set period use twice uninformative specification estimation trial limitation outside report incidence covariance accord study unobserved analytic solution trajectory transform age gender follow detail transformation specify situation population serious use mcmc mcmc learn recursively reject mechanism adapting thereby discover enabling mixing improve adaptation explore dimension epidemic model present detail iteration proposal construct markov simulation ode propose parameter vector ode correspond observation acceptance evaluate methodology internal base variant global proposal involve metropolis covariance learn explore distribution iteration proposal empirical parameter chain provide optimality present recursively detailed forward simulation study omit decide assume life long population simplification would procedure initial simulated system step year calculate generate age specify statistical ode take run forward obtain observation figure present posterior estimate ode epidemic parameter construct quantify statistical incidence prior specification derive mcmc sampling dimensional mmse posterior generate demonstrate work accurately synthetic study estimate trajectory disease state figure step disease observe trajectory within aggregated trajectory true aggregate furthermore versus fit incidence simulate agreement observation gender h ode real maintain duration individual choose well measure performance posterior associate gender incidence consider calibration except appropriate group think life incidence age acquisition rate age high capture incidence though incidence especially similar result develop model calibration calibration posterior take marginal forward result carlo forward calibration forward project ode solver sample state analysis distribution incidence post scheme receive year infection individual compare individual belong present predict response calibration epidemic level incidence develop ode transmission statistical modelling datum model ode model base projection ode static parameter together estimate require perform rejection demonstrate application estimation capability calibration incidence novel perspective studying estimate around equilibrium epidemic describe disease epidemic ode group poorly understand implication view relate epidemic infection attempt interest available aware publish calibrate incidence account confidence real epidemic consideration allow beyond currently available literature combination reason insufficient poor uncertainty interpretation notably well somewhat status disease short year ode appear use slow solution quality calibration concern separate group necessity deal nine prior want question ability accurately amount important address relate assumption illustrate property addition extension methodology trivial incorporate estimation decide maintain parsimonious like methodology mcmc automate additional suffer curse dimensionality metropolis hasting sampler mechanism solution affect change perform burden perform serious practical design facilitate framework approach epidemic ode grid increase epidemic represent place additional space suffer curse dimensionality volume add would ode solver sparse grid computational must explored contrast adaptive procedure computational effort ode epidemic mention straightforwardly bayesian paradigm selection select scheme perhaps goodness utility flexibility framework adaptive methodology appropriate effort statistically consistent weighted plausible calibration space sensitivity ode produce note statistically directly interpret advantage approach dr david discussion motivate conduct mr work conduct early project project ik research linkage project grant national medical program institute department south matrix widely disease define activity rate typically notation gender gender opposite individual belong activity group age brevity equation describe infection term person gender infect gender individual form acquisition call mix construct matrix construction study health survey despite limitation currently representative one acquisition age acquisition risk overall whole individual pool gender rate acquisition rate follow calculate gender individual gender gender calculate gender activity group opposite gender age group mix activity group individual gender activity active member gender become activity product side age suggest group able adjust sort depend need specify q help extent age activity fully age fully age fashion age popularity old reduce age age reduce age group age increase probability age belong age increase old group new population gender acquisition age multiply group balance balance demand want acquisition individual state acquire multipli thing imbalance establish demand gender gender gender e meet demand form h getting infect individual recover recover risk infection
ensemble expert weight forecaster forecast weight prediction weight bad expert expert individual combine draw ensemble expert track well place via introduce implement quite suitable mind however start low none much rather expert statistical actually share forecaster limit depend past grow ensemble add expert step stationarity expert fit keep update course prediction ensemble grows fit successively fit thus expert whole stream expert nearly rule whole nearly expert would fix share provide regret introduce exponentially forecaster grow ensemble modification fix share forecaster result two previous discuss significance sequence deterministic forecaster expert prediction forecaster expert forecaster prediction loss aim forecaster predict expert track cumulative good forecasting strategy expert ideally forecaster convenient sub observation likewise regret expert forecaster randomize forecaster little regret regret reinforcement negative evolutionary time fitness anonymous mention member ensemble actually expert forecaster keep even number forecaster describe next one manner update forecast initially eq control except weight share expert ever fall fraction weight exponential expert provide choose expert expert divide epoch length add expert c expert epoch epoch expert expert train hope cope course stationarity obtain tracking ensemble construction inductive write ti te I move weight te eqs expert within epoch track grow forecaster expert key prove e eq q action assume bound substitute latter familiar length achieve run modify fix forecaster make remark share probably tune well ensemble since term illustrate time great practical united record bank somewhat arbitrarily make year allow evolution growth stationary ensemble weight quite mis economic forecast smooth however autoregressive move fit residual give accumulate grow memory take advantage flexibility parametric non spline calculate good ensemble allow produce profile comparative loss also family weight new whose fixed minimum expert maintain share pre drastically cut dominate ensemble expert attain contiguous bound assume expert time ordinary loss able high model turn literature detect walk model specify seem ensemble capture process parsimonious concept drift machine deal input output unlike assume well nature expert incremental window grow expert handle mild specify able strong ensemble modification share still expert account grow arbitrary technical counting expert loss vary get introduce epoch evolutionary generative low deal character long even come fix system specify trend behavior well adapt automatically non il amazon com goal low expectation powerful guarantee often stationary ability expert expert combine plausible deal historical evolutionary system modify track cope grow fit new become available grow stationarity record gene prominent example problem anomaly sometimes process conditionally stationary historical though historical turning subtract trend analyze trend systematically
compact game heart matter minimax weak characterization either section opponent close study suffice deterministic high dimension ss complete closure manuscript term game term tuple player simply opponent similarly bound determine payoff payoff prevent trivial restrictive discusse satisfy state explicitly define follow exception manuscript iff iff analogously reflect suppose force force force repeat access selection tx deterministic suffice make rich tuple satisfy family opponent note player opponent assume play effectively strategy consider quantification opponent construct force analogously opponent quantification move move say move clarity definition effectively move correspondingly round criterion strong property respectively replace matching game grant every scalar final convention run strategy combine lastly hull notice straight force iff force iff force force property force force satisfy existence minimax suffice force construct b example demonstrate restriction fit becoming value say force subsection difficulty arise close nonempty q equivalent statement via duality z scalar hold convex provide player type existence nash equilibria suitable grant player whereas nonconvex structure allow side introduce major note repeat player projection equivalently hyperplane definition whereby either pass refer tuple force mean force force strategy opponent may force opponent greedy distinction truly early start allow opponent measure impact minimax equivalently force extend global property characterize compact continuous tangent grant set guarantee relaxed quantification strategy parameterize player eq opponent strategy reveal require convenience inductive term vanish otherwise grant inductive grant contain run unclear suppose union another nearly small piece miss way detect quantify benefit may force grant sequence derivation satisfy boundary away correspondingly ball boundedness pass interior player force regardless future produce allow various tolerance define remove strategy tool limit albeit construct opponent nonempty approach fail must infinitely choose opponent strategy follow compact hausdorff exactly nonempty opponent small provide next empty grant iff follow compact nonempty nonempty hausdorff nonempty os j contradiction grant way depicted eventually entirely complexity define player depict determine consist nonconvex theorem primitive opponent build opponent arguably constructive opponent importantly structure every encounter fact grant middle verify condition nearby actually location recall union center make thus take triangle strict treat cause fit contradiction since suppose extra guarantee nonconvex manuscript iff without generality compact contain contain gain minimax implie game statement grant strong minimax b piece throughout convenience map analog deterministic choose additionally strategy family augment member exact history clarity tuple eq result family effectively randomness reveal simply family adapt stochastic concentration independence suffice adapt strategy never care value rather sequence surely apply hoeffding inequality vary define even may across iteration independent apply direction establish opponent exist converse note bilinear compact grant present nonconvex small demonstrate relationship duration game perspective player minimal q minimal set proper consider every grant game grant particular eq play strategy determine inner remove arbitrarily stay piece begin outside opponent provide necessary piece require careful merely come z z consider boundary intersection satisfy second discard scenario take suffice third suffice satisfy b potentially perhaps adjusted length similarity triangle along away satisfy meanwhile mean grant property divide set intersection triangle rearrange desire mean similar triangle q b z every hull union h partial boundary actually pass meaning follow ball radius k exist force grant whereby strong force apply generality segment form adjustment placing force force appear presence optima lie plane lie must exist force force suffice symmetry lie appear plane force every satisfying set shift via shift grant guarantee satisfy nonempty close fall within force grant end inside appear incomplete albeit
group dynamical lead hamming quantile bp ar dynamical see share ibp behavior variation seq model trial e estimate mode sequence trial ar bp ar three top bottom panel display correspond colored place prior prior use contiguous predictive generating series comprise datum take mcmc iteration hold series likelihood conditionally independent bp hmm summarize hmm consistently identify ar mass ar shift positively roughly ar see hdp ar skewed towards low log whereas hmm key ar hdp hdp ar bp library infinitely behavior hdp assume select behavior transition ar subset behavior transition probability examine generate transition behavior ar exactly dynamical mode hmm aspect bp ar dynamical dynamic hdp regardless hdp predictive illustrate benefit switch dynamical successfully visual dynamical gaussian identify behavior amongst task ar benefit specify behavior illustrative examine six exercise combination arm circle raise two reach skeleton plot display trajectory contiguous second reduce box segment behavior color behavior analysis split behavior skeleton modification toolbox position angle select measurement informative behavior capture body angle angle angle angle frame preprocesse step window observation equal use data scale prior observation maintain hmms dynamical system idea herein relate limited easily specifie emphasize condition infinite reduce switch flexibility computationally future split merge proposal improve initialization time help splitting behavior problem upon switch consider splitting behavior occur split issue mix behavior dynamic behavior var problematic grouping address model behavior idea appear rate observation model grouping component portion allow flexibility increasingly important acknowledgment grant grant fa preliminary version detail present conference graph hmm mode bp ar hmm paper base feature ar hmms describe constrain along define running recursion desire likelihood compute sum message ar message hmm specify time birth move propose feature series namely previous contain parameter ratio note definition probability simplify ratio acceptance death move hyperparameter let recall definition write reduce compactly metropolis sampling simplify bp hmm series pt aa ff matrix create count plus appear appropriately index share consider pt plus create transition variant ik ff ik ff ik f calculate unique pt birth set death likelihood propose discard mm remove appropriately variable transition fix aa ff pt distribution update pt compute q working time start count plus increment imply implementation eq ks yy criterion nonparametric approach jointly model related time dynamical beta size carlo base process relying truncate sum hasting acceptance explore dynamical behavior death proposal base synthetic dataset promise beta markov markov switching series bayes generally focus inference daily stock index wish infer regime volatility grow field time financial eeg contiguous epoch complex e autoregressive exhibit among dynamic segment stock return might model volatility eeg pattern dependent type one like behavior among essence would combinatorial form behavior library behavior motivate paper multivariate velocity place go exercise routine series exercise g goal discover type behavior occurrence stream multiple multiple exercise take overlap discover modeling yet behavior assume series transition latent behavior individually model via dynamical system include switch autoregressive switching dynamical system prove useful diverse target track motion capture switch var underlie encode behavior var condition evolve discover collection behavior individually exhibit subset behavior relate series discover amongst series indicate vocabulary prior vector aim allow flexibility behavior encourage similar behavior motivate possible behavior model process indicate implicitly property beta induce encouraging share bernoulli specifically coin behavior series though clearly identical e ibp computationally var process partially review markov switch beta multiple markov switching process carlo truncation finite dynamical system feature reversible birth death proposal sampling rely ibp interpretation beta beta ibp outline section relate benefit propose dataset challenge motion process markovian markov conditionally independent emission modeling conditionally insufficient capture dependency conditionally hmm capture dynamical switching hmms many domain simplify property hmm autoregressive dynamical arise special case series represent realization dynamical phenomenon time switch dynamical mode var behavior behavior convenient shorthand behavior intend series dynamic behavior exhibit imagine represent exercise people expect behavior solely switch behavior time list set exhibit time define discover via interpret time one structure infer behavior share within maintain unbounded behavior feature nonparametric informally think coin bernoulli realization coin beta coin implicitly encourage sharing process realization inherent conjugacy analytic observe e beta process know completely measure sigma correspond real completely poisson completely assume yield collection completely arbitrary sigma improper distribution refer measure draw necessarily completely normalization constraint atom interval guarantee realization fig atom atom cc corresponding cumulative draw realization dot coin associate customer indicate feature provide realization atomic atom example realization show continuous independently sample realization series realization allocate bernoulli realization often summarize infinite ik feature variability employ I f I possible var behavior prior scenario beta behavior bernoulli bernoulli implicitly feature infinitely encourage flexible illustrate collection feature draw draw collection switching var define define dynamic behavior govern particular fact normalize gamma series delta define transition hadamard product define integer assign index amongst indicate precede generative distribution contain hyperparameter place extra component self analogously hyperparameter hdp imply j j jk really abuse infinitely reality examine distribution notation reader value useful likelihood hmms markov henceforth motivating series latent mode order ar hmm ik r var dynamical mode behavior pick feature behavior conjugate wishart cf place share comprised wishart conditionally mean define covariance separately latter library dynamic posterior pooling amongst consider bp result hmm set transition ar spatio comprise dynamic specification summarize eq graphical bp ar eq constrained transition j switching var develop binary observation hasting simplify auxiliary programming efficiently computationally rely predictive feature ibp ibp outline key birth death improve sampler ibp know ibp ibp arrive line case customer customer associate show fig q distribution prior process beta simplicity least separately location atom currently simply imply currently already portion generate follow argument simply choose ibp exclude series simplicity behavior ibp separately k ibp exploit exchangeability ibp binary metropolis hasting proposal mix fast give hasting proposal compute likelihood combine exponentially mode sequence variant ar hmms unique associate choose I unique integral intractable early ibp limiting process infinitely bernoulli trial metropolis proposal however move birth death reversible jump propose either single new feature exist eq proposal encodes create lead transition death reject maintain birth death eliminate associated gamma transition hasting compactly either birth move recall birth death modify jacobian term arise reversible share simplified treat assignment sequence message recursively backward pass detailed transition dirichlet prior conjugate derive transition mode generate transition equivalently solely update wishart comprise wishart normal prior I observation inference presence conjugacy equal prior hyperparameter ff conjugate follow hyperparameter hyperparameter assign prior generative conjugate hasting mean hyperparameter let recall definition eq hasting simplify bp hmm appendix bayesian building regime space recent series na I employ model couple share hdp assume space e demonstrate strict sharing ability discover behavior infer hdp hmm describe coarse
sample scalar impose require impose constant define ft trivially obey state need extreme opposite perfectly result generate trivially obeys important example instrumental follow density integral similarly absolutely dependence randomization permit specify absolutely nature admit either orthonormal j step eigenvector set preserve lebesgue dominate measure kolmogorov countable hold assumption final b j orthonormal r old side multiply iterate expectation continuously f w value lie intermediate orthogonal subset jk j b b j ta ta ta ta ta ta cauchy q enough triangle ta ta ta ta c ta g follow satisfied conclusion g bm differ row conclusion conclusion conclusion g g conclusion proceed condition check continuously zero theorem extension theory operator leave cone banach space borel equip consider point operator adjoint uniqueness example outline hilbert track derive outline uniqueness note remove complex need five cone nonnegative shall existence closure hull consider modulus eigenvalue least closure hull zero like associate simple shall compact solution constant sign vanish uniqueness mean contrary also theorem eigenfunction vanish r eigenvalue positive zero vanish everywhere imply eigen triple obeys multiply side integrate positivity everywhere appendix maintain cone ill pose inverse et relation among give fourth conclusion cone identify identification conclusion cone conclusion identification side find similarly conclusion conclusion jacobian rank local identification role restrict necessary sufficient relate subtract section classical identification moment condition differentiable true parameter model dimensional condition importantly show avoid linear give restriction value identification primitive identification condition model include quantile instrumental asset pricing asset pt important conditional eq functional work motivated restriction identification chen fan paper identification condition nonlinear include quantile restriction economic far identification identification condition local fisher identification pointwise sufficient identification restriction give curvature also primitive addition identification euclidean usefulness illustrate example primitive primitive regressor another identification identify condition consumption capital asset pricing literature condition conditional restriction restriction give identification apply moment instrumental general identification sufficient parametric model primitive condition apply quantile model section index asset pricing example conclude contain help explain brief identification parametric let vector context derivative exist condition state condition parametric local rank equal extend value depend structure banach banach I restriction local identification concept note identification ball strictly open bound calculate sometimes require fr functional empirical strong space analogous condition linear follow familiar identification completeness continuously mild model think regressor instrumental variable weak condition deviation linear moment chen discussion completeness bound completeness fr local open explain identification fr continuous fr neighborhood continuous unfortunately assumption strong nonparametric moment inverse ill pose survey obtain identification locally ill problem infinite consequently nonlinearity identify zero restrict restriction deviation relate nonlinearity q condition include number old continuity derivative continuous derivative condition represent nonlinearity restriction specifie happen include linear fr constant impose give link curvature scalar twice continuously differentiable bound derivative value r l r restrictive identification neighborhood richer restrict necessary real number let twice bound bound h moment norm moment onto even example locally kk k bound satisfied give f locally identify provide necessary condition focus associated locally easy result nonlinear complicated absence assumption involve nonzero result model assumption satisfied restriction impose possible give interpretable hilbert compact mild operator mild compact condition compact eq eigenvalue eigenfunction denote adjoint assumption part operator restriction boundedness smoothness assumption chen chen sense analog right follow select countable orthonormal equal accord marginal lebesgue measurable properly obeys example highlight include impose boundedness positivity restriction use randomization employ economic previously notion argue rich induce iv inspire absolutely need lebesgue marginal absolutely case perfect require density one choice inferential j characterize coefficient necessary fouri old j j ellipsoid nonlinearity allow well environment impose fouri coefficient smoothness derivative chapter sense identification impose impose van generalize hilbert scale chen reference parameter illustrate scalar space integrable conditional pdf pdf x link assumption complete add primitive local primitive identification discrete lee condition rate nonparametric estimator identification possibly decompose component assume banach eq vector residual inner g g also banach inverse theorem semi parametric moment ai chen identification consider lead identification parameter even derivative identification imply neighborhood effect identification g local neighborhood model often suffer curse practice thus dimension q instrumental apply show identification instrumental variable nonparametric model part separately projection vx w jk simultaneous equation still hence identification require completeness conditional instrumental give primitive condition variable hold completeness one necessary one dimensional strong restriction dimensional consumption capital asset pricing provide example lee chen apply identification utility consumption easily specification consumption consumption growth consumption give substitution asset q one external special case chen formation risk asset price consumption singleton economic market substitution solve asset return return impose helpful restriction moment restriction multiplying moment uniqueness differ nonparametric iv restriction equation integral nan operator dimensional kind example need completeness integral second kind arise model inside integral regularity give eq compact direction close specify identification give precise impose regularity condition apply map b primitive suffice let c c ac bc c hc hilbert compact impose completeness version previously note find separable weak identification away necessary small consumption growth amount infinity
research control dms dms award finding recommendation express view identity projection th matrix position zero elsewhere index position elsewhere decision counter subtract side expectation side obtain useful throughout proceed tuple update note depend let expectation final moreover convex turn line intersect schwarz multiply independent first cauchy jensen inequality final jensen obtain simplified recursion involve structure nonnegative sequence complete linearization put steady note sufficiently concave yield satisfy long processor fast set final step hence side suffice target value choice automatically suffice inequality follow proof section conjecture subsection computer sciences department st popular achieve state performance variety several researcher recently propose memory synchronization use without scheme processor share memory modify part scheme magnitude keyword gradient small rapid stochastic descent intensive sgd scalability inherently sequential nature processor web parallelization many large process parallel much recent parallel sgd focus tool develop extract web tolerance simplify ideally suited online numerically intensive ensure indeed google size machine preprocesse necessary statistical consist problem system performance advantage high throughput share processor share physical gb typical read less frequent achievable parallel synchronization amongst scalable overhead overhead call allow memory scheme might processor sgd step modify memory speedup processor occur problem formalize speedup canonical machine collaborative product convergence demonstrate implement one sgd practice sgd recently method significantly slow respective free variety induce hypergraph induce row edge hypergraph simply graph whose throughout natural function learn interest act induce node induce edge example suppose support sparse rewrite arise distance popular resp resp problem involve frequently arise comprehensive survey nonnegative matrix index goal graph similarity arc correspond nonzero list associate cut entity resolution string index author e propose precede formalize define edge determine fraction intersect intersect sparsity regularity feature appear clustered hypergraph one assume provide divide linear relatively discuss parallel setup memory processor processor processor read store share addition atomic perform processor hardware operation single atomic exchange purpose processor require auxiliary processor standard range equal component euclidean matrix onto coordinate I equal zero elsewhere multiply coordinate component know component index processor sample gradient processor leave alone stepsize processor processor decision processor bite careful definition break cycle compute subgradient yield individual asynchronous incremental hypergraph isotropic number counterpart since nearly speedup turn protocol analysis tractable update follow replacement processor uniformly subgradient stepsize completely notation replacement scheme gradient compute one scheme component carry decrease perform modification procedure begin processor respective mechanism analyze replacement tractable analysis replacement replacement step replacement sampling state result parallel scheme follow simplify lipschitz lipschitz modulus minimizer even ordinary main summarize lag integer reduce achieve sgd protocol achieve processor form bind plug bound minimize minimize equal set error protocol still logarithmic initial stepsize robust protocol curvature arbitrarily dimensional convergence factor curvature rate occur round implement agree processor eliminate band message processor reduce processor allow synchronization idea present seminal instance gradient computer master worker setting convergence extend sgd convergence variety delay computation stepsize protocol procedure theory theoretical demonstrate protocol provide propose setting machine average show serial scheme involve via distribute protocol propose implement machine communication require core average scheme et order decision writing speedup lag severe magnitude type c set netflix e epoch core implement fair code rr identical schedule update notable software release rr mechanism need round order magnitude increase discuss ground protocol identical line fine grain induce undesirable slow code run x core gb ram software tb never store disk file disk share read factor run pass epoch large look pair percent describe text training test demonstrate suggest nevertheless figure factor speedup rr worse indeed gradient implementation train average pass parallelization train gradient computation serial run serial ten parallel averaging run large netflix datum row reveal column reveal set note memory reading disk attain speedup take speedup rr completion experiment quickly rr serial serial netflix slow way segmentation cut popular two cut scan voxel associate connect see cut rr twice slow serial entity recognition entity list website entity datum entity string compute individual quite sparse core slow rr speedup computation slow outperform ccc three massive speedup parallelism epoch
et rank element moreover derive order gain great understanding mechanism formulae rank page every area threshold cut double ranking introduce triple estimator unique follow percentile rank side subset maintain rank non uniqueness classifier sd plug conjugate gamma level average replicate loss optimal column scale express indicate p p pt pt g ig ig ig n n ig ig ig ig ig close entry percentage percentage small percentage page report percentage regret particular plug proportion posterior simulation scenario plug plug performance ensemble compound plug compound four cb gr ensemble estimate scenario notable cb exhibit triple goal classifier compound gaussian superiority cb estimator gr trend cb plug outperform triple mle bad although plug rapidly increase poor plug describe observe increase result compound compound plug cb classifier improvement compound plug systematic parameter ensemble classifier plug model equation level replicate first column scale pt pt scenario n n ig g ig g n n ig ig ig ig ig close digit entry percentage close entry percentage point document different overall within percentage point optimal posterior cb classifier fact identical performance simulation page due cb plug preserve account estimator gr plug good cb surprising rank gr triple rank conditional upon optimisation therefore gr expect plug well experimental result substantial plug compound compound model regret severe compound bad increase explain term relationship factor large sampling prior plug note plug substantially systematic plug contrast different simulation ensemble spatial highlight however choice evaluate plug function spatially ensemble complete highlight aspect classification spatially sc compose isolated area isolate sc surface varied heterogeneity modify generating produce true recover risk laplace l fully finally level count modification analysis spatial loss criterion posterior moreover performance plug cb triple ensemble point plug interest weight iii function evaluate consequence posterior loss dependent therefore classifier respect optimal yielded rate laplace sd five plug estimator expect level scenario isolate isolate isolated spatial sc posterior indicate pt sc sc sc sc sc sc sc high sc sc sc sc truncate digit entry close integer percentage spatial simulation plug quasi conditional varie depend scenario cb plug sc classifier classifier behave poorly simulation plug outperform variability heterogeneity produce plug apparent examine posterior level heterogeneity plug appear ensemble percentage area sc easier classify plug consider extent cb benefit dispersion mle extent gr appear benefit plug prescribed observe systematic improvement performance ensemble note car car model spatial category scenario consequence quantile procedure assessment classification sc form evaluate classification chapter entire heavily dependency gr visible gr classifier increase five plug spatial scenario isolate cluster isolate heterogeneity sc spatial hide percentage pt sc sc l sc l sc sc sc sc sc sc sc sc sc sc percentage close decision adopt increase particular weighted introduce conservative decision giving false large area risk medical loss quantile different plug interest result car laplace plug percentage finding report plug consistent scenario overall sc cb medium observe two marginally cb estimator sc cb gr behaviour outperform counterpart heterogeneity fourth section constitute plug estimator counterpart positive addition false page plug term performance point cb classification necessarily find gr plug outperform mle spatial mention find small posterior ensemble risk cluster estimate reformulate theoretic posterior whose already focus chapter plug theoretic indicate constitute candidate routine may aid report public health estimate note interest normally posterior mean converge note weighted penalty give great negative plug experimental shrinkage constitute naturally conservative choice produce positive false negative gr cb produce posterior percentage classifier decision paradigm posterior substantial describe page optimisation adequate order reasonably precise estimation value make thereby candidate plug loss positive converse issue false rate area threshold area equivalent rule positive natural identification justify sensitive health media suboptimal classifier risk classify public health conservative strategy report surveillance finding function page pc est case expand risk arbitrary relationship determine follow directly consideration chapter review main finding thesis extension consider thesis generalise sophisticated serve address thesis thesis adopt formal inferential issue arise firstly heterogeneity secondly pressure report research finding various use ensemble inferential analysis estimator parameter qr contrast quasi plug inherent difficulty associate ultimately objective thesis set nonetheless investigate formulation decision introduce framework inferential objective allow preference one goal another manner thesis introduce non spatial estimator perform par mle hierarchical improve spread parameter estimation lack another specifically difficulty specification loss see symmetry weighting ensemble classification identification exploration chapter propose penalty negative potential classification loss ranking originally author suggest generalised penalty point framework basis classify element ensemble basic principle thesis extend fp equation page family loss produce estimate zero regardless correctness side amount amount proportional proportional permit vary penalty negative moreover adjust directly weight relative although theoretical family would implement practice family respect rank generalise necessarily generalise therefore need behaviour generalise affect processing ensemble interest good lead substantial heterogeneity ensemble already several instance dependence quality ensemble concern lead empirical prior assumption heterogeneity classify thesis three prior conjugate conjugate proper prior non improper prior interest one tend accuracy ensemble often qr inferential encounter ensemble element preferable classification addition natural modelling problem chapter cut simply interest threshold procedure interested se sophisticated adopt category instance use reversible decision ensemble chapter calibrate threshold differently different median may nonetheless specific mixture parametric context ensemble level could aid ideal exercise conduct spatial simulate datum simulate present normal level different average synthetic pt ig ig ig ig ig ig ig statistic equation gamma average set appendix simulated synthetic simulation sf controlling report correspond cancer adjust occur extract cancer pt pt scaling risk variability medium control scenario sc sc simulated model sc sc sc pt sc sc sc sc sc choice scale sf variability medium four spatial scenario sc sc p low sc sc sc medium sc sc sc sc sc sc low sc sc sc sc sc sc sc sc sc sc sc frame line car normal car v car weight alpha scale line car car convolution n u rr car car u alpha v v line label I alpha rr alpha v lemma conjecture remark proposition proposition ensemble application supervision david thesis college constitute modern hierarchical theoretic framework ensemble report important particularly ensemble unit interest estimation may vary inferential since satisfy range investigate set meet inferential thesis produce quantile ratio classification purpose review decision theoretic framework optimisation ensemble cb square triple gr ensemble maximum estimate ensemble posterior latter firstly set plug qr synthetic spatial spatial regret loss choose plug estimator plug quantile qr area weight unweighted false positive negative converse unweighted unweighted framework quasi scenario five also literature estimate constitute plug loss approximately cb gr functions surveillance report uk demonstrate classifier optimal study plug classify risk conclude chapter thesis specification tailor serve inferential goal pt dag direct iid identically york model car autoregressive cb ig gr integrate laplace york maximum odd ratio normal odd ratio pm quantile square qr ratio loss square smoothing square square parent acknowledgement write thesis always present long chance distinguish I great I understand stage thesis dr de grateful particular would like van suggest support uk research acknowledge college like thank addition department university lastly great thank throughout concern report mapping wish cancer cancer area public health compare different may interest indicator task summary statistic constitute complicated variety goal goal estimate optimally area alternatively select histogram true ensemble related point heterogeneity public health covariate naturally exist criterion report ensemble point estimate yield solely estimate certain ensemble within former inferential goal permit interest optimisation function point loss resource become area increase collect public health expansion especially use prior strength permit variability estimate bayesian optimal mean interest readily summarie choice author particular researcher demonstrate mean estimate true unobserved limitation motivate suggest function produce match empirical ensemble ensemble certain part specify element successful inferential objective achieve triple constitute good empirical ensemble goal however appear qr element amount qr good candidate quantify related absence dispersion ensemble frequentist little goal wish thesis interest education lead routine surveillance drawback method make surveillance particularly significance need uk public acquire development reflect journal advance disease surveillance publish international surveillance focus monitoring count classify ensemble classification could assign quantile qr optimally meet inferential goal central thesis inferential behaviour plug evaluate compute posterior specific plug estimator spatial non simulation thesis chapter describe principle specific commonly spatial respective estimator thesis plug chapter dedicate optimisation empirical qr dispersion ensemble classifying surveillance consider possible extension technique thesis emphasis serve inferential goal chapter briefly decision theoretic attention commonly absolute definition provide family type throughout thesis model emphasis parameter ensemble issue relate review brief discussion adopt theoretic ensemble estimator cb triple gr plug consist ensemble thesis particular optimal within inferential dispersion parameter ensemble ii throughout thesis criterion evaluation theory special difference frequentist point issue arise thesis throughout chapter rest thesis restrict introduce thesis assume real decision sense constitute opt among decision utility strength theory fully theory firstly denote state refer act reason generally term decision link define loss state consequence know theory replace order provide several property demonstrate game probably accept also decision thesis especially simply decision identical also line assume moreover consequence one albeit discussion description utility bound precede theory point naturally game attempts modern game select optimal decision follow rhs differ handle frequentist traditionally describe perspective sample basis denote estimator estimator take perspective rule loss population call traditional specify state reflect world specification bayes prior entirely distribution rule equivalent via equivalent bayes risk term action notational decision implicit thesis adopt inference operator respect posterior generally denote except define precede assume improper well case well context refer generalised commonly improper generalise bayes decision bayesian model world frequentist perspective true set constitute discussion decision put nature may specification consider everything else equal decision however facilitate problem function accept throughout science three constitute key development square ii iii review turn especially therefore respect statistical euclidean candidate strictly mean compare make quantify loss bayesian posterior median gray fill gray corner font rectangle draw circle draw circle draw node draw circle circle circle minimum denote parameter ensemble control dependency random variable traditional effect probability directly different hence joint specification hyperparameter specify distribution distribute acyclic dag refer iid nuisance albeit typically sampling simple assume compose function iid modelling relaxed thesis priors identity simple normal compound compound gamma eq gamma proper prior model logarithm risk second level sometimes conjugacy example conjugacy study link conjugacy specific sampling evaluate improper linear count dependence popular prior inter dependence intrinsic car car marginal ensemble therefore generalise detail thesis study population refer affected case divide briefly present conventional assumption model known binomial represent disease population thesis especially interested modelling rare disease rare relative reference count likelihood ensemble refer tend flexibility type specify thesis hierarchical formulate implement software improper flat vector capture define ni area normally area proportional tail specification smoothing area car laplace laplace car laplace identical car refer thesis refer parameter structure ensemble several parameter interest wish ensemble loss form eq use notation adopt ensemble optimum bayes denote sometimes compound compound use entire loss multivariate posterior constitute ensemble property empirical ensemble generally define indicator analogy context iid random effect iid may ambiguity posterior base discuss sometimes ensemble quantify distribution ensemble quadratic interest instance empirical ensemble bayes turn key study although theorem case generalise relaxed distributional retain whereas conjugate compose normal relationship density ensemble control two describe equality hold present lot total variance e total sense former conditioning side equation dispersion posterior mean ensemble commonly encounter shrinkage bayesian towards desirable property little justify modelling limitation especially affect count theoretic produce empirical ensemble concept sequel introduce definition element ensemble define equation ensemble notation function sequel percentile eq quite reference quantile percentile rank practitioner percentile quantity percentile rank quantity quantile inverse cdf monotonic among rare quantile cdf argument discrete variable right convention last infimum variable quantile variable ensemble define parameter abuse spatial structure iid satisfied understand reference mode distribution wish vector respect notation sometimes useful allow accept argument monotonic decrease particular transformation monotonicity integrate function sequel quantile percentile rank introduce different technique derive quantile particular permit address highlight three decision theoretic specifically construction plug space order address hierarchical point est subject mean true eq q function generally refer cb albeit empirical ensemble lagrange multiplier played explicitly transform equation resemble estimator element mean unity index compound cb estimate interpretation ensemble differ specification conventional posterior I constrain similarity formulae mean every equality controls weight take oppose produce point ensemble cb cb estimator gaussian cb suffer limitation influence functional particular cb first two cb ensemble skewed approach attempt limitation ensemble triple constitute cb solely moment successive goal turn result successive gr rank note gr goal good consecutive step equation order obtain ensemble posterior follow secondly rank rank ensemble est rank true rank wish vector compose integer used estimate rank arise successive produce triple estimator triple heavily rely parametric permit case joint conduct use simulated rapid inclusion quadratic loss denote element strictly constrain satisfy square take weight reduce weight rank entire notational convenience ii bayes whose element formulae law every equivalent addition moreover trivially generalization inferential goal adjust shape extreme quantile adjust emphasis parameter symmetric consider highly describe specification middle rank specification thesis ensemble estimator function variety straightforward several inferential thus performance particular use way compare sub difference associate incur formally loss hold concept although expectation specifically point concept setting finance consider plug value plug classical loss denote review conduct parameter take constrain triple goal moreover ensemble useful benchmark compare clarity experimental vary useful achieved generally report distinction sometimes absolute ensemble firstly determination either demand unit high risk surveillance cut interest follow choice heterogeneity chapter treat element ensemble threshold point summary heterogeneity present instance indicate quantification heterogeneity link ensemble chapter heterogeneity qr quantity function loss estimator ensemble obtain cb also measure evaluate plug non spatial simulated gr plug scenario performance noted car tend optimal quantity basis recommendation qr frequentist main effect attention turn ensemble quantify heterogeneity present whenever quantitative statistic constitute useful group nested model n heterogeneity general measure generalise generalised interest response parameter effect dispersion ensemble interest generalise effect include crucial risk modern non function effect unit area exist control unit specific variance several constitute heuristic solution quantify interest model commonly heterogeneity attempt heterogeneity effect conduct study potential dispersion individual model effect constitute dimensional factor control develop disease exact relationship variance level difficult infer consideration address assess amount variability computing ratio choose absolute iid normal variance k moreover choose random suggest aforementioned specification truncate gaussian logarithmic dispersion ensemble meet singular literature adopt disease monitor health setting available available rarely distribution exhaustive evaluation require quantity within function mean compute median absolute randomly fitting become rapidly grow intensive quantify dispersion ensemble ratio parameter measure heterogeneity show statistic datum modelling especially generalise linear regard dispersion qr amenable standard optimisation decision approach quantify interest quantile ensemble computation describe goal thesis optimal ensemble conduct comparison posterior regret chapter assess simulation experimental particular construct use spatial simulation sequel reproduce structured surface evaluate impact plug estimator estimation dispersion ensemble analyse describe important hypothesis status recently family history may area year consistently rate city relationship report suggest set task hand shrinkage use ensemble provide dispersion estimator construction plug first theoretic estimation quantile empirical qr simulation plug conclusion introduce two loss ensemble quantile qr straightforward easily addition discuss evaluate quantity empirical especially relevant dispersion optimal quantity quantify loss function take note posterior expect depend joint quantile q ensemble dimensional expectation posterior plug quantile ensemble solely quantile plug quantile optimal bayes ensemble cb gr quantification dispersion ratio qr quantity first denote ratio likelihood likelihood combine intensity qr relate section qr obtain qr formulate quadratic qr argument clarity refer particular quadratic qr define ensemble qr posterior interest plug compound numerator denominator estimate alternative qr quantity q obtained estimate qr posterior numerator distinction ensemble qr dispersion constrain normally element however context estimation conduct loss mean qr quantity estimate qr latter difference posterior dealing ensemble replace quantity posterior loss optimal qr compare assess lead comparison optimisation qr describe follow take empirical empirical wish qr equation become estimator equation sequel optimal qr qr regret function introduce qr cb gr qr respectively quantile qr evaluate using assess quantile firstly well addition level heterogeneity iii concerned assess quantile qr estimation distributional generate estimate generative identical description test effect qr compound compound skewed conjugacy two case regard conduct prior every bayes give shrinkage every simulation control specification however specification study compound gamma ig specification conjugacy obtain posterior simulation choice detail synthetic model report table simulate observation variance different effect distribution choose interest size simulation choose take experimental factor variability aspect make select ratio sequel formally compound compound gamma level similarly model order shrinkage variance compound gamma respectively choice approximately variance keep comparable compound empirical gamma adequate choice prior result replicate factor aforementione identical model compound burn conjugate compute quantity various plug distribution non simulation page estimator compound compound gamma different replicate set entry scale percentage p pt p scenario ig n g ig n ig ig truncate percentage truncate small point page compound compound posterior percentage posterior percentage correspond denominator formulae posterior remain regret gr plug quantile gr estimate simulation column plug gr cb mle ensemble increase outperform point page impact b increase behaviour explain ensemble recall set choose tend since control hierarchical yield phenomenon visible panel tail distribution explain rapidly increase plug perhaps gr compound gamma model percentage indicate compound exception trend gr plug systematic trend identify ig optimal ensemble study page point remains center thereby tendency distribution ig skewness make affect cb plug plug increase systematically effect term add variability associate directly cb effect term hierarchical shrinkage subject large low sampling variance change overall shape cb thereby lead trend gr gr difference definite conclusion plug increase regret ig finally size ensemble result systematic increase percentage plug early relative estimator empirical ensemble plug quantify ensemble table remain cb gr plug absolute ensemble ensemble scheme qr plug model replicate estimator first scale express p pt pt ig g ig ig n ig g ig ig g ig percentage entry percentage qr table page compound loss replace order plug percentage qr order function exhibit percentage across scenario except n plug worse notice qr size plug explain modification increase ignore estimator bad mle base qr cb plug triple goal match simulate scenario systematic percentage regret identify gr estimator q level heterogeneity mle cb estimator systematically decrease increase cf behave direction whereby yield gr appear level gr qr regret ensemble gr loss model strong mle cb five nine table ensemble estimate compound gamma yield qr systematic identify estimator dependent level triple plug ig plug contrast low percentage compound gamma ensemble bad estimator cb plug goal systematic estimator gr plug performance qr scenario simulation evaluate plug realistic modelling study firstly spatial parameter secondly consider modelling parameter overall cancer west occur expect vary across simulation assess influence plug plug interest spatial main simulation smooth car scenario construct single isolated area risk simulation scenario situation contain relatively spatial structure sc area sc spatial occur overall count keep scenario west uk generate protocol one isolated cluster sc isolate isolated pattern function pattern covariate produce medium sf locate west uk protocol medium sf west uk protocol variability four spatial construct situation isolate five isolated isolated area pattern spatially three level across scenario specific scenario replicate consider isolate area randomly select cluster area sum count entire remain cluster denote whereas area e level stand risk choose thereby create rr datum create situation heterogeneity simulation risk risk firstly area index contiguous area describe first scenario ensure expect count variability vary every define rule except risk sc randomly area adjacent overlap buffer around region exclude ensure adjacent third spatially risk surface specify symmetric denote region convention total particular moderate low twice value specify spatial structure third set overall variance spatial vary surface heterogeneity region create q social intercept throughout produce set generation rr variability simulate multinomial trial count denote scenario variability produce replicate thereby set spatial set posterior parameter ensemble evaluate scaling factor sf combination factor additionally varied q generation consequence scale datum sf example produce simulate true ratio medium sc sc sc sc table page table statistic model use combine effect car car fit car prior stationarity specify flat car normal car hyperparameter specification constitute common choice code report appendix simulate qr estimator quantify estimation estimate respect choose choose evaluate theoretic interest express parameter dependent square qr respect qr qr square qr discrepancy laplace scenario q posterior level variability spatial isolate sc isolate area sc heterogeneity sc surface covariate sc average set express low sc sc sc sc sc sc sc sc sc sc sc sc sc close first digit entry truncate close simulation car denote car posterior plug note symmetric model overall gr plug estimator outperform plug derive ensemble cb well empirical poor estimate outperform cb scenario sc simulation heterogeneity sc lines sc sc different one sc spatially risk scenario page typical discrete nature sc ensemble behaved shrinkage function moderate whereas cb gr result ensemble different shrinkage maintain typical property change dispersion disadvantage quantile ensemble sc sc property true superiority level variability substantial plug percentage scenario spatial cb estimator heterogeneity ensemble mle contrast appear performance although trend mainly restrict sc effect decrease percentage plug systematic trend mle plug positively increase rr ensemble car laplace mainly affect cb estimator performance car sc simulation trend verify cb estimator produce bad regret gr plug marginally laplace column page indicate inferior normal qr qr plug column variability four scenario isolate isolate isolated sc highly structure heterogeneity sc surface generate covariate replicate scale posterior p p pt posterior sc sc sc sc sc sc sc sc sc sc l sc entry percentage close percentage qr plug level experimental table page precede dispersion sc variability property advantageous variability converse cb three plug percentage sc increase triple find outperform ensemble yield car result loss consider qr percentage cb plug summary behave similarly qr percentage simulation htbp q five posterior report spatial isolate sc isolate isolated sc highly spatial heterogeneity surface average replicate factor posterior percentage loss optimal pt p pt scenario posterior pt sc sc sc sc l sc sc l sc sc sc sc l l sc sc sc sc sc sc sc sc l sc sc percentage close integer htbp qr six loss first spatial isolated sc isolate cluster isolate heterogeneity sc surface covariate sc replicate scale factor posterior optimal pt pt scenario sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc l sc sc sc sc sc sc l sc sc sc sc sc sc close percentage expect page qr percentage table qr small result optimal cb plug comparison scale percentage gr plug qr function sf large counter comparative loss estimator although gr plug correspond posterior loss qr describe low induce high base episode design investigate differential episode ever episode conduct uk date analyse solely centre individual area contact service probable broad country incorporate wide variety address pass scan history schedule schedule record international disease broadly detail provide count deviation area count car laplace specification five datum discard burn computation plug estimator previously simulation plug estimator spirit direction thus separately qr qr five ensemble point qr similar fashion qr empirical qr equation estimator pt pt pt pt qr laplace qr optimal qr column p pt p pt car car laplace qr car normal car laplace truncate close digit percentage page family histogram ensemble car empirical distribution ensemble distribution estimator sensitive hierarchical cb gr balance ensemble cb gr behave distribution gr modify behaviour estimate observe panel car car also page car summary page empirical estimate modelling plug quantile whereas mle plug quantile opposite relative hierarchical quantile posterior counterpart plug quantile cb gr function difficult cb appear empirical qr plug likely qr overall ordering plug estimator qr follow order report non spatial simulation exception parallel cb empirical qr qr modelling albeit posterior qr slightly car importantly interval quantile qr car laplace thereby suggest quantity car page report posterior qr quantile order plug estimator study plug estimator yield large posterior triple empirical quantile quasi cb plug outperform triple qr goal cb term however large ensemble set region affect plug remain qr appear agreement conclusion particular counterpart analysis heterogeneity qr yield approximately upper risk exhibit heterogeneity area variability finding good quantile qr triple gr plug behave throughout spatial
permit derive production rule structural show syntactic arc syntactic branching rate language present syntactic syntactic generalization model give syntactic filtering band numerical study syntactic experimental syntactic trajectory pattern use conventional processing sentence structure dependency make hidden stochastic naturally language major tool biology dna sequencing protein hmm insufficient capturing spatial effort enhance incorporate state attribute target span engine utilize target trajectory velocity cross attribute track track feature also find complex recursive example plan infer generate computationally intensive surveillance track sequence directly level inference drop person person lot target turn infer measurement track coupling track loose temporal inference single perform surveillance sense identification via ground move array processing filter conventional wiener form single hand wiener filter form component receive move dimensional although use conjunction base motivate syntactic model syntactic tracking algorithm present overview tracking language behaviour syntactic syntactic pattern express decompose descriptor trajectory primitive pattern line primitive target syntactic support syntactic syntactic aim ground surveillance move classify trajectory pattern motivate syntactic tracking syntactic approach security gate turn around circle around build move track recognize spatial identify pattern syntactic tracking example high description motion surveillance characterize certain formation level inference geometric interest pattern filtering enable characterization identification trajectory stream approach interact recursively compute q exponentially merge instance hypothesis mode merge merge syntactic filtering keep apply prune specifically syntactic filtering term probability keep bayesian probability likely trajectory track syntactic pattern computation probability discuss sec perform syntactic system framework syntactic five processor move target optimizer assign measurement track keep track target compute target sensor pattern knowledge base production pattern geometric pattern enhance characterize transition refer characterize geometric remark already exist association probabilistic evaluate track association tracking assignment solve optimization field syntactic modular well suit data move stop target complete detail syntactic trajectory provide discusse estimate syntactic geometric pattern prove specific parse classify sec motivation outline spatial description syntactic formal theory finite set production leave paper letter letter string markov termination rule specified terminal generate denote production form specify denote exclude case length string include indicate terminal side production rule independent contrast free embed embed represent chain show compactly generation segment production use symbol symbol string derivation production rule eq b illustrate possible acceleration depict mode model ground velocity transpose uncertainty direction indicate orthogonal noise switch remark ground target compare acceleration acceleration semi jump ground exhibit model equal direction mode assume different mode ground target move particular observation matrix element range rate angle measurement platform motion sensor time mode sophisticated pattern clarity focus line line arc generate regular target move arc associate syntactic processing save arc align axis trivial trajectory extend etc chinese character language include string string derivation hide arc compactly matching mode stre number forward mode basic arc parse inclusion cause parse case production rule arc regular self need eventually move language rectangle string rectangle side comprise turn side trajectory coincide reason comprise second view recognize target robust rectangle end coincide begin may language operation arc opposite continuous syntactic arc comparable identify location together within moreover event syntactic tracking trajectory trajectory identify open detect move toward ready formulate syntactic filtering mode filter model previous starting explain generation language encode rule line respectively generate arc point fig counter turn length production arc capture note illustrative exhaustive application development production rule practice production assignment keep next provide language use language generate sized sized arc course regular arc amongst trajectory algorithm incorporation syntactic semantic summarize process current scan geometric input sequence keep likelihood mode evolution production rule syntactic filtering track sequence iteratively build geometric estimator compute implement particle nonlinearity need describe mode feed continuous mode denote transpose approximated associated mode resample avoid degeneracy step involve generate base matrix involve transition calculate importance mode yet normalize equation use becomes normalize generate resample replacement necessary effective compute eq resample degeneracy occur certain one kalman process q noise convert convert measurement sec terminal parse store mark enforce consistency two interact mix match probability combination ready syntactic processing need integrate tracking describe system framework illustrated assume datum association module syntactic extend keep parse robust data module track syntactic unlikely track computational largely specific syntactic measurement extension later parse introduce production track scan scan let spatial correlation many spatial correlation experimentally production move production generation add modify include map module parse give intuition parse short string bb production list production parsing extract state entry table applicable string state place build detail produce previous operation state contain operation search marker marker symbol operation predict production rule please state look index marker terminal terminal input string index position index marker state please entry terminal lastly index marker state advanced position please completion completed completed discuss associate line applicable explain string empty add production predict closure leave corner compute significance reader state discard forward threshold value track form detection capture start instant operator string state generate filter update update parse operator advance marker state string marker end rule state predict generate state derive correspond match terminal explain marker accordingly associate accord closure unit production compute cyclic complete threshold trade track completeness parse incorporate knowledge human operator logic add probable add whose production yield symbol compatible string word purely parse incorporate parse parse collect experiment setup pre summarize numerical finally sec level syntactic substantial possible collect band mode wide imaging collection mode surveillance feed e carry enable collect surveillance unit near centre embed system centre undesirable deviation ideal hz angular increment yield velocity gps internal kalman result position velocity external kalman give stability phase correction trajectory result coherent collect scene move knot position discuss ground move trajectory geometric ground point radial velocity could continuously move point angle angle length frequency coherent duration acquisition second fairly snr velocity move consider feed consecutive tracking input target need similarly require several eliminate eliminate although tracking range range respectively use component sensor platform provide give range rate angle local coordinate plane coordinate order tracking plane centre syntactic illustrate deal arc numerical filter kalman similar extend kalman show track illustrate fig trial real track dot line extend perform quite turn constraint impose mode noise state kalman syntactic parsing mode bottom panel four mode easy display syntactic parsing mode soft soft hard hard parse soft parsing focus parse arc pattern parse trajectory arc two arc detection track parse sec arcs arc identify arc irrespective orientation illustrate rectangle panel maintain high support segment terminal string drop drop certain terminal syntactic syntactic could completely could greatly parse signal processing geometric syntactic fed syntactic arc useful algorithm syntactic sec fed mode weight mode output instant offer mix equally mode geometric move dotted line correspond time sharp turn turn covariance syntactic tracking goal syntactic filtering making behaviour syntactic signal geometric trajectory word syntactic geometric pattern form trajectory provide target parse trajectory implement parse parse bayesian tracking track extract track anomalous spatial trajectory model move move stochastic context free adaptive parse tracking context
solely university ny depth arbitrary without skeleton configuration correctly pose degree freedom depth dimensional pose advance image convenient challenge provide training estimating pose arbitrary skeleton pose estimation evolutionary algorithm rather belief train solely explain vast skeleton survey two method particularly successful human tree approximate pose core hour million randomized pose domain explicit additional comparison motion pose complex model underlie structure representation object tracking estimation optimization skeleton model observe angle branch link parent robot camera depth sensor drastically subject capture robot link second link degree arrange robot distinct pose collect range angle pose set eight subject pre background run evaluation approximately core intel processor minute evolutionary successful pose place qualitatively score survey hill baseline superior sharp evolutionary hill optima drastically
parent option depict grouping coefficient explore htp new represent pattern commonly adapt advantage pattern fast compressed improvement encourage coefficient sparsity parent persistence regularization child group denote group group application disjoint couple lasso offer deal deal introduce coefficient group certain overlap propose analyze subsequently column replicate shrinkage overlap lasso treat coefficient parent group persistence transform motivate cause scale modify master copy copy variable force agree encourage strong persistence scale previously overlap lasso quadratic penalty apply henceforth lasso lasso fig organization right invariant preserve encourage child persistence organize penalty child lose randomization ratio less group ratio small indicate strongly reconstruction htp evaluate real equation haar wavelet sparsity induce transform explain htp illustrate deconvolution fig compressed variance corrupt htp improvement penalty noise scheme variance varied error toy fig average trial toy trial employ applicable reconstruct piecewise length jump child measurement signal implicit oppose conventional see propose coefficient natural coordinate wise penalty traditional group overlap address new penalty demonstrate penalty compare partially support university statistical deconvolution sense reconstruction work issue greedy suboptimal reconstruction propose modeling group solve perform deconvolution coefficient deconvolution sense commonly tree graphical superior situation inverse deconvolution sense linearly mix dependency exact algorithm past issue greedy iterative pursuit modeling group penalty exactly compressed efficient standard approach lasso modify dependencie nature structure widely use g structured admit pruning pass operator strategy general unlike motivate main pattern sparsity wavelet capture model persistence large small
rule set existence uniqueness bregman entropy projection geodesic affine arise third taylor relative accept unique make appeal derivation cox might derivation accept relate mathematical update operational description specify choice decision consistency person principle choose arbitrary inductive inference opinion inference oppose construction accept member give range decision information crucial observation see always frame specification outcome allow experiment variability fact determine consideration inferential experimental operational relationship entity theoretic scientific fact relative decision individually frame decision fact instance completely consideration outcome configuration range allow outcome theoretical provide inductive inference outcome historical development separate operational define operational inference justification rule give operational language justification choice constrain assumption express operational language operational description latter operational language provide operational experimental inductive theory mean shared decision construct experiment verification prediction particular give model setup type inductive inference simplify operational definition fix operational use theoretical operational experiment correspond theoretical plausible outcome interpretation passive static define relational relationship construction use experimental setup logical neither experimental jeffreys capable solution neither objective justify evidence procedure behaviour relationship operational construction experimental well theoretical operational relative user agree agreement meta theoretical character describe rule inductive experimental arbitrary scientific objective remove criterion coherence experimental establish link inductive propose inferential insight kolmogorov approach replace measure measure statistical directly define specify deviation affine prior eq algebra quantitative integral represent triple specify give triple lack existence say initial prior general consideration distance operational conceptual meaning require accord information description provide operational construction experimental give description correspondence impossible inductive operational verify instance mutually abstraction twice stream abstract structure criterion category experiment type give category abstract subject consideration active configuration input outcome association outcome course instance experiment experimental map treatment act configuration outcome decompose part part provide allow deal category experimental observation projection layer equip triple hypothesis assign equip geometry preserve hypothesis respective inspired interpret passive active passive layer call respect context rule iff replace absolute evaluation correspondence analogy object entity geometry consist integral understand quantitative describe setup type description determine inference relationship inference mm section probability aim mathematical conceptual logical reasoning sentence provide mathematical difference inductive form logical procedure specify base inference inductive depend reason inductive quantity choice statistical element everything single leave arithmetic around frequentist probability outcome limit influential year sound separation formalism inductive theory statistic frequentist interpretation consideration subject change change moreover inference frequentist approach principle justify convention possess mathematically justification frequentist beyond successful theory theory quantitative used need choice particular drawing require justification requirement conceptually relate experimental justified amount justify nature reality theoretical construct syntactic amount calculus formal sound calculus make syntactic irrelevant early jeffreys cox e inference attempt provide observe le replace description term integral function boolean normalise integral expectation characteristic hand finitely additive equip dynamical bayes continuous domain concrete provide inference notice bayes laplace precisely validity necessary frequentist model lagrange multiplier careful look note cox type derivation bayes equivalently
break degeneracy posterior observable central prediction inclusion dot uncertainty helpful break degeneracy conclusion analyse situation rapidly curse dimensionality markov carlo represent respectively universal choose wrong model parameter wrong take sign strategy refer permit detector plus subsequent counting one construct prior except hard adapt improve analysis accumulate realistic assume reference compute however long realistic generate multi rest prior mathematically propertie difficult begin simple construction together function proper physics study serve subsequent density index build reference yield credible frequentist robustness inference weight multi exponent permit reference flat physics model hyper surface three illustrative question mcmc seem little tune markov chain suit severe accumulate remain bayesian discussion grant de interpretation term parameter bayesian prior construction bayesian reference likelihood physics reference induce physics model posterior density continue indistinguishable model start difficult practical physics depend multiple landscape possibility another construction model hundred thousand million shall difficult task accomplish another physics become applicable multi three availability increasingly computer become routine technique monte development show qualitatively use base benchmark study frequentist method frequentist construct confidence project parameter flat albeit construct parameter profile likelihood fit profile fact conceptually detail extract credible region datum accord uncertainty irrespective systematic good guess etc conceptually coherent unified construct investigation difficult circumstance parameter place physics model choice prior therefore purpose paper prior difficulty conclusion draw new conclusion extract difficulty obvious prior lead successfully recent top production cdf namely multi intuition bayesian approach view extremely powerful put mathematically place method call use invariance excellent latter bayesian credible moreover perturb control way check robustness conclusion significance analysis prior publication analysis parameter propose rapidly prohibitive parameter solution begin proceed describe detail step compute interest prior marginal posterior study clearly step apply single count yield simple possible calculation prior parameter detail count multi prior sec three parameter universal generalization conclude describe count pass expect give purely additive event physic expect value letter encode function density pdf discrete expect signal lie interval result count remain cut however physic devise event rather background hypothesis readily generalize count count experiment event make q mean factorize two way ref likelihood marginalization permit avoid technical ref evidence expect expect signal derivation arrive next section nothing expect signal flat act ill equip parameterization flat new experiment single interest prior maximize intuition construction great separation quantify separation kl invariant zero identical gain count wish influence prior depend enter density twice satisfactory integration practice frequentist approach un thing however inference sense perform unknown completeness key algorithm appendix simplify considerably coincide jeffreys prior model adapt ref coefficient reference eqs reference integrate detail assess physics traditionally propose reference composite signal hypothesis favor alternative background thing normalization pn ds incur background outcome experiment significant decision reject thereby accept alternative physics choice kullback leibler divergence hypothesis specify ratio counting characterize search take bayesian analog calibration independent count count signal significance generalize fact product task map predict predictor determine plausible specific class would every indistinguishable look constant hyper arguably however information surface simple something challenge functional practice simulate cut f term sect application bayesian model important experimentally describe suppose wish rank candidate observation model distribution aspect need specify chance however reach possible rank model necessary use prior integrate improper latter prior proper construction reference publication prior physics dimensionality provide simplicity illustrate consider free fix lm observe grid lm
integral fy dy continuous function equip order corresponding eigenfunction function consequence representative furthermore verify yy consequently product linear rkh operator pointwise comment scope involve pointwise assume rkh co norm f precede definition example recall orthonormal underlie hilbert operator act generalize k hilbert domain scale give variation obtain I x case uniformly spaced approximate development various previously square matrix involve principal equivalently operator case strictly sharp minimax meaning hold take infimum bound infimum hilbert iv chapter meet illustration truncation bound theorem tight show fouri truncation geometric interpretation tb l f generalize operator general although sharp semidefinite eigenvalue trace q relatively partition sharp semidefinite c leave consequence different hilbert illustrate subsection reproduce rkh operator assume eigenfunction k condition decay decay exponential decay r corollary concrete meet sufficiently constant sufficiently correspond express uniform operator let e lebesgue explicitly eigen integral q write easily verify n example share involve periodic eigenfunction away introduce periodic row column periodic vanish row k q constant class find particular imply c eq display type periodic absolutely convergent uniform define simple explicit practically function integer elsewhere identity obtain principal index row column periodic suppose example decay polynomially apply combination kernel decay hence obtain correspondence diagonal eq define bilinear whose argument space decompose similarly term partition partitioning argument yield assumption control inequality duality put piece together note notation show note rhs problem detail particular change proof geometry tb convex hausdorff toeplitz range projection bind piecewise display bound norm ball restriction act norm particular estimate norm certain infinite eq eigenvalue depend sampling fourier hilbert fourier case sample eigen decay n technique sharp acknowledgement aa support grant devote reproduce base auxiliary begin state eq claim j x symmetric adjoint bind consistent look whole principal submatrix k let principal previous notation name block similarly denote event event k boundedness k apply c summarize show extend lemma pc sampling pi satisfy th basis note boundedness n eq bounds union q impose exponent furthermore eq hold sum furthermore bound recalling use converge partition index submatrix submatrix combination perturbation write fix argument note double sufficiently double get part quantity simplify look slightly version r convert low bound properly invertible hold contradict b constant noting right translate subproblem discuss explicit formula generality solve picture arise depend whether fig case plot equation claim generalize fourier truncation u ts select v appendix derive inequality use consider semidefinite partition q q matrix observe condition differently find another sufficient corollary example remark norm acting surrogate study subspace include usual empirical encounter rkh implication pack quadratic space function l metric regular borel continuously within design
distinguish time target time property track joint filter particle filter give position candidate instant decision neighbor sophisticated approach important disadvantage determine drawback motivated conceptually heuristic usually limit scenario difficulty presence trajectory significantly decision available exclude cause batch avoid track deal come even furthermore require evenly process gps apply gps component role gp responsible I force gp approach inspire association particular multiple perhaps object construct gp associate target component ambiguity target reflect simple model scope localization mixture gps propose similarity tackle trajectory demanding oppose trajectory standard variational standard tight organize brief review gps introduce discuss hyperparameter year gps attract attention nice art performance brief summary input predictive gp observation model noiseless plus independent proceeding power k yy property function unnormalize one fast correlation output decay input grow unknown function nn conditioning predictive computable time inversion typically evidence analytical available carry take pc set correlation expect covariance overlap mixture latent trajectory association latent determine trajectory entry model trajectory trajectory handle trajectory matrix place prior q e multinomial gp different trajectory multinomial general form impose hold clarity omit conditioning assume moment analytical intractable resort approximate hyperparameter jensen approximation search distribution maximize analogously analytically latent nm increasingly therefore compute term divergence affect trajectory computation gps mn inversion assume however practice select change slow solution maximize tight improved bind propose low divergence kl correct arise jensen constitute correct analytically depend possible decay amount posterior process equation implement instead numerically though orient towards use datum grow new arrive start effort use update latent fully markovian property hold update instance slide rank option behavior association regression show implementation ghz gb yielding time experiment perform association toy circular available source observe source trajectory circle center source make difficult successfully identify illustrate specifically decrease whenever source come air tracking scenario observe interval unity detail refer pose tracking estimating scenario pass one source instant sir filter sir joint particle filter combine technique instantaneous operate sir initial vector regard ht trajectory sir initially perform difficulty source come one trajectory mistake mainly prove insufficient multiple solution perform entire trajectory rmse trajectory assign wrong trajectory observation version sir furthermore sir initial state knowledge l rmse rmse sir batch interestingly find multi communication third experiment wireless interference ia wireless network idea ia spatial interference align receiver interference filter possible smooth frequency implement datum association setup illustrate fig datum distinguish smoothly noisy matter fact implement ia set project application multimodal come source htb observation three gp fail multimodal previously mixture restrict gps depict b interpret along one dimension horizontal snapshot noise measurement snapshot gp capture gps ard covariance noisy identify mean finally correspond mechanism produce observation gp track previous mixture rely
inference conditional investigate surface location variation much remain main difference occur location height height height height height height location height height height height height height height height height height height height result simulation setup paper exception fix knot figure compare residual surface knot knot surface knot model knot describe generate mean dataset knot knot use knot loss knot model knot improvement knot covariate fix knot equation theorem axiom conjecture remark division computer science link se mail explore sensitivity inference variation aspect prior shrinkage influential prior flexible display density variance row large knot partly three place prior mean
strictly point lemma invoke return rx e suffice turn n j rp e form neighborhood around fact complexity noiseless uniformly dimensional possibility calculation assume similarly total variation contain contain radius upper bind similar noise region give volume calculate respective notice identical reproduce particular suffice eq claim derive denominator numerator nb depend increase small outside keep keep apply conclude choose select condition choose return nn particular large procedure correct homology least deconvolution bernstein hoeffde crucial clean remove point region notation analyze indicator mean case case eq remove bernstein case bernstein inequality region remove bernstein put piece procedure least x measure deconvolution provide completeness let z px qx qx qx px qx remark lie understand geometry homology group manifold topological provide algebraic contain connect estimate homology manifold noisy bound risk bound ball appropriate radius establish le dimensional manifold homology group homology manifold connect cluster order homology topological extract beyond topological homology group application homology rapidly homology application medical imaging shape analyse microarray book study range homology homology appropriately bound homology show condition homology manifold perturbation wasserstein study homology geometric prove theorem homology homology induce literature focus aspect manifold homology reference therein upper bound mainly generalize henceforth establish sample dense thin region size homology high variety clean essentially remove sample thin sample fall deconvolution concentrate around sample procedure deconvolution reference use deconvolution cluster tree support knowledge homology exist previous result outline describe brief homology ccccc noiseless paper unknown dimensional manifold whose homology seek depend describe manifold clarity positive constant different specify riemannian volume manifold ambient paper regularity impose condition open bundle self collection manifold volume clarity treat constant low match noise noise clutter observe distribution clutter noiseless case noise q qp dimensional whose add restriction manifold comprise manifold use along consider vanish get let difficulty sample small permit call use phrase homology ball usage discuss briefly see detailed computation homology group topological might imagine union geometric underlie infinite computation tractable computation homology particular see discrete classic homology identical homology complex describe homology e triple triangle sum integer ease form denote simplex boundary simplex boundary chain p group z z ht boundary collection chain cycle cycle chain th homology homology cycle homology correspond next homology cycle loop homology equivalence homology collection homology cycle cycle boundary chain triangle fix radius ball since field compute homology polynomial size reduction variation supremum show respect make le le iid n q infimum le member close two different manifold homology manifold establish task underlying manifold lemma describe apart smoothly end red manifold pair apart radius inner radius smoothly end manifold manifold correspond identifiable recover homology impossible establish bind case thin region ball carefully around sample far away manifold threshold region include edge concentrate manifold deconvolution densely construct ball homology high review statistical fan modify account deconvolution kernel noiseless take dirac generally specify construct union ball around appropriate homology probability homology one matrix homology compute homology approximately homology quantity minimax similarly write constant analysis typically upper factor correspondingly factor despite two manifold le result manifold get carefully upper specific estimator homology rate provide upper bound carefully choose manifold specify manifold describe make low calculation show low sketch noiseless densely concentrated manifold union risk straightforward result reproduce inference choose c appendix clutter sketch lower via construction calculation variation le sketch preliminary clean far away close analysis show probability ball appropriate radius give homology outline invoke clean careful region show retain thin b rx thin dense cover homology formally dense upper minimax resolution appendix noise get width around highlight phenomenon dimension concentrate around care reconstruct homology rate identifiable depend tight address dimension identifiable consider manifold place manifold center manifold clear distinguished old however important manifold differ appendix still recover identical bind noiseless sketch suffice involve two sketch ball compare ball inside mass concentrate around manifold large disadvantage homology around right homology balance consideration homology cover n homology bind minimax resolution argument case derive additive somewhat separate mixture collection distance clutter clean ball consider manifold deconvolution draw clean around show homology satisfy density j transform deconvolution require divide transform bound behave noise say broad deconvolution satisfy take fix bind aspect manifold accurately reconstruct homology use thus draw measure appendix ball radius remove ball radius
edge mesh onto solution eq q equality property whenever complement inner property noise integral assumption functional basis compute piecewise function basis computation appear structure work hard relate problematic good number mesh require sum piecewise mesh row basis consist basis zero everywhere tp I solid kernel dash approximation principle function job careful depend check sensible ern respect space get effective range knot obtain furthermore computation integral figure non stationary sphere still explicit element compute extra ability stationary continuously markovian assimilation trick equation integral integral sum functional quadrature infer primarily replace progress overlap mat ern markovian mat ern model smooth integer markov essentially powerful non mat ern material surface field demonstrate property greatly boost modelling capability ensure combination modelling markovian random continue produce useful markov field frequently unfortunately traditionally traditional property markovian practical markovian gaussian offer parsimonious interpretable practical viewpoint statistic computationally speak spatial advantage linearly rather success series markov spatial field connect jointly equivalent iff problem usually concern spatially domain fundamentally reason field paper space markov use markovian barrier conceptual non possible construct sphere show difference infer cox observation infer non convex multiply sphere flexibility construct field paper review deterministic approximation insight theory markovian field review take detailed gaussian begin lead differential markovian gaussian insight discuss gaussian field possess spatial facilitate quite explore markovian investigation one quantity instead significant quick computing special amount dimension key cholesky efficiently book cholesky cholesky sample dense occur instance however conditional usually significantly dense conditioning technique whereby unconditional conditioning krige cholesky krige inefficient sparse conditioning datum subsection construct note krige update solve auxiliary definite reveal krige augment system tucker constrain unconditional unconditional measure method cholesky mind iterative solve system work need property conditioning discussion tie especially play set bayesian jointly joint implicitly equation eq sparse structure availability inference come section likelihood field group mcmc sampling posterior easy exercise marginal give take compute gaussian perform integration moderate inference show construct scheme integrate laplace successfully spatial art user interface aim statistic infer field gaussian gaussian I non dense like transfer computational outline gaussian random barrier classical tie discrete discuss broken barrier spatially markov simplify directional nature distinction future allow less exactly generalised obvious informally field markov set separate independent spatial obvious spatial property ignore markovian gaussian gaussian show markovian stationary markovian elegant tool correspondence differential operator variate k lf infinity covariance call operator q l follow previous isotropic stationary markovian operator symmetric dt univariate critical markovian sense value neighbourhood depend everywhere covariance locality field previous form markovian random field set greatly noise field proposition stationary integer markovian random field somewhat surprisingly back part show mat ern mat isotropic mat markovian ht consider sensible approximate necessarily class approximate reasonably demonstrate deterministic linear mesh mesh mesh functions grey pyramid figure piecewise jointly gaussian approximate markovian mat ern ern field stationary let right hand side integral green formula derivative vanish boundary sensible
asymptotic construct plug preliminary allow projection regularity condition ensure convergence estimator subsection shall normality efficiency notation integrable cube decompose dx dx triplet belong resp denote convergence choose subset index sequence partial projection three assumption subset find constant ellipsoid necessary condition control square queue grow decay density basis assume great regularity property function function derivative exist assertion regular conditional two allow infinity asymptotic respect control normality entail theorem er er rao bound consider density estimator family neighborhood rf efficiency define summarize estimator asymptotic p extend result whole half column matrix formally x generalize estimator taylor lowest precise handle also cause er rao er rao taylor help expand normality aim estimator context similar result behave reasonably despite article project would semi regularization constitute simplify explore like take derive equation get argument decomposition eq first part constant term shall classic variance establish q converge remain h use usual derivative equation imply I dy I dx ij result quadratic locally normal b eq uk er recognize follow random central cf n tf j dx dx mu nh dy hoeffding nk nk term give constant equal equation prove end proof da du paris universit france regression set methodology joint du paris france universit france noise face call develop overcome reduction density projection refer effective dimension approximate covariance denote transpose therefore refer parametric form vector lastly method element asymptotic high quadratic integral semi estimate estimator alternative plug organization section notation coordinate estimator state expansion technical lemma proof section square joint density index triplet density finally dx dx variable center give aim similar whole extend write integrable functional specifically parametric identically sample split subsample
tolerance tolerance final tolerance properly algorithm algorithm tolerance smc couple replicate rejection start particle tolerance support bin lead grid cell sum difference histogram study yield unimodal model fig fig logical number child straightforward partly census peak approximate toy intermediate value lead recommend require reach gain simulation algorithm versus simulation replicate bar replicate circle triangle plain triangle smc grey plain square grey star plain rejection depict plain circle smc target equal tolerance square time depict average replicate choose appropriate interpret stop sequential performance large modal distribution benefit improvement automatic selection summary step regression processing contribution directly comparable would straightforward different publication collaborative project european union contract description cm simulate sx ti nx acc simulate u sx sx acc ti x k cm keep particle I acc acc acc mini k j fx acc xu proof exist associate tolerance define identically define particle aim level easily interpretable toy example complex social fast abc currently computation particularly relevant easy lead proximity approximate lead run demand exponentially tend limit paper minimize run reach abc subject scientific version regression improve informative chain improve sequentially approximate use sample set focus part avoid systematically apply abc rejection method however bias approximation population hereafter perspective sequential deriving difficulty remain tolerance bad modification monte abc algorithm population monte hereafter tolerance stop furthermore approach avoid particle mcmc algorithm intend stop value apply toy significantly less simulation population monte carlo hereafter hereafter smc hereafter smc implement package approximate present abc currently available limitation population hereafter carlo sequential abc algorithms appendix predefine tolerance parameter get target particle methodology step use transition randomly proportional inverse eq reach procedure draw prior particle introduce newly density probability reach e attribute newly particle major decrease tolerance optimal sharp importance sampling chance possible indeed arbitrary level smc algorithms level sample define smc concern smc algorithms move particle mcmc jump accept probability metropolis mean limitation smc mcmc kernel drawback particle jump particle accept keep new particle occur particle appear strongly solve jump evolve course weight attribute draw algorithm generate scale step throughout ensure weight produce distribution stop criterion proportion particle particle stop rule additional marginally change ensure proof toy toy study structure comparison indicator perform density histogram particle tuple sized bin choose compute indicator choose tolerance level equal sequence tolerance algorithm explore perform average deviation simulation twice effect particle distinct course show number cause decrease particle evolve towards ensure relatively distinct particle maintain reasonably run see versus average replicate represent replicate blue circle plain triangle tolerance smc grey plain square plain depict black plain circle distance times abc smc target tolerance distance algorithm cell depict ccc per accept new make couple couple distance net distribution impact studied improve quality total slowly numerous stop toy explore good acc
I f calculation across subject describe classification column equality appearance fit parameter age cognitive test score school intensive make outline essential acknowledgement thank associate suggestion correction respective length constant eq q minus derivative derivative minus example fitting marginal marginal lagrange produce identically covariate infeasible size base categorical penalty constraint year marginal generalize parameter multivariate flexibility enable conditional especially graphical na I algorithm estimation several challenge closed equation raw log consuming newton scoring try general constrain context show equivalent second less deal level unless size small variation marginal model penalty section marginal basic algorithm effect individual method categorical joint determine probability entry strictly positive frequency entry arrange likelihood p vector whose design determine matrix score expect take eq enable simultaneous modelling specification suitable complete restriction family ordinary group subset interaction margin possible margin call hierarchical later fitting describe study likelihood show exist outline maximize constraint derivative multiplier proceed suppose current reasonably replace second minus quantity exploit solve explicitly get eq sort length multinomial fit constraint constrain jacobian invertible rank thus crucial completeness necessary condition smoothness choose remove remove exploit computing note design equivalence matrix follow parameter respectively score alternate combined column definite u v rewrite second give update proposition step neighbourhood function respect neighbourhood expression add subtract least solution design implement constraint dimension parametrization computation making usually ordinary take detailed asymptotic existence maximum might observe zero converge jacobian matrix ill make unstable concern note modify newton modification elsewhere update converge stationary addition consequence tucker ensure look know local efficient observe matrix give model concave value order one probability need define parameterization example matrix row homogeneous simplification mention everywhere ordinary extension first equation substitute similar covariate available
boost receive ix ix ix x mm output q deterministic cutting deterministic proximity fx fx become extra jensen strongly strong convexity side inequality condition optimal deterministic quite programming case stochastic q assume clearly ix iw prevent asymptotically cut plane apply option low q accordingly guarantee constant lose cut plane validate clearly already deterministic unbiased bounding bit either condition independent bf algorithm summarize convex stage yield upper constant call first optimize least time option previous instead decompose side oracle updating prove asymptotic behavior apply algorithm rate hand recursively induce worst four gd cut plane method fast sgd result corollary yield markov loose demonstrate probability strongly convex respectively expand right w immediately sum consecutive property accordingly eventually together convexity fx f even easily decrease exponentially reduce average parallel distribute order four strongly norm continuous higher discover loose possibly fail likely fy n isolate parameter q unfortunately alternative g smooth reveal completely principle good ever still time raise
dynamical visual repository illustrate ability develop extract storing object speed architecture connection feature repository formation recognition capable development experience stimulus extraction som system ability extract feature frequently collect aspect extension neuron inferior modular like topological representation domain capable operate multimodal integration moreover modular imply flexibility architecture plausible configuration area capability stream superior etc serve platform network like mirror system intelligence school valuable comment comprehensive discussion focus development object recognition system general module module correspond connection level match stimulus short stimulus distribute spike reciprocal spike initial wave spike test information consecutive extraction accumulation rarely activate update repository update illustrate dynamical topological connection area broadly discuss visual object subject artificial intelligence ai create exhibit achieve parameter architecture replicate brain responsible visual recognition include area inferior temporal structure visual area visual size field neuron neuron decreases tune bar orientation tune various form degree translation invariance visual recognize consecutive area along stream object system ai architecture resemble visual area either task vary ms human process feedback speed implement feed furthermore rapid neuron produce feed top circuit reciprocal brain efficiently concept reciprocal bottom top connection spike reciprocal numerous present area broadly several address matter development orientation primary visual modular structure inter development key mechanism formation implement self extraction area object mean self grow feature reflect inclusion development regard growing establish intra development neuron hierarchical extraction level process visual recognition consist module som radial neural network level inside area group neuron mutually head line send som connection head line schema translate rbf grow som unit receive input grow layer well rbf exist signal dash unit mark head double solid som serve visual development som head dash processing intra layer ten fig black white describe v orientation wave equation filter relationship input different therefore four neuron cell densely implement result orientation map orientation contain neuron selective orientation location employ v area connection indicate orientation prototype specify empty absence orientation v neuron several neuron support neuron integrate orientation process complex different neuron type neurons rbf current rbf prototype preferred stimulus neuron term variance origin feature visual vertical integer rf vector c map wave wave serve feature repository update reach indicate signal spike inter connection inferior neuron consider rbf area address various area visual experience change apply neuron neuron neuron neuron store neuron perform stimulus stimulus center tuning estimate neuron feature dimension absolute depend activation actual stimulus normalize threshold detection store recognition processing however code store regard store neuron cause neuron rbf class densely cover produce response grid densely spike response grid scientific paradigm stimulus additional stimulus confirm reject process formation refinement orientation hand object hand spike step curve corner plane component show axis axis horizontal vertical dimension original image correspond magnitude information wave spike object study step follow grid store neuron calculate identify object neuron value present back purpose coherent identical neuron map actual localize map detect localize activation belong wave spike exclude call identical element signal stimulus internal stimulus spike major accumulate wave spike form occur refer hand object spike produce stimulus distribute spike shift spike initial hypothesis store stimulus effect stimulus rbf object feature stimulus significant effect take stimulus rbf center many common acceleration neuron spike send generation retain change procedure match identical element repeat back iterative continue spike wave wave dynamical intra inter connection change extraction mean exist rarely hand processing stimulus distinguish frequently counter good counter less chance remain feature extraction extract
definite approach efficiently surrogate concerned matrix might rank exponentially trace argue norm sign situation quite surrogate surrogate control e entry explicit section norm norm norm specify entry bound trace entry magnitude fix maximal minimize reconstruction guarantee absolute reconstruction measure impose scale word think measure relative constant obviously multiplied constant notion magnitude magnitude entry simplicity replacement whether choose whether possibly apply entry magnitude summation repetition uniformly magnitude ss either theorem hold probability without replacement nm obtain sdp potentially requirement error exponential subgaussian yield guarantee complexity similar ij previous yield max upper actually identify predict entry reconstruct observation observe entry n equation high uniformly assumption replacement positive always yield remainder organize prove sample bound square norm compare establish replacement set replacement follow regard determine either norm norm particular hypothesis error lx ij absolute rademacher trace ball rademacher trace norm index pair bound give prove loss immediately ij detail reconstruct max trace norm choose trace norm replacement square loss least infimum predictor upper size class ab rademacher last assume increase set theorem remark dominate instead square might hope similarly square use unfortunately fact demonstrate square rely specifically arbitrary reconstruct choose regardless control expect reconstruction regime relevant generalization obtain nm oppose favorable theorem dependence dependence technique rademacher even give norm factor come elsewhere literature familiar max norm square max factor theorem replacement without observe intuitively replacement sampling replacement state briefly notation denote lx sx os without replacement class define rademacher derive sx take must imply well replacement similarly rademacher lx hold subsequent remark replacement replacement error j draw z set aim observation intuitively replacement although tradeoff independently prove matrix square error entry lm ss depend noise future zero reasoning q turn without replacement notation sample add time without show nm long nm might require different sampling incoherence see g relative maximal magnitude nm compare approximate difference require comparable essentially guarantee rely compare guarantee discuss approximate theorem near exact state recovery exact otherwise identifiable matrix reconstruction aware immediately excess bound reconstruction restrictive guarantee replacement replacement entry e entry identical I independently difference quality sample recovery literature error minimization penalty method cite prove approach exception mostly trace norm use approximation entry elsewhere remove three provide nm generality subgaussian sample replacement sample time give independent recovery meaningful specifically size bind assumption bind error actually meaningful subgaussian replacement weak subgaussian similar fairly different method rectangular matrix see scale improvement much hand include compare theorem pt recovery assume identically subgaussian recover possible estimation significantly approximation know believe technique quadratic result match norm relax scale absolute strength replacement include entry noise replacement observed entry draw mean exact near recovery exact corrupted subgaussian near recovery result recovery work fundamentally bind definition approximate recovery type guarantee link generate nonetheless comparison magnitude recovery near approximate rest follow literature complexity section section recovery complexity result incoherence sufficient getting norm section incoherence condition clearly approximate recovery norm recovery svd also denote improve prove recovery low set subgaussian incoherent improve noisy give additional meaningful therefore regard subgaussian reconstruction algorithm find observe subgaussian simplicity relaxed require ignore sample ignore dependence case exact incoherence complexity establish exact less recovery bad recovery approximate recovery km light complexity result rescale contrast recovery incoherence number g may attain incoherence enable approximate relative average incoherence entry perhaps extremely subgaussian mild dependence complexity complexity several max excess method result obtain loose align writing number rank slightly singular perturb matrix complexity trace norm matrix carefully paper string rank show rademacher combine argument guarantee rely point superior commonly able reconstruction study replacement establish generic relate setting bad approximation although dependence rademacher consequence rely mean assume max rely rademacher max careful rs complete proof matrix time bound lemma draw replacement element appear exactly time obey convenient might write regard list order format let follow equality arise recall next etc time appear rescale treat therefore eq observe rs conclude except entry follow statement sample vector sx unlike lemma therefore apply index
obtain vc conjecture theorem department query optimize execution operation major contribution interest vc outcome dimension operation join operation individual query collection collection tuple database query build database high database evaluate query need major change compute small store present analysis advantage technique complex microsoft server advance technology collection storage vast need work database crucial query execution plan optimization database task efficiently obtain range table database yet powerful commonly inherent involve table query datum cost query collect disk large storage medium therefore expensive provable running sized database concept vc dimension novel technique query hypothesis sect theoretical vc dimension class indicator product dimension number join adapt vc query database execution query provide query database hold collection evaluate query major change query measure vc experimental accurate estimate use surprising concrete memory significant execution apply vc dimension uniform practice efficient store table construct present experimental validate technique microsoft server give accurate prediction create multidimensional histogram join estimate sample possible solution rest paper organize relevant work sect sect introduce develop analytical contribution bind sect sect query plan explore range offline histogram datum mainly tuple arrival discard sampling database disk operation cardinality significantly sequential keeping improve apply cumulative inversion still expensive offline et use systematic require tuple tuple et use obtain belong expectation query inference bind conservative join sampling point number table table priori together common join sect explain pre compute statistic predict histogram used construction histogram examine rigorously size building estimate query extensively although offer query efficient storage need histogram inherently limited query drawback histogram intra uniformity assume distribution inter independence correlation suggest use multidimensional memory updating literature seminal computational encounter success related database context database aggregate vc need unbounded privacy purpose column tuple appear select join operation define consideration impact column operation tuple tuple combination clause exclude discussion category table column return tuple join basically join condition contain involve mean tuple join one correspondence correspond join set table return subset tuple set selection return join direct whose elementary join operation decompose node output table operation definition combination select join nevertheless define query plan leave internal join throughout tuple tuple join join table execution sample operation execute function equivalently subset bind vc bind learning outline adaptation specific work vc define space range finite infinite point range set query input tuple query combine join involved class vc range range vc cardinality let range vc range query database table tuple query range set family interval set point interval vc observation vc dimension half dimension theory relation size sample let ax rr b construct vc subset q relative least interesting sect probabilistic challenge vc dimension range fundamental result size define integer growth lemma range space vc finite sect combination range range space intersection member combination intersection q hold vc dimension start complex attribute query join query join operation table column tuple tuple equivalence previous paragraph range vc dimension tuple extend query query selection combination selection predicate note query either rewrite clause either vc number predicate clause space align box output include disjoint align intersection overlap collection technique see combination query intersection operation imply let table select tuple product order pair form join simplify query predicate r r j join query v jj car ap j ar r identify b c c p r eq hold plan select internal join tree range select query table join table define note extension join query set car ap ir r vector see I ia tuple actually choose choose operator join q join operation involve boolean operation suggest result vc dimension concrete optimization compute class class involve query execute query execute def eq within table plan query rooted internal query run obtain define correspond thm application prove maintain easy independently write tuple product table assume table tuple form tuple tuple ix query join count tuple ji kt op tuple tuple op alg query table table add index tuple concatenation tuple element note major point general impossible join join require size result join vc size identify plan may standard plan generation execute multiple common candidate overhead execution approach likely technique well overhead store intermediate significant extra present microsoft goal usefulness theoretical assess run query database size use random join operation previous estimate thesis large query sample commonly histogram briefly sect histogram fine grain histogram soon column long give modern database system query optimizer histogram statistic help determine efficient use histogram information item frequency default store bin list histogram histogram list table tuple histogram tuple column develop sample require join attribute therefore complex database million tuple distinction run selection query join run different category independently treat contain normal I value correlate join query consider join common table treat tuple replacement table join tuple table independently tuple base tuple form sect table original sample fix thm million tuple vc size fix
grant office two grant powerful effectively extensively area trick inner feature without know trick extend spectrum space eigenvector sense eigenvector explicitly lead eigenvector base subspace spectrum sense spectrum simulation correspond lead eigenvector matrix segment kernel pca generalize ratio sense cognitive availability band secondary user without interference primary user spectrum match covariance spectrum nothing receive hypothesis user involve present sense alarm rate alarm extensively apply especially support machine svm counterpart kernel diversity algorithm inner implicitly infeasible inner point become product mapping generalize space nonlinear well hardware pca white wide signal prove stable paper spectrum lead propose employ map inner lead know eigenvector say eigenvector pca generalized algorithms spectrum propose hyperspectral detection background subspace span target background projection assume combination column kernel match employ consideration background hand operator assume generalize space determine simply speak lead pca product value projection space spectrum organization spectrum sense lead review eigenvector propose eigenvector kernel modify base match simulate receive primary user signal transpose lead eigenvector large assume eigenvector eigen decomposition diag corresponding eigenvector template pca likewise number receive simplicity denote lead eigenvector sample I pca desire alarm eigenvector call pca kernel classical employ kernel implicitly feature need increase computational receive kernel obtain x last linear imply combination side ij eigenvector matrix positive eigen normalization eigenvalue point know however compute template detection x coefficient lead explicitly lead eigenvector sample know likewise eigenvector lead product orthonormal vector white obeys receive hypothesis q conditional gaussian eq approach explore cast eq substituting take operator onto subspace detection threshold value operator feature eigenvector nonzero projection operator assume project pca eigenvector nonzero eigenvalue covariance x x eigenvector nonzero hypothesis gaussian claim still gaussian kernel employ approach center spectrum gaussian kernel consideration center kernel j normalize receive eq compute determine threshold alarm detect presence chart spectrum kernel detection threshold determine ec ec obey ec optimal hypothesis ec totally blind without design maximal minimal pca matrix primary signal assume amplitude receive length receive ec implement implementation kernel kernel pca varied pca ec fig pca method kernel ec know knowledge notice affect varied snr ec still ec ec gaussian actual perfectly subspace model factor affect calculate kernel kernel divide maximal method test choose kernel operation center similarity kernel linear test verified correctness capture sense first sample user matrix test framework divide segment segment eigenvector first segment hand varied snr compared ec fig pca db db correspond ec ec fig kernel
topic relative frequency query access plot probabilistic ir oracle produce vector evidence section design end refer ir report term occurrence exclusive e presence ir pearson occurrence partition disjoint include frequency rejection document absence region subspace span entire symbol partition include accept rejection observe operation subspace span subset explain vector illustrate law vector space span plane dimensional note provide since vector occurrence due must measurement finding via device occurrence physical device read text frequency sufficient optimal vector much difficult physical measure vector report comparable effort oracle text limit actually informative content observe document question automatic indexing retrieval van book introduce formalism ir within uniform space theory book quantum phenomena ir interested investigate ir probabilistic powerful book provide theoretical foundation vector particular optical communication optical signal improvement domain quantum theory relevance parallel detection weak relevance play role quantum statistical ir paper obtain characterization measurement sense distance relevance justification angle unitary justification excellent quantum particular explanation illustrate difficulty however mention e interference superposition retrieval find quantum phenomenon aim leverage ir quantum formalism ir abstract formalism exploit formalism illustrate improve author conjunction angle quantum interference interference suggestion probability induce different induce traditionally concentrated extract combine evidence accurately ultimately scalar structure retrieval implementation thus implicit achieve possible ir incremental achieve answer end suggest ask proposition accord document high optimize retrieval accurately principle subset theory name vector classical subset mathematically verify suggest retrieval effectiveness condition measure define probabilistic ir probability event distribution use set measure axiom state theory stem prefer event set classical probability latter interest whereas ir scope replacement ranking ir report result date ir result rank irrelevant often unit document list besides weight result research ir view new perspective describe probability accordance effective rank accordance classical evidence effectiveness correct result prove mathematically experimentally quantum show superiority retrieval investigation phenomenon argue quantum ir ir give intuitive contribution quantum outperform classical central introduce vector exploit rank probability occurrence optimal document address theory outperform model feasibility tell vector occur document conclude appendix proof depict intuitive relevance occur presence absence symbol sake clarity side ir b act detector b relevance probability appropriately receive symbol ir implement relevance transformation symbol figure whereas implement transformation symbol straightforwardly receive symbol equip ir implement classical theoretically hold ir describe ir compute symbol index relevant document relevant detector symbol occur non call document retrieve rank document relevance operator hermitian unitary operator rule equivalently namely document subspace optimal document end define represent relevance uncertainty condition take upon illustrate appendix probability acceptance leverage rule high optimal span relevance determine geometry decision geometry correct decision probability relevance relevance geometry vector clarity generalize angle vector angle orthogonal angle relevance rotation hold vector locate relevance note ir like detector compute symbol occur relevant document call mutually exclusive yield latter refer refer relevance non suppose relevance p vector eigenvector main show latter exception probability latter term find detector sake clarity e bm slightly theory bm illustrate find underlie normalization estimation depict ir equipped bm differ ir evaluation already ir detector would oracle
denote also xx jj proceed construct outline extra ensure recovery key recall row define magnitude excess residual element either least element role role block matrix construct pair follow significance define ss subspace matrix orthogonal complement ss projection motivate bb subspace support projection define p bb index achieve sign p technique consist construct dual candidate theorem step differ condition impose follow primal candidate design candidate restrict problem subgradient specifically optimality oracle problem substitute step would require recovery result since candidate pattern make follow specifie problem problem p assumption c ball respectively fact eq saddle point strictly dual equivalent uniqueness matrix solution show primal condition differ complexity theorem result assumption positive guarantee property suffice hold hence unique inequality result hoeffding part p equivalent hoeffding happen k condition primal projection eq moreover eq thus nx state regularizer nx happen probability pr state condition distribute tx hoeffding bind result least rsc solution unique b nc min provide proof part separately result eigenvalue hold share portion large portion assume equality dual recover strict variance turn fix j row sub norm j conclude assumption condition hold constant projection notice concentration theorem union conditioning mean value gaussian get eq j fraction large task gaussian q go substitute ds rv union discussion solution f require investigate characterization convex derive sub differential differential cc rt belong sub denote jj condition uniqueness jj provide proof case j j j j prove establish contrary j index k contrary exist element except j entry j j trivial suppose exist row j j conclude ratio regularizer since optimality integer unique row equal except row contradict optimality similar uniqueness fact j kk j solution contradict characterize introduce take tuple initial guess warm stopping update subroutine update block matrix pd x j break cyclic descent keep unchanged k mi ib return cyclic keep unchanged block sparse cycle unchanged vector fix subproblem vector j kx correctness directly correctness descent argument correctness unknown measurement like multiple sparse several partially recover whether decrease line recent actually separately ignore sharing depend level parameter pay differently show theoretically except case perform well multi learn dimensional increasingly problem high hope statistically consistent lasso rank high dimensional context task task estimate leverage aspect feature sparse problem simultaneous arise variety context signal range sparse represent column row block sparse zero mostly lot focused encourage sparse example norm regularization encourage sparsity assume share suffer arguably realistic depend feature addition concern row nearly identical far original feature method regime support vary widely parameter thus ask leverage overlap parameter identical instance general fall block might bias first focus simultaneously superposition search overlap sparse theoretically extent support remarkably follow sec basic setup present sec denote row sum absolute regression k task independently target notational convenience interested estimating leverage extent simultaneous sparsity certain entry share row row would feature adapt level guarantee successfully recover sign sufficient merely recover row support particular support column error interested superposition feature task two encourage simple would sparsity regime interestingly block sparse clean statement recover support scaling regime provide sign row scale require sign recovery third explicitly quantify lasso block find early theorem correspondingly follow case task design two fraction interesting phase transition scale successful sign particular rescaling sp rescale sign converge scale converge regularize rescaled sp show probability success near rescaled sharing perform share perform show rescale sharing sharing everywhere else detail respectively outperform require sign support consider specify analyze normalization incoherence denote incoherence minimum consequence finite regularizer require regularization hold unique false false b b remark guarantee relevant addition require recover exactly need enough matrix ensemble graphical row condition curvature impose matrix row regularizer require regularization pr least guarantee estimate finer precise quantitative gain experimentally regularization except share match result sample also generate submatrix regularization scale choice c reflect balanced share synthetic datum consequence theorem theoretical digit task digit show practically via cross discuss generate use search regularizer program find three sign sign decide program successful sign explain simulation three generate entry support multiply cross k kk train generate recover descent appendix tuple guess regularizer set good guess stop algorithm inside update objective algorithm completely optimality thus partition logarithmic filter coordinate minimum run recovered otherwise element different sign predict stack observation sign regularizer lasso less observation performance grow well lasso still change theorem versus
variant detail follow mod sec reconstruction result key sec iv cardinality also value mean clear sub element great strictly zero element less equal pattern transpose matrix denote also reconstruction problem reconstruct length denote denote contain signal estimate along restrict c sc restrict orthogonality roc small real mc ss frequently roc matrix rank mod put explicit order small similar constrain suggest mod achieve reconstruction exact mod put encourage close nonzero instead distance add least exact recovery mod cs mod bp recovery mod bp exploiting result discuss implication sec lemma lead proof sec iii b outline inactive find set maximize constraint bad size unique u notice large reason fashion notice ensure first ensure practical active nonzero deal signal take signal sec iv take expect hold occur recursive video original video project base foreground person move camera large correlate noise similar look slowly video rate common small extra requirement bad case mod cs result hold recursive mod recursive recursive compression extraction correlate moreover active mod anonymous exact knowledge recovery small reconstruction error mod mod cs recovery subset observe probability reconstruction increase size check remove element hold check complexity roc condition result loose bp mod cs set sufficient unique next measurement fashion theorem later suggest tucker kkt set necessary condition keep us condition let find recall mod cs measurement roc lemma help hold ensure third apply iteratively notice ct notion satisfy mt b appropriately condition bound size ii ks ks ks c positive similar give disjoint condition iteration disjoint theorem satisfy probability good nonzero mod compare mod cs higher alternatively significantly reduce probability compute various exact recovery also second simulation signal step equally scalar variable range notation mod cs weight mod bp mod mod bp bp mod cs mod rmse normalize unit norm size uniformly element support n cx tx ki thus set mod bp mod weighted choice package primal equal choice record table record record good root mean squared rmse record next thing half increase increase decrease increase mod cs realistic scenario bit set x zero clearly draw finally n since expect mod well bp mod mod work sufficient exact recovery mod active subset active satisfie give mod achieve weak happen estimate weak mod cs either bp simulation mod bp minimizer fact follow scalar equality eliminate equation tw w hold equality equality support xx simple fact let minimum I semi definite matrix mt mt mt ba mt definite b b numerator denominator mt mt mt na mt prove try find construct lie nan satisfy infinitely closed mt mt mt mt mt definition theorem satisfy lemma similarly equivalent rest follow beginning follow ii mt iv mt u mt mt x otherwise take e mt thus satisfy material finally theorem zero combine simplify summation absolutely convergent equation w rt give condition satisfie theorem definite ta symmetric let b since form I thus
finite due result possibly separable hilbert space except adjoint definite identity outer linear u uv operator frobenius mm denote large vector form orthonormal since let eq copy square uniquely eq specify condition dimension covariate moment apparent size accuracy technical merely remark naturally ridge analysis kernel easy product equality expectation mapping q leverage finite surely sure bind relaxed analysis sake simplicity subgaussian requirement subgaussian lead ordinary surely satisfying typical approximation error eq subgaussian subgaussian normally distribute mean minimizer approximation surely q sure relax sake appear low ordinary requirement exist surely surely inequality triangle third schwarz argument satisfy eq ordinary study design covariate response independent analysis review section control bias ridge essential compare ordinary estimator random design square covariate bayes relative error plus predictor approach beyond analysis often additional dependency spectrum moment comprehensive survey excess empirical satisfie certain boundedness bound objective approximation result square computation work provide ridge regression bound essential assumption fail give review analysis assume surely assume dominant explicit assume almost surely explicit surely lead factor specialized estimator depend moreover ridge therefore vanishe bias analysis expect sharp ordinary mild assumption bayesian ordinary probability least rely assumption tail arguably reasonable quantitative establish comparable although understand ordinary mention number work considerably weak section present result square ordinary square review design review set deterministic response variable x eigenvector v design modeling decomposition proposition ridge effect fix error effective notion level ordinary square fix analysis ordinary exist specify bound inequality simplify constant spline ridge also specifically periodic smoothing spline th map condition satisfied remark datum behave lack approximation reveal crucially control fix overcomplete approximately solve exactly square prohibitive approximate satisfactory randomization rotation replacement n square problem rotation randomly matrix apply create square quantity solve sample leverage accurate leverage direct generalized orthogonal rotation matrix arise randomly row na arbitrary rotation operation distribution see considerable retained ridge operation suffice approach treat regression produce pair row choose rotation matrix problem b ordinary guarantee probability consider constant proof fix random orthogonal q th coordinate nu vector union simple condition distribute random follow jensen eq last hadamard vector rademacher component especially present operation significantly fast run na I multiplication therefore last hoeffding bound lemma due slightly argument reduce difference recall basic used bind design ridge employ matrix matrix contribution probability well effect proposition inverse normal w j follow w expand square equality shrinkage error consequence second norm define condition pick probability consequence tail adjoint ab product condition pick definition q claim observe condition triangle assume observe ni jk define I nonzero eigenvalue observe spectral version definition cycle property denote von lemma inequality cauchy schwarz acknowledgement david inequality choose order availability subgaussian generalize gaussian vector form subgaussian value matrix let variable q expectation moment generating degree fact tail application bernstein vector bernstein vectors eq spectral accuracy moment bind surely independent copy ordinary least regression estimator set mild sharp reveal error neither decomposition concentration inequality matrix set linear regression sample population vector random typically
recovery universal nearly optimal universal bound necessarily rip qualitatively rank component reconstruct nuclear norm frobenius frobenius f good state reconstruct frobenius lastly rip insight design measurement setting repetition experiment essentially element small operator rip actually strong result appear elsewhere rip gaussian strong rsc somewhat weak show completion follow cover argument concentration failure union seem randomness phenomenon weak union result account favorable correlation behavior different matrix behavior positively correlate good transform bind use compressed rip fouri key nuclear norm ball use entropy duality also sometimes set I interest call state gain complete element nothing element turn rare body work regularize program see organize discuss appear section vector p denote act product element measurement incoherent speaking must I proceed sketch quantum dimension semidefinite trace convenient experimentally feasible matrix tensor kronecker matrix expectation matrix scale incoherent general uniformly unknown minimize nuclear constraint note convex relaxation thus efficiently approach selector alternatively regularize lasso discuss estimator isometry rip say rip denote norm notion rip rip large measurement orthonormal incoherent depend let rip x remainder section rank previous obtain selector measurement near measurement incoherent operator rip result sketch couple quantum real arise situation state low instance pure decaying typically obtain good ignore secondly inherently noisy observe entry copy state copy average approximated gaussian noise noise one reduce repetition experiment possibility choose elsewhere reconstruct frobenius let estimate allow mechanic distinguish error implies contribute quantum sketch apply value residual part ideally estimate selector frobenius say strength seem norm particularly strength nearly fall tight nearly theorem couple besides f even nontrivial result incomplete consider tail know nuclear small bound let choose sampling operator mm rd selector bind interpret bias second term norm tight well decay term adaptation bound covering summarize x adjoint norm banach respect suffice first hilbert schmidt inner adjoint prove rademacher small fix k universal algebra gets find root dr desire result concentration bound rademacher norm comparison iid get index upper bound supremum cover metric need cover actually elementary calculation I cover notation unit norm observe u give number restrict next use universal covering number cover banach modulus let eq duality reduce dual basic denote ball need cover empirical dual covering inequality detail lemma banach norm want work infinite measurement restrict isometry property rip low technical nuclear ball et al manifold space one strong rsc manifold embedding try generalize rip metric embedding rank likelihood variation notion g fully david anonymous suggestion work california berkeley grant institute technology measurement supplementary overview claim defer later section involve proof cover use restrict isometry claim rip suffice imply adjoint suppose rip rip norm element term precisely standard basis vector norm write ab ab cd ab true must ab ie cd write q argument get straightforward finally banach complete respect return concentrated around claim rademacher view hilbert schmidt adjoint element equation bound fix lemma uniformly norm mc lemma operator operator side rearrange root bound simplify write c c c concentrate concentration banach jx x copy inequality use constant write eq plug inequality failure decrease rademacher quantity iid lemma equation index supremum fact pp cover radius metric need simplify norm actually semi b upper metric simplify I gx x x cover number plug change covering introduce ball observe simple eq
agent b lastly check reveal always satisfied imagine pool play repeatedly desirable failure avoid agent resource sufficient equally utilize common would solution issue protocol communication decide channel slot channel occur need access protocol protocol probabilistic work consist power transmission power able otherwise transmission straightforward independent constant play measure utility agent agent update update agent keep track perturb payoff history agent select provide satisfied independent identically refer ni topology borel field measure endow topology topology value chain denote let banach q verify admit stochastically state cardinality identify dirac characterize follow collection probability eq finite chain asymptotic therefore ergodic process proof require proposition refer govern element sequence notation induce initialize operator x decrease satisfie eventually hold initialize path let c x prove define tt define recursion length rough induction eq side transition eq positive q evolve necessarily yield property prove let obtain decompose perturb also define similarly transition simultaneously lf invariant invariant side multiply side invariant iii limit right convergence dominate ss unique eq constant transition support also note continuity theorem q definition show restrict irreducible proposition unique finite incorrectly strong characterize asymptotic behavior note game characterize stochastically end denote clearly correspond profile important strict game decrease dominant facilitate start govern introduce payoff profile dominant profile receive within let game hypothesis u c induction set recall simplify follow obtain secondly possibility profile decrease value play gets define thus strong hold pick level perturb level satisfie lemma let profile respectively j j j obtain sn collection pure satisfie player kk kk game mutually coincide accord profile terminate let j contradict assumption partition strict game either h partition family transition decomposition equation positive get last proceed complete combined theorem provide game behavior theorem formation strict apply htbp h nod neighbor dominant network single network network plot typical setup level well denote convention graph plot run inverse I dominant profile play show pool access resource remainder establish provide invariant irreducible chain extensively show stochastic argument particular ratio etc satisfie equivalently point define hold state every finite distribution chain theorem game profile two profile k kk equivalence profile write u kk correspondence profile strategy pure identify path provide enumeration path moreover also path since game state profile sequence profile lead therefore hence graph use game pool set successful state pure strategy action I mention respect relation since pool pool game lemma I recognize disjoint invariant put weight either establish moreover put state outside state theorem behavior game accord percentage perturbation failure setup action demonstrate learn either agent succeed increase joint neither succeed zero htbp pl ac algorithm analyze game multiple contribution relation allow apply payoff become arbitrarily correct learning game two player demonstrate simulation network formation game network interest condition outcome player show multiple utilize limit expect resource divide among increase remark ari outcome game base player continue exceed memory become level proportional chain characterize include pool game game profile outcome desirable game common pool game fair sequence highly simulation fair outcome game include pool game game learn interest engineering distribute formation medium access control wireless utilize resource desirable formation immediate possible link schedule I resource avoid achieve adaptive distribute couple adaptation motivate agent endow utility depend agent previous experience receive challenge optimization impractical inherent number action lack form reward theoretic obstacle utility include adapt agent consequently effective issue consider agent scheme simple stay shift successful drop desirable return level update prior experience agent learn play history scheme explain decision simple decision stay reference action select similar contrary incorporate update effort game payoff profile payoff action bad profile player game keep track satisfactory action satisfactory payoff benchmark stay benchmark pareto efficient payoff play b make stay shift level payoff game payoff converge neighborhood pareto payoff surely paper also focus achieve pareto efficient action agent characterize game condition select characterization asymptotic induce chain extend game player also call particular unique put weight action level sufficiently formation formation primarily response dynamic dominant action profile desirable game interest profile might action profile similar interest weak direction change also existence equilibria furthermore definition write profile b payoff profile namely desirable action profile namely nash profile outside row bad satisfy definition might multiple choice selection desirable table alternative profile case property word game desirable action profile exist terminate profile u repeat generate sequence necessarily terminate consequence claim acyclic games formation interest wireless communication model development also network formation model introduce formation motivated plane agent contain combination link link establish point direct direct link start end e integer joint action nash equilibrium show unique path neighborhood action profile set network payoff dominant exist connected link dominant
semi method first demonstrate supervise methodology label allocate label functional procedure value figure ratio assign unlabele seem expansion supervise select criterion simulation microarray show modeling strategy relatively rate supervise functional work support education aid cm cm center q smoothing consider procedure assume independently normally distribute function dispersion dispersion follow spaced knot regression maximize log semi smoothing criterion matrix diag methodology circle discrete solid solid line et give label predictor previous indicator first logistic label functional group al introduce expand logistic describe depend multinomial introduce label unlabele functional l k belong unlabele n label unlabele ill pose problem functional employ regularization regularize positive numerical identity log explicit via label n help fisher method probability lx replace likelihood parameter scoring satisfied statistical include selection regard introduce construct semi supervise logistic theoretic propose aic evaluate likelihood field various estimation supervise functional selection criterion diag diag diag block diag l z hadamard product bayesian schwarz bic viewpoint likelihood bic cover et criterion estimate regularization generalize model al statistical model semi model tuning criterion derivation selection investigate simulation modeling demonstrate model discrete two g u g w I case implement supervised randomly divide unlabeled label semi functional logistic model et rbf svm knn functional machine et al consistency yu inductive transform smoothing semi supervise et al five fold cross knn select leave validation comparison average parameter observe evaluate superior method almost seem error case method outperform svm knn situation
vanish involve outlier outlier base new analogy q indicator cell definition base freedom goodness outli base anti diagonal ideal base table ideal ideal increase instance without key algebraic lie mean linear model way cell avoid indicator cell model cell parameter behaviour immediate connection cell definition pattern outlier parsimonious set procedure step residual definition flexible show analyze form classified gender social network small count ccc base write log base markov basis form move quick inspection residual suggest run test monte carlo case notice helpful test show behaviour pattern namely separately exhibit strong evidence homogeneous additional cell interpretation model common cause scope description geometric technique table exercise logit multinomial contingency table refer contact influence management ccc ccc medium medium medium medium medium base degree fit poorly form computation ti analyze residual table base residual absolute cell count low equal bold two approximated carlo value outlier look table cell special inspection relevant useful address outlier contingency efficacy direction recognize view algebraic outlier theory research question remain mention need problem level multiple widely article cite see statistic yet explore connection mixture characterization outlier yield useful structure correspond outlier basis already currently acknowledgment acknowledge help several suggestion clear algebraic anonymous associate valuable quality collect presentation result algebraic definition ideal ideal ideal generator exist write parameter act constant notice sufficient define ideal ideal generate associate know ideal finite pure generator computer implement ti ideal major side markov contingency sign eq markov move log associate identify eq actually polynomial generator relation ideal following hold statistical form corollary pattern table goodness test outlier essential clear applicable several example word important contingency recent author estimator cell count high expect count present account application find difficulty contingency table contingency present appropriate later notion outlier detect standardize note notice respect univariate gaussian sigma criterion outli contingency less variable categorical cell contingency count model one use quantile use outlier presence use adjust residual chi square detect decade algebraic statistic research major table natural contingency structural actual algebraic statistic contingency cell quasi symmetry reasoning algebraic contingency table outlier term fit table algebraic tool test easily describe structure outlier useful example analysis candidate appropriate goodness later material recall study single monte carlo notion pattern two conclude remark work help experience polynomial decide main avoid possible group technical fact normalize probability classical sample contingency cell contingency extensively section wide statistical contingency table scheme cell count random log lie give linear design obtain parameter representation minimal sufficient simplex table issue e implicit eliminate paper follow connection implicit contingency outlier present contingency table residual column suitable approximation adjust residual detect outlier useful ml poisson tail appropriate analyze residual test high residual critical evidence cell level region adopt outlier contingency define goodness nest algebraic statistic understand outli give contingency cell table model representation name outli base well negative base eq indicator th goodness fit compare avoid sufficient goodness test point impose candidate easy algebraic statistical variety state negative row virtue proposition show system generator definition inclusion theorem outlier goodness fit log statistic maximum must chi alternatively observed contingency contingency eq contingency
gradient prefer iteration obstacle reason describe piece block correspond realistic reason describe arrive datum network equally necessary problem characterize use cd remainder mix claim relevant outline contribution basic algorithmic strategy name linear linear gauss subspace domain impossible reason iteration focus single block conceptual cd literature reference therein survey block cd semidefinite never focus cd recently area machine machine learn regression closure partly change describe efficiency cd depend balance time spend choose update one possibility descent prohibitive due compute seem compute away surprisingly cyclic cd yet know author proceed sequence theorem iterate gradient cyclic describe realistic well effort perhaps block intuitively randomize strategy preferable randomized case possibility block goal speed adaptively availability block choose current partial derivative usual sometimes computation seem promising candidate describe line implement entire reasonable value algorithm use problem smooth nonsmooth proper precisely cover decompose block machine elastic solve consistent linear enjoy global author claim conjugate motivate study give randomize cd quadratic function coordinate proportional define interpret derivative onto dimensional consider lin iteration several descent smooth apply coordinate nesterov descent smooth improve way understand result rare good cd composite analyze case unconstraine sum c rewrite box extend simplify nesterov treat sum convex nonsmooth separable focus accelerate accelerated scale problem website author solve iterate composite block coordinate descent case ht ccc objective complexity composite convex c symbol precisely far encode iterate minimizer strong convexity nonsmooth briefly outline expand find composite cover minimize make detailed use argument remove nesterov norm approach norm experiment focus regularize support machine powerful adaptively change speedup shrink organize basic state describe generic uniform variant composite study cd regularize support follow identity vector write u I euclidean identity matrix vector wise lipschitz separable decompose readily eq ready describe generic iterate pick block update upper directly closed example ix k iterate random clearly scalar define subsequent denote element eq iterate technical enabling improve ii suffice expectation case notice well precisely since hence albeit result run process technique shoot two argue choose use case technique ii example name ix compare therefore define minimizer use repeatedly generate fx ix ix tx ix k estimate fix ready need logarithm attain counter choose f q lemma result follow c theorem assume optimality lemma useful prove convergence strongly convex define lemma analogue logarithm respect convexity random theorem first add thus regularize approximate regularize quantity advance fix regularize convexity rest subsection show recover minimizer analogue apply compose suffice section much simplified section smooth norm let usual constitute euclidean believe smooth euclidean nevertheless subsequent claim give depend rewrite ix fx fx h fx purpose analysis come objective euclidean result target kp us decrease fx u ix get fx rearrange result first notice states lipschitz set assume strongly respect definition convexity estimate objective accuracy decrease ix fx f rearrange fx compare paper endow table look smooth nesterov contrast brevity strongly convex nesterov n ix fx nesterov lx euclidean strongly convex comment table detail hence lead logarithmic line coincide lead term note term table cover set eq ignoring bind note additive decrease result probability uniform result cover fine logarithm general term fashion hence result summarize characteristic complexity coordinate case cover general grows increase randomized give method constant block constant schwarz coordinate variation choose lipschitz c choice yes greedy expensive cyclic separable yes impact estimate approach individual constant coordinate fu fu l c iteration counter section sparse group size gb ram problem induce result moreover lipschitz constant explicitly thresholde give modification direct throughout refer generator control instance generator propose generator detail refer reader paper versus complete iteration iteration table roughly linearly depend per iteration proportional computation htp typical performance start instance big huge first size ccc c c time r column correspond iteration coordinate counter coordinate correspond respectively residual iterate residual decrease additional factor first look problem table half support solution residual decrease factor surprising linearly iteration decrease decrease factor convex iteration guarantee able hour performance strongly less minute take second convergence htp b nesterov constants vector power instance produce generator correspond htp tendency small choose switch effect influence effect coordinate vs lipschitz work multiply constant interval exhibit find tolerance fast fact lipschitz choose well order phenomenon lipschitz constant whereas able quickly get update method solve eventually iteration htp position resp lipschitz remain resp summary solution accuracy recommend switch shrink speedup shrink increase encourage increase setup certain predictor
economics unite objective conventional evaluate efficacy numerous fail cycle period formulation fit absence prefer aim distribution observable series term feature dynamic cycle long short want name challenge misspecification paper attempt develop systematic address aim match rigorous propose conventional fluctuation direction development dynamical widely describe change salient feature observation capture demonstrate sir newton newton explain concern motion series make interest long future salient approach often prefer available mention discussion evolution population delay take death specification suggest reciprocal unknown nonparametric estimate consider similar discrete biology see nonlinear time cycle cycle cycle suggest although possibility consider dynamic underlying cycle sir sir investigate many disease member variation individual epidemic fall infect enter recovered disease break back death birth death sir dt dt contact recovery investigate successfully time version birth rate basic transmission disease understand birth force rate transmission disease also guide policy maker spread pt concern objective observable salient cycle secondary mean interest truth model fact improve capability innovation recognize generally matter note setup different error happen coincide ahead condition minimize mathematically box wrong know boundary wrong unlikely remain restrict estimate preferable box spirit diagnostic goodness device recommend box device development development however spirit stage rather worth recall classic autoregressive ar capture business term letter respectively observable stochastic constitute realization observable throughout start model say match series perfectly conditional almost surely call approach version subsection name quite possible definition broadly step intend answer step involve model economic unlikely economic calibration matching difference methodology methodology bayesian usually wrong seem end methodology private requirement vector practice value prediction norm ahead idea depend speak complete ahead model composite q expect arrive suitably multiple ahead accurately sense measure weight innovation difference instead development worth exploration suggest speak shape tend emphasize pass filter slowly vary si tend filter balance pass filter pass filter series square ahead cox xu chen clearly former ahead different extension build shift effectively panel square error stress primary good sometimes medium long member panel recover formally setup weight unity leave zero observable stationary weak form difference distance explanatory difference two difference example distortion discussion difference observable fy ar ar setup classic describe way implement criterion innovation observable estimate second reason estimator lead series innovation e shall illustrate minimal observe give analysis case involve measurement relate linear skeleton special misspecification measure version measurement measurement old least early assume recursive namely equation write give independent mean variance long denote series yy determine estimator equation p matching incorporate lag beyond may minimize denote minimizer regularity condition far smaller substantially decay slowly decay decay cause shall return exactly equation difference show amongst mse method lag extent lag method find autoregressive plus additive white noise approach essentially treat attract attention li autoregressive match wrong consistently possible lag contact call next stop ahead estimate typically reasonable autocorrelation observe property discuss dynamic measurement apply system example example skeleton observable employ ease explanation start dynamical lyapunov exponent predict lyapunov exponent system x admit cycle derivative suppose lyapunov neighborhood x summation take noise origin fy fy x fy fy used mle indicate ahead noisy second part suggest reduce bias cause also past offer remove statistically interesting skeleton value nonlinear white ahead consume numerically ahead implement suggest ahead skeleton sometimes helpful especially turn statistical parameter parameter sense tend absence specifically achievable observable precise infinite parameter define unity ease exposition infinity see panel figure useful time panel observation nonlinear eq show figure skeleton replication match period column four infection massive considerable prediction year trend http sl txt panel error average ahead error take mark select mle ahead density function method fan estimation matching capability investigate span ahead step period display bottom mle superior short reverse b pt pt surface cycle notice cycle cycle fall regular helpful weather help historical recorded world number period http www autoregressive lag higher suggest well notice dynamic self threshold autoregressive regime delay equal recommend try delay horizontal solid respectively post skeleton period model frequency fit fit generate skeleton draw short one ahead panel long fit step beyond ahead appear line series avoid unstable observe time line datum consist population control laboratory datum employ obtain daily biological major series population bi population bx bi day differently usually discretize ahead mle base skeleton cycle extent day bi day cycle capture period grow bi day bi day last panel clear cycle period light population twice period panel mark axis vertical color code blue higher thus indicate period infect sir ordinary qualitatively behavior however make bridge early epidemic general form liu di sir force bi base infection year explain massive statistical improve directly observable distribution reporting propose reporting rate line equivalent reduction adjust multiply un show panel unknown panel adjust birth panel line skeleton list calculation simplify solid panel figure show two panel bi important birth middle birth massive relate cycle birth rate relationship source theory york year year cycle birth piece cycle five equivalent birth transmission massive quickly change birth fail capture worth aid show satisfactory matching investigate cycle birth peak code peak correspond birth birth become birth cycle three year cycle even cycle fitting model perhaps capable birth rate thereby support sir box either parametric free instead way match notion absence early attempt estimation ahead method one ahead prediction application absence prediction secondary evidence stand medium aim horizon say rest nothing take observable give estimate minimize term sample design k rare condition estimation alone extra information exactly indeed exploit approach concrete approach issue specification suggestion experience would connection bayesian statistic need experience especially area relevant reference include al al form criterion model comparison multiple series among fit might appendix theoretical justification however se strongly mix sequence exponentially moment w marginal fy moreover linear fy x marginal g h old cx w ease otherwise large x e note w cx distribution hold triangular array due lag introduce denote quantity pn
appropriate value freedom play response vector modeling produce freedom fit define element estimator combination response matrix effective select tuning parameter ridge lasso penalty express form unbiased degree regression take proof suggest e unbiased expect model select minimize estimator freedom criteria consistency bias correct cross cn generalize introduce furthermore modify ease burden algorithm generalize path closely variety condition condition member absolute value example penalty include elastic minimax elastic net many convex include denote tuning path start solution value continuous coefficient monotone jt jt remain unchanged e jt jt condition first jt I update direction apply produce iterative algorithm loss kt orthogonal ng jt orthogonality times absolute monotone square follow degree freedom iteratively calculate trace initial mt kt freedom freedom easily derive orthogonal orthogonality define small close get degree increase degree freedom degree equation little square th coefficient become kt ng kt coefficient appropriately freedom algorithm freedom jt ps jt kt jt compute equation suggest step matrix inefficient simple cost want generalize degree freedom store qr implement write orthogonal triangular write th equation freedom extension bayesian criterion generalize validation via degree package comprehensive investigate effectiveness generate datum set correlation simulation degree adjusted tuning compare degree freedom penalty selection criterion give elastic elastic net detail present generalize degree et freedom freedom selection result deviation sd se th zero say coefficient et procedure tend incorporate et deviation sd percentage non propose mse sd selection criterion correct aic bic generalize cross information yield paper bayesian criterion need square complex leave validation expensive validation compare penalty elastic net elastic net elastic net elastic eq produce ridge elastic approximate penalty difference elastic similar lasso al directly elastic net deviation correctly say criterion elastic follow yield elastic sparse correct good mean perform g dense performance bayesian bic cv elastic family excellent validation estimate error unfortunately generalize elastic solution penalty case aic sd nn sd mse sd ex sd mse sd nn ex nn sd nn ex cv sd sd nn ex sd mse update na I various change penalty carry os produce solution speed base na I slow degree large na I average I na I modify I na I modify diabetes ten baseline predictor tc response baseline penalty elastic elastic degree freedom decrease penalty increase degree path nd helpful degree rapidly update algorithm large make substantial change degree estimate standardized diabetes elastic net elastic degree generalize elastic net yield elastic intercept age map tc procedure via method wide penalty carlo simulation procedure tuning yield especially penalty elastic parameter include generalize linear multivariate tuning parameter future research large mathematical appendix us te new expectation te equation new algorithm first consider write small function sign jt programming mm principle international statistic l heuristic instability reweighte p apparent error rule penalty cross least angle fan li tool h ann classification report university regularization descent j software fu regression regression improve freedom multivariate department stanford team language foundation schwarz ann assessment stein ann n correction shrinkage wang li tune smoothly absolute deviation li shrinkage tuning measure effect model lin group zhang nearly penalty ann g yu ann regularization net adaptive lasso oracle r degrees ann guide journal author manuscript option head source file please examine carefully follow formulae reference etc closely please acceptable requirement large amount manuscript delay publication attempt correct production team paper please department country mail address address publication put california sp publish side probability drop beyond paper elsewhere document web essential place side subsection use divide appear subsection end paragraph mark extra character line line correct production english use mark direct attribute mcmc ml dna ordinary upper efficient method method well special symbol text mathematic rather symbol comprise letter english autoregressive capital letter aic macro start publication delay failure journal write idea clearly reader time sentence nr cg alternative force energy parse focus underlie mathematical expression please carefully display display expression brownian motion derive proper name square error mean square minus sign uses join double person user range join name people minus mathematic remark place phrase cut equation refer number place formulae save line text display likewise expression leave expression avoid g matrix unless vector avoid preferable capital care please iterate sign power complicated quantity multiple avoid symbol align var trace expectation please point use list give range integer denote whereas range number appearance expression display place display mathematical reference production elementary comment find standard reference axis label page use avoid axis graph environment roughly usually page ensure label letter font production panel type symbol appear please figure colour essential care journal figure refer reduce error sentence example pc refer check effective page avoid reason character wide include minus sign character arrange without table improve multiply power ten respectively save effectively code example table digit justify gain clarity error measure common phrase standard census percent percent r black white black american u united comprising stop description line symbol appear verify agree line correctly line symbol body brief
broad insight range achievable suitable review relate binary parametric family accommodate motivate type quadratic adjust family specify compare correlate index sub vector denote arbitrary I moment coincide first moment inequality know several article map I I ii dm ii exactly fr symmetric definite derive moment maximize valid specify da multiplier complete coefficient valid elaborate compatibility seem easier express binary coefficient feasible correlation coefficient difficult statement notion independent dimensional uncorrelated uncorrelate p p p mutually uncorrelated independent since correlation binary form ff u use term coefficient x I regard metropolis hasting mention result proposal auto metropolis hasting let metropolis hasting auto definition expect value obtain triangle inequality yield ix structured dependency average ij article correlation family sampling correlation high require therefore conditional value triangular conditional monotonic straightforward wise one auxiliary allow come back idea ht q p discuss conditional truncate link structure matrix fails accommodate complicated conditional valid structure conditional structure differentiable cross value triangular link probit dy proof moment b moment r monotonic jacobian proceed induction scalar conditional construct conditional since suffice show column contain inductive family general suggest motivate logistic arise triangular h design parametric marginal distribution family q identity non q quadratic procedure logistic conditional original exponential behave function absolute thus exponential hard fit correlation dd cross moment propose fitting dependency space limit low dimension sequel structure enumeration secondly approach adjust cross add new moment triangular exact expectation expensive replace remark ij become limited solution fail converge conditional moment rather usually virtue lemma version feasible cross assess di adjust parameter newton suggest bivariate distribution proxy might start solution copula attain small alternatively near frobenius integral therefore easily incorporate introduction conditional logistic copula moment fit parametric record sample meet alternate sd dm di dd dd replacement consideration inverse dim di dim ii ii ic dim dd I permutation approximately uniform draw matrix sampling moment might scope parametric introduce difficulty shrink uniform cross matrix entry moment parametric adjust might numerical experiment find frobenius qualitatively picture conditional replace exact carlo concern conditional family explicitly sample cross feasible loop dimension order random median show group family scale axis represent em suggest scope conditional far practical logistic accuracy compete conditional family fast grow flexible conditional besides allow evaluation conditional far finding comparison carry particular part author ph thesis supervision family specify mean correlate pt pt definition procedure class family
coupling succeed recall event coupling fail imply combine prove proportional coupling pt comment second coupling prove pre cutoff careful actually prove cutoff continuous total stationarity choose since coordinate time improve bit vector start chebyshev observe entry prove mix worth point argument go change large distribution include step cdf show total essentially argument analogue indistinguishable know analogous markov box hold ball every putting remainder give greedy give wide couple create practical way obtain point simplex obtain simplex much hard past monotonicity anti monotonicity begin detail choose start markov chain space couple single couple markov coupling eventually bad inefficient property property chain easy coupling track start maximal minimal markov infinite course keep start monotonicity obvious tracking little copy start couple chain chain jt jt jt tm jt jt jt jt tm jt n j x j give q apply q tell extremely second identical run choice representation attempt chain merely substantial coupling use variable perform determine couple otherwise n n j analogous coupling subset probability measure metric coupling chain couple subset fail perfect coupling succeeds start epoch record information fail epoch rigorous rigorously estimate fail run distribute binomial stationarity author thank david author thank lemma support determine gibbs sampler conjecture couple contraction markovian coupling perfect gibbs sampler hard analyze measure obtain monte mcmc period analyze difficult receive exposition sampler general geometric coupling technique build technical understand simple detailed analysis nice analysis walk walk application analyze namely addition use powerful non markovian coupling past initially toward sampler complicate contingency author thesis technique paper improve analysis still substantial room improvement paper popular variation measurable start simplex whose stationary take move chain choose uniformly first show confirm demonstrate cutoff moderate simplex satisfy ccc step transition measurable hand prove bind closely relate chain idea wider marginally work coupling coupling assume accord tx throughout marginally sampler joint evolution time method markovian final move likely distance large amount probability coupling describe coupling coupling always update entry coordinate describe couple update complicated chain couple always specific perform proportional coupling otherwise assume generality condition list read start coordinate chain burn copy markov proportional generate measurable pt expand collect induction obvious n decide simplex go infinity first pair vertex jt jt jt repeat define construct partition let nest either define mark mark marked time whether graph connect follow place enough graph fix connect ng ignore vertex p couple coordinate update update coupling mark coupling otherwise proportional coupling mark coupling proceed accord walk time coupling begin show probability coupling satisfy bf assume succeed fy n f bound remain close closeness
bi local recent approach involve type sometimes bi degree work multiplicative merely way two multiplicative relationship frame video also class relation multiplicative bi circuit multiplicative interaction static thought relation product intensity example interaction transformation modern term mapping briefly feature discuss model feature graphical bi connect observable connect pixel unobserve one separately parameterized variety boltzmann machine learn hidden extract image capacity simple become application constrain capacity activity capacity mind common generation synthesis vector minimize reconstruction combine encourage optimization end alternate optimize inference minimize test fix avoid eliminate define implicitly common linearity encoder minimize reconstruction code one penalty term encourage alternatively train auto de corrupt achieve corrupt turn feed technique encoder rbms amount mapping linearity tractable gibbs order training ica one complete encourage response becoming enforce inefficient practice eigen generation mapping typically refer appropriate train advantage stacking eqs affine shall bias avoid clutter alternatively think extra size around pixel reason computationally demand important image deal bag interest sift corner add representation vector classify classifier variation histogram similarity collapse descriptor vector classify classifier scheme local yield competitive performance recognition approach extend encode oppose naive two coding receive would transformation suppose detect net receive show equally logical see example content every constitute act detect equal c less receive spurious activity depend pool amount compute value datum pdf shift secondary evidence hide compute model suggest transformation typically variable figure alternative sub slice connect pixel matrix stack gray intensity commonly define slice tensor leave image turn map shall correlation pdf coding consist could bias connect shall drop clutter simple bias bias dimension coding model sparse code almost code group variable example infer recall code turn function form show inference amount define one simple linear bi inference differ standard meaning versa image analogous expression quadratic function commonly represent function representation factorial factorial contrast mixture make cc xy assign quantify useful calibrate achieve use training dependency discuss detail section standard difference code outer way utilize view conditionally input case contribute possible similar cf iterative coding like rbms machine change normalize consistent define joint training difficult involve auto define learning encoder acyclic use train see figure probabilistic make quantify image model change way rbm simplify long gibbs train model relational auto q force like auto gradient optimization reason allow image intensity optimize joint contrast hybrid train unit higher learn show toy example boltzmann image dot copy shift center plot figure infer vector infer infer strongly figure transformation top image illustrate decompose factorial end parameter thereby parameter cubic assume pixel easily highly suggest reduce inner q factor like choose illustration factorization interesting note factorization law factorization group onto cc reduce project claim find frequently restriction optimally allow multiplicative interaction example factored encoder factor multiplicative interaction restrictive connectivity equivalently advantageous show factored filter optimally class component rotation figure affine top bottom natural training field obtain light filter take page correspond detection plausible motion project image onto shift function add square sum square model temporal sum square component energy thus cosine pair quadrature shift set different translation pick get pdf suggest like layer filter may along function image view layer use suggest adopt ica avoid degenerate ica cf approach shall hide layer analogy factor factor boltzmann art motion closely connect term eq activity mean unit think unit variety simplify interaction practically field filter slowly maintain way maintain normalize every connect top level full connectivity help unit virtual grid space neighbor popular mainly learn common patch center contrast usually also detect rotation train transformation code represent shall restrict attention transformation identity also know practically translation shift transformation important eigen complex eigenvalue multiply amount complex plane pdf orthogonal thus equivalent project real eigenvector rotation invariant iii projecting decompose rotation contain secondary elsewhere fouri rotation subspace amount fourier norm fouri signal translation part matrix fact come commonly refer quadrature pair detect eigenvector rotation refer eigen transformation central way transformation angle rotation within share well set eigen transformation less spatial rotation set eigenvector transformation eigen differ angle result may angle rotation eigen projection onto denote projection axis plane projection cosine angle normalize projection cosine rotation need normalize amount divide unfortunately case invariant rotation close axis rotation ignore provide recover impossible angle q rotation allow preferred angle denote maximal angle rotation idea well subspace rotation meet projection represent subspace via ability suboptimal representation detector take compatible detector image computing sum subspace encode subspace rotation pool stack eigenvector respectively subspace appropriate meet row row filter rotation inference cf pooling identify thought multiple learning involve orthogonal subset learn dimensional subspace form filter albeit canonical correlation cca canonical tie pca tie weight model simultaneous transformation imply length impose non compute mapping interpret provide content representation help noise rotation detector equivalent square subspace thus projection contribute transformation make response response invariant projection detector prefer rotation detector concatenation model transformation modify term adjacent markov alternatively energy compute square implement train video move dot auto encoder concatenation constrain frame dot move speed speed vary movie auto two effectively implement absence structure implement figure model learn spatio temporal fourier frequency orientation cccc concatenation random dot main utility feed network may play task cross requirement currently correspondence task extraction descriptor removal system replace single train help visual variety homogeneous module correspondence invariant input orthogonal encoding amount graphical manifold distribution feed effectively transform canonical pose represent canonical pose help explain filter recognition selective similar rotation feature show feed recognition somewhat recognize static biological move pose compute product automatically associate object train frame analogy argue analogy making phenomenon question make module building capability sparse energy standard cf
n xshift q xshift cm n n xshift cm r edge double n double xshift edge double n edge n xshift edge present obtain p select path fully length first line enforce choice completeness make heuristic identify make enforce among left pt maintain throughout return minimal input equivalent lemma ps take unary view substitution replace decompose atomic brevity rule child edge p atomic identify long would discover incorporate query convert line contradiction argue lemma p qp learnable example set challenging unary unary select indicate path query example boolean query construct problem devise satisfied recall query class head boolean boolean query boolean would path xx xx xx xx xx ps ss trees l p r r return I path result element arbitrary late presence singleton ambiguity xshift offer n item n n offer edge clarity presentation path item item query whose string choice negative item sp boolean query tree sound completeness algorithm sp completeness sp prove class learnable resp return minimal boolean necessarily query minimal query boolean conjunction fix q ci boolean path em example q consist another path query reconstruct query speak formally path query note become embed ignore htb xshift double edge xshift edge double xshift double edge free element path algorithm present extend query xx xx xx xx xx ss l return satisfied never sound exactly time query illustrate consider article book xshift cm article title article author title title return yield give title author title consistent title em query minimal produce minimal perfect tree child parent path completeness construction minimal element algorithm query fusion moving never leave finally exist leave subgraph path leave claim sp xshift cm xshift yshift xshift xshift yshift cm xshift cm x edge yshift yshift xshift cm yshift yshift xshift edge yshift cm cm yshift xshift edge x x block v x separate one construct consist formula tree tree encode leaf ensure separates form encode input remove example part technical separating hold edge free consistency pattern np adapt np remark expressive interpret unless class learnable polynomial learn language inspire tree area language learnable g language language learnable fact class finite learn restriction g reversible language occurrence expression important query come expressive path query relatively responsible learn path query unary query inspire survey extra character instance match nonempty possibly empty match unary need use match instance pattern query behind inference unary node infer schema information pruning handle annotate advantage expressive properly query drawback scenario heavy formalism exist easy visualization infer include class path language technique infer convert unlikely successful translation task consider beneficial avoid line restriction would easy translation approach would require modification would constitute language learn begin construct generalization pilot generalization similarly word inference query incorporate example program inference use direct membership query refine infer framework query contextual query contextual language language condition exactly relative child tree language order finally study relational database language offer structure database instance query construct condition tuple table study infer query investigate several boolean two one example sound complete path path query hand inclusion sample believe informative investigate advantage allow query example learn produce direction add union quite decide clear technique interested operator impact restrict path define semantic query match connect path query semantic notion match produce moreover finally enable take schema cf schema explicitly result algorithm testing general hardness modify limited schema tailor section institute investigate learn query query take annotation must general user annotation one always return every practical unary learnable add picture application basically leave store easy document need require exist core query allow file language framework user formulate knowledge specialize address gap remark however set exchange query define mapping specify pattern new source discover another query annotate query select accordingly annotation produce annotation may current identify type annotation node must select annotate every term computational setting one instance document annotate htb library book edge book edge r title author title edge capital edge n n edge edge n receive annotation properly annotation fit work consistent consistent annotation mean influence computational inference learnable take I query sample trivial discussion learn query sufficiently informative precisely role teacher rather add satisfactory commonly sample I return extend restriction impose ensure unary query investigate unary document boolean certain case purpose boolean positive simple feed offer web site annotate htb xshift cm edge edge xshift cm offer item edge edge xshift cm list type n edge pc boolean annotation select unary query example presence path query learnable algorithm inclusion minimal algorithm try construct respect algorithm learn positive reversible regular language occurrence return consistent sample query consistent minimal learning algorithm query see approximation path query identify enable path give annotation trivial example query select consistent result consider query restriction admit positive define establish theoretical addressing additionally investigate construct minimal check consistency learnable unary path base nontrivial way remain nontrivial organize introduce notion define learnable query present learning discuss negative outline direction restriction proof acknowledgment would anonymous helpful comment thank insight improve support education de region project education research n throughout document order extend define canonical iff document tuple tn child acyclic require non relation tree cardinality node tree leaf path node view path word term n n n n xshift represent answer distinguish distinguish tree select node indicate rarely make ambiguity structure query query may additionally distinguish child correspond axis unary distinguished selecting n n n q q acyclic require exactly boolean query boolean distinguish edge line line box restrict unary query one unary capture exactly positive sequel use unary node unary query virtual way mapping query say relation symmetric match embed match simply embedding tree htb n xshift edge bend densely dot bend bend leave densely dotted bend leave densely xshift bend densely bend densely dotted bend densely edge bend densely require embed node query map path query note self typically semantic unary query nod unary define query naturally notion closely belong notion extend condition replace n p converse general b complete whereas identify minimal tree term inclusion ss standard adapt query comprise concept learn instance concept case tree possibly semantic concept tuple example unary tuple define setting auxiliary notion query take learn two return special query extend often learn fitting requirement insufficient eliminate instance return universal sound require complete analogously sample later exponential size section learnable essential enable implication search refinement logical capture property polynomially query serve characteristic property learnable merely direct learnable open question example set form learnable query match characteristic construction match generic symbol construct match contain child every path label contain unary ab edge edge xshift edge xshift characteristic construction seem artificial match simple sample path query path query node main reason work embedding restriction node boolean query impose technical lemma impose restriction edge unary query b restriction impose node boolean ab unary query properly query limit query basically discriminate depth alone restriction boolean query equivalent note however boolean query query technique unary boolean lemma claim structure path b boolean unary path query occurrence claim unary let construct map root map b node belong map preserve label follow take therefore node
utility ir observable property observable usual observable value intersection hypothesis use state formalism quantum non state ir relevance call alarm call false alarm false alarm acceptance induce acceptance maximum power note pearson ml define powerful acceptance decision provide nan alarm detection q space vector field set linearity set scalar vector vector transpose product dirac notation reader refer brief illustration dirac observable observable vector correspondence correspond observable vector state define distribution hypothesis play role ir suppose variable relevance alarm document principle ml test cut false recall detection equivalently document region rank relevance recall observable observable vector probability relevance however scalar clarity binary relevance relevance belong either term basis establish relationship observable orthogonality superposition physics superposition observable much relevance illustration distribution binomial approximate observable document mean term document poisson give large equal observable term collection indeed probability frequency encode parametric term pt false alarm acceptance relevance difference subscript swap correct decision coin discrimination alternative minimum minimize maximize yield power minimize curve region relevance region rejection document possible I term alternative complement pearson subspace orthogonal subset yield operation orthogonal suppose yield new subspace reformulate observable something three subspace ray subspace plane dimensional span provide l l thus hence subspace key optimal way powerful region subset yield ml powerful state high eigenvector optimal region observable vector spectral vector geometry decision state suppose observable vector define span relevance correct angle observable vector vector figure illustrate geometry observable reader generalize angle observable relevance relate angle angle vector hold illustrate replacement observable angle correct q cosine angle poisson shape correspond shaped curve plot relevance relevance less high relevance prove side lie angle relevance poisson apply tell whether observable even estimate tell false alarm correctly acceptance induce observable poisson term thing equal estimation outcome observable term either term relevance function induce observable corollary compute optimal computed acceptance region observable observable distribution document false coordinate orthonormal basis observable coordinate express angle provide angle error error minimum observable observable vector detection maximum observable thus accept p p observable span optimal actually sort every test illustrate aim realistic experimental computed state measure rather aim probability every superposition curve like label probability shaped curve part different word never shape curve curve label error label figure topic type exhibit pattern issue optimal observable finding absence device frequency straightforward occurrence physical measure device sufficient much physical correspond frequency document outcome correspond observable answer automatic indexing retrieval observable retrieval indexing interpretation observable provide superposition interact place observer upon whenever trace induce valid observable subspace span however logic combine subspace combine subset say relevant relevance measure say possesse measure observable set describe document term index orthogonality vector implement mutually observable hence say union intersection complement ir capable observable occurrence document specify thus achieve accurately possible ir capable observe observable ir observable document decide observable shall pay attention would ir van book book theoretical foundation optimal book quantum quantum retrieval find formalism ir aspect paper research
isotropic therefore u estimate logarithmic moment line let eq inequality fix subgaussian design response satisfy eq subgaussian zero mean invertible square eq jensen inequality q define triangle inequality follow bernstein martingale follow variable follow give tail freedom q eigen decomposition orthonormal eigenvector thus ax freedom random variable proposition remark quadratic form subgaussian setting quadratic spectral following provide multivariate appendix invariance tail random due slightly weak sharp gaussian tail sum sum martingale I square much large fall show completeness appendix difference proposition give constant subgaussian result subgaussian bind deviation simple proof explicit
tt tt tt u tt straightforward covariance function pointwise quantile identically representation sampling lin validity enable confidence band one band substitute investigate sample procedure design normal conditioning failure censor censor study pointwise moreover simultaneous band outside present survival censor smoothing necessary usually square plug lead infeasible say minimize integrate behind censor validity censor use censor table average coverage pointwise confidence computed variance tend coverage deviation decrease numerical reason compare subject availability expand hence bias indistinguishable deviation become satisfactory empirical interval good performance select automatic point probability censor reveal pointwise nominal simultaneous coverage coverage probability simultaneous confidence band stay entry diagnosis death depend receive patient prior rest receive study patient association decrease marker death meaningful analysis simplify marker dependent n far variation provide dependent simultaneous band week summary confidence band classify patient time within non conclusion detect week week respectively figure decrease patient close figure accuracy measure time along line appendix derive converge pointwise simultaneous reveal figure negligible word might lower discrimination ability cd classify subject week explanation homogeneous survival time cd count estimate survival cd irrelevant receive variability significant enable traditionally derivation load expensive develop technique rigorous justification estimator well variance application detect propose stable estimate sample censoring marker censor propose simple easily derive bivariate estimation estimator always meet advantage totally censor empirical censor survival form occur pair usually focus survival period marker censor make practice flexible assumption conditioning estimate article quite study appendix ix tt tt cf direct calculation side express u h theorem asymptotic establish hadamard result delta van weak van eq taylor expansion mapping theorem imply q reference distribution bivariate censor wang expansion statistic consistency neighbor berkeley auc diagnostic marker measurement parametric roc curve censor diagnostic marker lin functions journal theorems mathematical van asymptotic method diagnostic york sd se cp se time se sd cp sd se cp se sd se cp sd cp sd se cp sd mean sd pt se cp sd sd se cp se cp time sd se cp sd se national assess diagnostic result operate characteristic roc curve widely summary measure generality life extension event usefulness disease detection compute expression process provide essential evaluate cd cell patient survival clinical inference dependent patient receive signal detection diagnostic clinical beneficial sake improvement diagnostic compare roc false purpose diagnostic test advantage curve specify moreover invariance characteristic roc scale suitable base move application auc area roc might entirely capture roc curve auc totally performance furthermore drawback range practically sample property estimator comparison future address estimate develop procedure loss derive censor represent censor status ty perfect interpretation rescale explain value marker censor
return high feature lead place lead high whereas retrieve observation pair thus design translate provide occur event partially satisfactory cause reach optimality without cost superposition coordinate unit quite provide expression reach optimality basis therefore feature new value region superiority pure rank induced region follow pure case differ answer mixed region acceptance whereas pure counter rank discriminant associate associate pure equation rank detection false alarm state find density define coordinate weight estimation occurrence hypothesis ir whose state bm weight drawback weighting bm tuning database thus rather problematic illustrate follow accept true false alarm threshold formulation rule indexing currently bayesian estimation quantum mechanic book quantum example interference processing start area ir inspire report within paper however management context classical quantum principled retrieval ranking retrieval request decrease request effectiveness system user principle classical probability keep subspace acceptance rejection probability unit yet address paper although perhaps rank probability interference estimate dependency relevance rank rank quantum effective one ranking contrast address interference estimate classical probability effectiveness probability stem region acceptance quantum interference contrary ranking optimality upon subspace relate framework view paper formalism feasible ranking formalism comparable bm classical database query survey query plan execution classical concentrate answer tuple assign probability database paper incorporate management information analogous management future unit may insight quantum investigate crucial understand paper theorem management database index retrieve information tuple document pattern profile meet relevance usefulness unit unit thus law admit total law perfect optimal question ask improvement obtain theory set unit differ know false quality ranking give false alarm detector classical within management implement effectiveness recall estimate probability management retrieval ir ie store retrieve tuple document system management uncertainty rank crucial deal uncertainty aim assign relevance usefulness unit place measure relevance theory measure end necessary represent prediction management retrieval database measurement predefine hypothesis calculation probability false alarm management thank rank perfect unit theory event kolmogorov axiom theory describe hermitian quantum entail classical set region rejection indicator base detector detector quantum imply phenomenon investigate ask whether quantum theory outcome principle available known theory compare quantum detection classical ranking quantum rank section interpretation region provide work also quantum view definition exclusive event event sake clarity introduce simple occurrence binary relational usually represent mutually exclusive scalar scalar product dimensional vector event must inner event binary vector symbol representation must event difference start point paper algebraic hermitian self adjoint operator use quantum mechanic prefer matrix sake clarity prefer operator hermitian transpose hermitian important hermitian quantum eigenvalue hermitian correspondence vector conjugate transpose event mutually trace axiom function hermitian particular space collection event space unity orthogonal latter term resolution unity dirac notation appendix represent mutually exclusive hermitian event system particle system system color ball internal device e color management physic management query click tuple attribute state review allow algebraic approach rule introduce consider event occurrence alternative list arrange matrix matrix zero probable otherwise mixed pure correspondence outer transpose concentrated certain help event provide hermitian finite rank mutually spectrum theorem pure one say hermitian decompose pure matrix hermitian probable mixed pure whereas estimation another management predefine categorization whether display engine page put engine page database decision computation associate illustration necessarily brief simple aspect bring elementary retrieval quantum information unit number review make number example calculate engine function clarity frequency interest customer store mathematical implement decision unit hypothesis generate hypothesis system hypothesis density label management customer irrelevant engine item customer irrelevant user depend density decision pearson instead without whether hypothesis threshold reject otherwise accept nothing accept high alarm threshold pair give size curve receiver operate characteristic pearson partition distinct region accept acceptance region observable region pearson instead utilize acceptance subspace follow region spectrum identify theorem function positive observe place usual deal one however mixed eq eq q absence acceptance correspond respectively never accept accept feature accept topic categorization furthermore build describe either include unique feature occur yield pure replace distance density cosine justification fact angle space subspace unitary pure mixed detection alarm pure curve counterpart yet evidence alarm interpretation interpretation tie pure follow rule decision rule estimate alarm let probability order pass lagrange interpolation pass lagrange power plot produce decision suppose ten management binary h computation follow interpretation latter
various optimization plot result size pass method poorly even pass hybrid solution visually nearly indistinguishable still possible obvious deterministic stochastic inversion van image place surface emphasis application analysis objective function prescribe depend accuracy implementation rather intermediate approximate particle particle thus provide acknowledgement thank suggest le valuable grateful anonymous comment lead detailed mistake theorem l l grant schmidt author grant research involve offer subset progress approach solution contrast steady expense hybrid exhibit benefit incremental gradient rate quasi experiment illustrate potential benefit often function measurement choose square q gaussian logistic rise equation often amount uniformity measurement mean evaluation unnecessary progress motivate method evaluate iteration gradient time incremental achieve rapid initial progress accuracy higher full dominate incremental exhibit benefit two incremental preserve incremental version iterate update q residual gradient deal typically converge suitable gradient estimate residual specify iteration characterize convergence deterministic noise context refer influence scale account curvature g quasi newton minimizer overall strongly continuously ratio number imply value characterize every iteration asymptotic example characterize emphasize note imply converse imply convergence analyze iterate sufficient strong convexity imply eq rate uniformly positive definite constant euclidean replace scale divide imply sublinear convergence describe ensure rate achieved arbitrarily require convergence deterministic gradient sublinear show maintain sublinear size control achieve effect choose size allow control error implicitly idea line implementation fitting compare incremental base spectrum full expensive fast incremental sublinear iteration dependency give stochastic oracle contrast convergence satisfy iteration incremental gradient converge incremental achieve close spirit convergence treat pass gradient difficulty et memory grow strategy practitioner explicitly mention grow aware present propose quasi newton previously analogous weak extend work require strict decrease allow realistic lead asymptotic stochastic source file reproduce consider basic iteration iteration inequality q obtain eq low left hand side give aside require convergent sublinear reflect noiseless convergence absolute suppose iteration recursively choose trivially arbitrary implication linearly zero q yield scenario gradient appealing achieve nesterov strongly possibility bind imply limit error sense counterpart expect expectation proceed construction come require convexity gradient lipschitz trivial current optimality positive control strong bound imply case computed expectation low linear follow expected rate bound fast extreme mean long bind seem allow iterate maintain strong iterate solution calculation fast trivial bind valid although practice well norm gradient deduce proxy magnitude gradient residual decrease heuristic decrease increase likely remove deterministic sublinear assumption sublinear preserve average sublinear rate bound lipschitz continuity convexity deduce hand simplifying obtain result side nonnegative show auxiliary enough ic recursively recursively side get eq imply thus get imply sequence convergence achieve deterministic constitute typically single element choose cyclic sample sublinear implicitly improve sublinear complement term portion sample thus leverage result next optimality linearly increase induce sample towards q obtain furthermore nk difference consider control size ensure bound iteration fairly use irrespective manner g sample obtain tight however suitably modify related gradient use eq q q side bind initially zero illustrate sample size requirement e b size schedule cc far make modification scale search attempt curvature quasi newton approximation hessian system approximate maintain use limited maintain recursively formula recursive bfgs update ensure bound condition scale hessian gradient analysis requirement part ensure sufficient decrease design attempt balance rigorous none procedure quasi newton method initial trial represent observation yield decrease increase eventually reduce conventional true guarantee eventually couple choice newton hessian local convergence summarize incremental gradient grow sample series fitting application summarize structure random general first outcome choose likelihood maximize regularize satisfy function upper available particular application least last datum central relevance digit structure crf noun crf image nonlinear compare conventional limited memory bfgs hessian study type memory quasi newton logistic incremental constant method prove advanced crf compute satisfy interpolation number grow linearly include deterministic plot progress pass evaluate model application feature outcome goal parameter inner give q typically add equivalently take value compactly form lie nonnegative hessian rank combine binary logistic regression describe token email whether email spam datum plot figure across pass power give sensitivity advantage deterministic hybrid hybrid make rapid progress unlike method make steady progress deterministic agree multinomial
cm xshift cm elliptical vertex box box specify box specifie uncertain calculate calculate see right leave plot next find fortunately see minimum conclude proposition node leave x node interestingly low actually cumulative expert say along although name water water level study length parameter uncertain expert scale calculation introduce uniformly transformation coefficient mode uncertainty water measurement triangular distribution actual equivalence box box calculate depict immediately apparent node log form need safe side traditional instead box independence expect much wide lead lead weak maker verify method box assume analytical uncertain course box operational modelling inference box totally represent coherent natural consider box totally space thereby box infinite box space number paper calculate box prove natural extension box additive component respect topology equivalence corollary completely calculate natural suffice follow proposition box whose mapping box particularly attractive correspond region shape combine box box rule combination thereby dependency consider extreme theorem formula derive arithmetic obtain approach demonstrate inference harmonic assessment calculation generally open box even marginal choice lead dependency arrive bound p box possibility measure acknowledgement grant grateful extremely suggestion various finally constructive comment presentation substantially example remark cdf curve I I curve I type I I I type I I I I curve point type I lower cumulative function box theory expert quantile formal already new construct inference theory heavily extension independence particular arbitrary totally thereby p bound set dependence fr extend arithmetic practical feasibility uncertainty information allow include upper possibility necessity probability coherent unlike classical probability complex functional description possibly expense ease upper box cumulative distribution essential aspect box include arithmetic efficient inference box connect technique applicable box arbitrary yield problem signal coherent work generalize many equivalent set finitely study box lower enable p box problem concern come tool reflect consequence extension many extension box new tool inference arbitrary bound quantity box low mention box totally box box box continuous perhaps even importantly handle consider admit p box space box build dependency express construct unlike box restrict rather box space exact inference extension arithmetic real provide multivariate study model point natural extension event topology equivalence natural extension gamble via box real multivariate construction box coherent dependency model special unknown probabilistic approach demonstrate infer harmonic parameter box expect box end introduction model shall also brief summary call gamble denote context value elsewhere confusion subset interpret transaction assess collection gamble argue subject belief usually conjugate upper subject infimum price low gamble coherent see ff f pf linear event finitely additive lower let dominate coherent agree envelope see consequence envelope lower coherent pointwise coherent agree pointwise natural extension conservative coherent thereby minimal consequence extension natural extension set gamble gamble include natural gamble usually coherent interest lattice gamble wise wise formalism totally finite space instrumental natural extension restrict box write exactly similarly simplicity always moreover mass happen make sense topology yet denote induce element cumulative distribution definition effectively impose impose decrease thereby box interpret interpret box straightforward box arbitrary gamble natural extension box circle circle draw smooth node node node node node yshift node cdf smooth yshift xshift node node function envelope natural extension gamble conversely belong belong coincide cumulative belong shall proof sensitivity regard knowledge cumulative trivial yet useful box natural obviously extension instance extension dominate least conservative devoted natural simply expression clearly event field extension introduce element however apparent model stick coherence nothing gamble shall finally gamble precise box finitely additive finitely include proposition box eq cumulative secondly satisfy kx k satisfy deduce fx k expression calculate aa use conjugacy determined box coherent field unless shown given define aa deduce eq coincide even define total eqs denote cumulative box arise event calculate cc supremum event arbitrary calculate interior closure respect closure also component operation commonly addition complete component interval immediately yx z constructive calculate case connect case usual case precisely continuous immediate take immediate eqs conclude form component complete full particular square naive way decrease constant element think write box triangle whose diagonal box need define marginal box partition topological interior eq illustrate inference probability ordering poorly choose approximation event interest cycle xshift cm node xshift cm cycle scale cm yshift right cycle instance ordering square course entirely obvious discretize natural strategy reference modal correspond interest p box totally natural example give gamble express simplify low gamble topology show totally space finitely completely complete monotonicity envelope monotone necessarily completely envelope monotone box result box space coherent clearly monotone denote restriction completely satisfied monotonicity gamble gamble immediate event coherent monotone gamble coherent low monotone turn every fortunately way topological eq natural interior gamble easily interior closure ff eqs use calculate must simply component cut upper calculate integral monotonic way mapping whenever element think cumulative function think box type eqs topological closure call multi see sometimes inverse full c corollary eq result union equivalence component clearly large large must contradiction gamble immediately q section construct box coherent hoeffding assume factorization arithmetic special natural coherent low construct box induce perfectly mapping induce marginal box induce roughly speak relative effectively mean affect inference choice nothing prevent theory induce p box quite complicated see feasible box represent coherent low consider rule coherent coherent gamble couple eq conservative theorem box whose natural dominate follow envelope distribution linear compatible completely structure rigorous discussion eq conservative box natural immediate box box outer close domain consider extension event monotone box monotone contrast independence refer rigorous box dominate joint box outer approximation joint domain extension writing dominate dominate
dirichlet monotonic strictly monotonically define ff f mx mutual depend concave eq eq lemma let q upper increase trajectory minimize eq little irrespective process namely expect information thus subsection bounding subsection lemma np horizon namely abuse time horizon follow inequality decompose capture difference second capture side decrease simplify let operator look k maximize deterministic history trajectory probability ta clearly put constant thus central follow eq compare let proof difference decrease however alone converge environment consist connect component environment markovian transition kernel ensure go half imply always state positive propagate state iv choose decrease increasing imply finitely simplify fix visit happen finitely imply visit probability get visit finitely ia must reach discount visit choose number turn last union nan event nan one hold surely value bellman let initial follow bellman contradict law union combine produce iv agent act illustrate environment consist densely clique agent action clique generate goal clique consist state compare select information gain reward iii action iv programming exploration follow mdp reward gain four clique greedy dp clique exploration clique reason gain decrease attempt plot early accelerate history lot attention artificial progress divergence back optimize future expect differ explore environmental design focus query affect environment principle highlight necessity balancing gain progress conceptually motivated agent environment external permit present exploration environment center sound foundation design dp optimal mdp gray need initially answer environment send environment agent carry build select greatly measure shannon agent least optimally choose cumulative maximize special mdps experiment exploration approximate programming follow review basic establish terminology exploration focus exploration mdp briefly review suppose agent environment cycle action receive string refer facilitate environment initial capable light summarize bayes dp environment worth require certainly sign gain history grow kl respect write gain additional action view distribution represent encode element sample optimal coding cause update recursively information proceed action mean action environment life extend agent action action agent gain action formally recursively note definition action reinforcement particular h expect gain notation equation immediate situation end cumulative give history optimal namely monotonically length increase always close never reward accumulate p recover mass dirichlet kl compute accord gain gain increase exploration constructive readily implement finite maximization compute respectively immediate large look ahead infeasible heuristic practice restrict maximum span define infinite limit life without coin cumulative increase simplify discount reinforcement force environment without discount ahead exponentially policy eq optimal discount add discount value horizon case however scenario discount fail
ahead horizon cone specialized set horizon cone large hull affine denote small contain hull cone transformation convex cone get cone corollary transformation close b bn b k n bb k b k b v k corollary subspace theorem characterization effective domain represent many relative iy iy side converge stay clear union finitely many unbounded subset elementary must integrable tu cone k kk yu b b cone lagrangian feasible cone point since cone active column rewrite explicitly slack slack nonzero equation kkt entail operation reduce triangular always ip iteration follow diagonal positive diagonal invertible proof eq immediate diagonal diag diag diag I ia kalman smoothing context diagonal density moreover invertible finish reduction upper triangular definite tt block invert form time remark conjecture penalty optimization huber interior reconstruct dynamical classical penalty robust fast system dynamic e jump addition penaltie huber loss paper model computational kalman smoothing develop establish interpret penalty present kalman maintain extend computational efficiency broader contain mutually interval unconditional estimator column become compute kalman provide circumstance linear penalization difficulty reconstruct jump great impact machine selective shrinkage compressed circumstance smoothing replace insensitive support regression interpretation machine year lead review representation relate scale fundamental condition allow view ensuring penaltie huber penalty penalty derive new smooth penalty measurement deviation point perfectly degeneracy solution theoretical foundation interpret generalize formulation theorem tucker kkt penaltie necessary result support theorem remark recall definition hull affine cone cone cone horizon quadratic penalty nonempty familiar penalty huber hypothesis proof quadratic density always contain characterized th piecewise piecewise function dim true density affine adapt order consider case sequence notation definition also q block kalman problem mean def decomposable prove derive recall extend work tucker interior ip relax specifically decrease ip relaxed theorem main show computational ip time kalman come e penalty appendix show solve preserve usually practice motivate huber verify ml huber respective kalman nonsmooth penalty
apart past explicit quantify decay move definition mix strength dependence mix regularity speak precise lag separate independence numerous learning rely dependent focused derive via consistency approximately prove iid remain pac bound rank finally stability input generalize exist iid mixing mix coefficient knowledge researcher specific coefficient mix risk empirical generalization compete prediction machine know satisfy explicitly mix accord mix begin derive addition convergence markov application result desirable series mix rate certain see overview rate give mix regime present mixing time define coefficient consistency estimator intermediate convergence histogram input conclude proof fix sigma distribution denote many definition however intuitive variation say absolutely speak say total joint separate supremum unnecessary time negative integer subsequence nearly iid block entire odd even block sequence see main two stage recognize finite lead version infinity observe dimensional histogram joint dimensional construct histogram detail grow bound induce approximate establish consistency provide bound error measure theoretic mixing mix coefficient process know case mix integer n histogram leave proof proper choice prove process consistency prove datum histogram let sequence respect essentially apply mixing block widely spaced block particular expectation mix density consider center let union bin absolutely absolutely partial derivative ik taylor fp integral bin proof histogram estimator nh indicator field block sequence allow proof distance density respect dominate let joint process create separate distance target theorem specify rate infinity histogram histogram lemma eq optimal determine n n nd nh nn k gives plug necessary grow supremum unnecessary sigma rewrite notation consist close dimensional marginal eq restriction nest sigma field monotone sequence additional rr supremum negative loss generality theorem exist must triangle eq remain consistent learn know ability despite obvious exploration well drawback convergence provide well understanding immediately quantity define rate tail variation notably mix estimator trick well use kernel joint density
connection identify redundant appear reflect structure interaction value connection demonstrate total correlation measure illustration use information analyze al available activity detail regard production analyze minute minute information create examine train identical label spike record throughout spike joint calculate measure double group assignment group order fire rate development information statistical create randomized randomization accomplish splitting spike remain spike almost preserve fig day essentially total develop information decrease slowly increase maxima day interestingly g steady fig mutual measure among mutual significantly interaction positive exception value group positive investigate relationship redundancy fig goal however redundancy portion redundancy provide distinction redundancy consider appropriate perform unlike develop show correlation throughout redundancy logic find like interaction unable redundancy able detect unable subset variable interaction partial decomposition provide useful unlike measure gate find redundancy interaction indicate perhaps system decomposition entirely highlight define redundancy redundant quantity decomposition similar show redundancy development able several attempt explore property produce similar gate examine difference gain able information illustrative interaction information produce hope system goal would thank comment rewrite add subtracting entropy substitution substitute total applying probability text show ex quantify rise although shannon commonly upon measure develop differ subtle review examine apply usefulness analyze neural early stage aid seek specific research goal user apply area include data compression code name broad applicability part rely value variable although measure approach one relationship measure physical biological information differ subtle way inconsistent throughout array measure clearly system difference information information goal facilitate information measure matlab software available calculate herein previously crucial related information redundancy regard though researcher begin understand provide know something portion know say receive know instead redundancy portion alone alone provide redundancy receive measure quantity researcher create distinct different result overall clearly measure limitation use attempt difference multivariate discuss multivariate theoretic name literature consistent name name name involve call entropy entropy quantify probability near uniform examine quantify give kullback leibler wherein product marginal measure among grouping treat value mutual calculate way consider information variable activity neuron consider relationship amount information neuron neuron consider group condition yield conditional quantify information variable attempt quantify among three attempt gain information knowledge third give eq widely refer redundancy positive imply among interaction imply redundant interaction thus information correctly multivariate redundancy take exclusive entropy entropy interaction denote interaction refer ci clearly interaction even equal odd become co information number variable present co generalized conceptual mutual information information gain mutual arrive correlation tc introduce vector entropy spatial integration correlation write series correlation dual introduce entropy beyond entropy condition refer excess introduce coding relate another signal stimulus compare associate act beyond conclude relevant correlation assume act state q bayes independent give leibl yes know correlation would question guess one nothing everything name change redundancy create redundancy index maximal provide negative index equal redundancy refer redundancy al original refer refer possible information positive say provide decomposition al partial overlap provide partial produce unlike second possibility redundant unlike brevity partial general original four variable describe redundancy work motivation specific quantifie specific provide individually minimum take consider separately redundancy calculate partial note information expansion information decomposition interaction imply contribution great contribution partial interaction exclusive accord redundant remainder article partial accordance term interpret interpret redundancy redundancy refer new expansion total decomposition contain variable contain equation interaction information summation multivariate measure simple system similarity system contrast measure produce offer note due boolean logic total equal boolean redundancy directly information measure ccc c result gate exception total interaction redundancy index partial indicate bit might expect know gate gate decomposition equal bit entirely determine confirm table decomposition difference gate produce bit variable amount decomposition find mutual individually redundant uniquely subsequently information amount interaction redundancy return gate produce beyond mutual individually take together beyond individually also gate gate examine value clear use present conclude chance chance specific must equal subtle multivariate measure interaction value whereas discussion correlation gate example demonstrate crucial difference dual treat case gate bit dual great redundancy portion redundancy surprising redundancy redundancy incorporate different quantity section discussion crucial intuitive variable immediately well examine express rhs though explore fundamentally complete alternative measure correlation datum emphasize provide useful information fundamentally ccc c interesting information decomposition indicate gate variable bit information redundant despite fact distinction variable amount contribution redundant mutual individually partial conclude unique bit ccc ex c c example interaction consider relationship know uncertainty additional gain lose know individually simultaneously example demonstrate consider produce eq total correlation relationship unable dual information pass individual together value variable measure ex c entropy either relationship focus interaction involve logic example comparison redundancy decomposition gate produce gate information interaction able difference
algorithm depict visually allocate produce rejection express rejection free present minimize rejection next construct kernel describe maximum arbitrary configuration weight one rejection reversible break depicted maximum subsequent reversible net heat spin transition probability concrete always minimize average rejection rate method construct reversible first method construct show net stochastic matter efficiency however flow introduction extend importance powerful tool degree bioinformatics economic method satisfy principle central concern achieve autocorrelation key effective choice extended method replica exchange method successfully protein problem selection wang loop overcome critical configuration physical determination focus probability total balance impose although attention transition principle minimize rate stay implementation metropolis hasting transition balance balance simple elementary easy find reduce autocorrelation concentrate within far however construct graphical instead surprisingly satisfy balance impose balance sufficient distribution find beyond balance balance picture explain concrete reversible spin configuration locally run whole discrete elementary spin ise candidate next weight configuration equilibrium balance express weight quantity correspond raw flow law total rejection flat w detail I symmetry task minimize visually move amount weight box picture ise fig allocation heat average rejection balance fail rejection mention besides easily geometric
figure show dimensionality reflect class structure clear rand load mat n noise tp tp nan manual eps eps manual eps eps converge noise noise sx end sx sx sx sx sx alpha alpha alpha sx ny ny w diagnostic ny lx w lx w ny w noise sx w w diagnostic converge limit converge false alpha cca end figure alpha color legend xlabel ylabel legend location xlabel ylabel eps figure hold alpha color alpha alpha ylabel retain eps seek representation spherical noise partially explain g covariate leave residual variance decompose call cca algorithm illustrate pose probabilistic analysis covariance datum assume impose center maximize variance principal principal independently place element likelihood form recover isotropic inner rather express model gaussian involve log likelihood low spherical dual paper form motivation partly study residual idea representation form information wish include motivate covariate could expression patient environmental well remain unchanged see place invertible log entire eq respect proof substitute eq see rely regular form substitute singular value dual follow solve symmetric definite invertible define via generalised algebraic likelihood easily primal maximum canonical covariate solve rewrite cca matrix individual covariance generalise generalise eigenvector direction pair space respectively feature rectangular projection equivalence generalise inspection generalise diagonal show cca maximum graphical correlation rotation likelihood subtle incorporate noise project covariance see cca block diagonal component explain two covariance choice residual case expression predict partially explain variation space graphical standard via turn share illustrate treatment series cell alone come group point gene group complexity uniform medical gene profile could represent imply covariance matrix covariance compute residual force find residual component function simplicity provide bandwidth roughly add noise project profile generalise norm projection retain decide cf increase eigenvalue become retain similar definite always ground list bind tp compare area roc curve tb microarray detail explain
begin specification section element domain database output count description qp ad class algorithm finally release later access private release private definition class predicate set thresholded sum set nf tn thresholds output output work distribution class distribution draw example definition full formal reduction differentially release learn description predicate result database depend mild polynomial release threshold simplify universe description distribution release provide free threshold release general statement overview note obtain query distribution accurate answer clarity simplify assume work access label enable get distribution addition access note literature particular harmonic release provide restrict slightly rich allow new release work reduction add grow private theory work connection technique correspondence database want target explicitly e view release set correspond release satisfy requirement distribution release obtain conjunction count threshold throughout fix query consideration monotone conjunction iff general monotone unchange original datum monotone locality conjunction unchange monotone arbitrary original free release differentially private accurate release way monotone runtime size release algorithm query boost release accurate big formally release free release database release differentially accurate release monotone conjunction database boost improvement work consider differentially release marginal contingency table corollary mechanism et release way table run however note guarantee distribution specific general database distribution specific bad differentially yield algorithm expensive database roughly privacy show mild release hold release release database k consistent range formulation give data release algorithm answering query polynomial fix datum universe description semantic description learn target use fouri view query count query fix many database use release w harmonic release query differentially query database size runtime q quite class namely query circuit data release database database polynomial release counting query boolean circuit differentially private release query database runtime q result majority circuit release structure non box note class predicate quite rich example include approximate count polynomial uniform query set query description amenable self answer w leave large query predicate may amenable computation hardness al data release count synthetic universe database universe order consist sometimes think tuple item distance map abstract range differential privacy pd pd class query specify fix database fix item query item satisfy database indicate satisfy close concrete equivalently query u section release set stage definition handle formal definition non interactive giving reduction give allow give item specifie receive back type query however potential access access release universe query description distribution u sampling database database database database release access release unchanged get sometimes release focus release release differentially private must randomize release differ free release boost datum release whose boost result release distribution single distribution uniform query access simulate low need domain unlabele predicate tn access take accuracy parameter integer output g oracle establish private release threshold next capture notion access definition sampling evaluation black oracle oracle access let restrict evaluation access access example answer remark release evaluation weak appropriate simplicity perfectly evaluation access hope threshold need learn pose stronger define next smooth extension reduction combine result concrete new release result private via threshold universe description threshold tn bn smooth running release next point two modification theorem use label use sampling query precise hold requirement e g query answer task predicate database predicate want allow bound query scale laplace approximation bound query differentially tackle learning predicate ii noisy answer sum ignore concern straightforward predicate give oracle thresholded predicate produce approximate thresholde sum privacy consideration thresholded predicate use oracle sum noisy answer threshold sum close restrict margin say threshold w still high reason every threshold threshold h specifically call roughly speak threshold oracle formally query suitably scale ensure return return large output oracle return condition remove conditional distribution useful oracle noisy answer allow privacy fact subsampling threshold privacy problematic complexity predicate predicate try predicate fact sum predicate close subsampling argument hope may namely rather notice threshold oracle goal well roughly query differential privacy avoid margin oracle n continue threshold denote algorithm exposition detail distribution release access denote oracle case sample margin approximation accuracy threshold contain extension theorem suffice receive oracle access overview section formal proof formalize analyze begin describe oracle purpose ensure differential noise access give threshold enable argue agree set terminate ask otherwise previously create fix two query answer differential mechanism answer give answer database database existence subsampling argument main state element far suppose event datum sample oracle magnitude tail laplacian happen two statement either thus else convert private threshold privacy preserve function call distribution learning threshold require denote ik il terminate pose hypothesis equal parameter description parameter reduction keep fix satisfy assumption make query call make query query ever prove produce formalize overall satisfie distribution chernoff terminate put terminate satisfied terminate put mass terminate choice third terminate assume terminate remainder generality sequence place particular learning reject step union bind induce enough lemma choice randomness randomness think answer variable remainder example iteration note since condition z p answer query precision chernoff fact choose assume argue condition internal randomness guarantee probability example sample correctly query ask learn sufficiently invoke claim query case first note put query desire claim direct consequence conclude probability randomness complete proof assume oracle occur three event claim proceed three event occur claim follow property must differ one classify spaced claim finish conclude proof privacy factor label universe description provide learn threshold g u bn algorithm indeed sense privacy correctness identical theorem release theorem release conjunction base threshold throughout query monotone I result differentially private monotone probability accurate runtime class boolean provide boost accurate establish thresholded value arbitrary section boolean multilinear denote hamming main subsection degree lemma follow claim give quantum give claim slightly univariate chebyshev polynomial desire degree denote corollary every integer desire polynomial integer consider previous subsection imply represent polynomial threshold entire boolean hypercube degree closely difference need use notion place width suppose proof follow construction subsection place throughout variable contain sketch directly width conjunction fix predicate replace size width represent proof identical theorem change width leaf get threshold pac see class sum terminology universe query description monotone conjunction distribution threshold bn approximate threshold conjunction learn q u bn ok label recall together reduction release begin data query reduction base algorithm free section count subsection query query differentially uniform accurate database theorem denote datum universe description threshold u u learn algorithm harmonic explain harmonic terminology smooth extension black sampling access sense constant output powerful allow recall precise difference black oracle access priori point query allow harmonic never query fortunately harmonic work box actually extension section equal harmonic return formulation subsection private release broad query namely circuit computed circuit release let differentially release database observe
many type see law rather number power law plot effectively nonetheless compare simulated right simulated point connect value distinguish display across unlike predict lie curve since process necessarily small small blue represent green probability beta actual examine probability blue represent classic black upper due fact process aspect simulation randomness bernoulli power law fast two law predict q merely create horizontal law beyond rank decay fast poisson formulation easy section model empirically handwritten digits repeat process handwritten sample digit mnist handwritten handwritten digit project component apply beta mcmc original two parameter represent three hyperparameter iteration draw left parameter draw discount autocorrelation parameter discount burn model show see round exactly index eq length round indicator rewrite v usual indexing expression th point express feature quantity step feature indicator step integrate break proportion round remain product factor write stick integral sum trick beta second factor dependence k r feature agree calculation value normalize feasible value fall pre latent integrate conditional likelihood integrate n nx nx final second factor pz k pz numerator integral beta discount three parameter indicator assume scale equal generation occur g round indicator variable calculate monte approximation discretization monte sampler tail fall pre complete discount lie behavior p prior describe sampler feature conditional variance proposition involve feature draw beta collection probability draw stick beta representation dirichlet break directly motivate law process beta stick set arise population involve covariate capture within assumption assume exclusive model parameter former principle model dirichlet typically whereas typically combinatorial encode binary capture bernoulli extended draw draw distinct value treat value random subject inference deal connection obtain stick collection distinct obtain connection place drive development wide algorithms model prior suggest generalization slice possible bernoulli bayesian family measure beta survival model hazard focus beta contain measure view infinite repeatedly entity random view integrate analogy chinese restaurant combinatorial previously process binary collection break particularly important algorithmic exploring generalization yield recursive case explicit recursive size evy yield construction stick break break ready power generalization stick recursive sized biased evy rather recursive representation stick break making break beta conceptual derivation elementary permit unify perspective generalization beta behavior particular law illustrate concrete previously factor loading dimension via bernoulli th binary encoding infinite modeling remainder organize beta conjugate stick behavior review break power law law come section stick break expand beta theoretic poisson process process aspect beta exhibit illustrate power simulate experimental result mcmc inference appendix bernoulli random measure measure poisson let denote product realization countable measure denote discrete measurable set completely random construction completely random measure deterministic completely characterize surface illustrate uniform line segment plot poisson process line segment realization denote continuous function refer nonnegative density illustrate draw beta draw index exchangeable indicate horizontal axis make white indicate infinite coin correspond draw infinite treat potentially atomic component define bernoulli discrete necessarily atomic corresponding atomic non link bernoulli draw random hyperparameter draw discrete draw draw bernoulli equal know construction overall beta draw depend exchangeable class exchangeable binary underlie beta process analogous chinese restaurant dirichlet process stick break broken middle broken remain stick recursively stick v form partition break simple dirichlet update stick extension yield power turn consideration break law beta stick break recursively break interval countable fraction remain break break stick break fraction remain stick stick broken stick break figure sample process heavy exposition wish obtain law break representation law notation draw draw dirichlet associate obtain cluster exactly total cluster behavior law constant limit hand value power behavior law law recall variety world problem specie construct internet document paper publish prefer reflect distributional attribute neither power law though dependence stick representation cf note bernoulli provide kind study power law model consider exactly elsewhere place non surely elsewhere sum case type law number say positive last type type procedure include inverse evy procedure stick process analogou procedure however worth note tuple subscript across additive one generalizing stick break breaking arise process poisson proof rely argument shorter demonstrate ease incorporate analogous thereby motivation stick toward break begin three generalization side notation say discount show mass turn represent process identically distribute poisson countable union poisson poisson poisson measure finite distribution enough end line follow monotone convergence term outer equality I substitute back copy bias take measurable size stick stick biased rearrange du du du stick break conclude assignment law big three parameter beta distribution obtain asymptotic behavior constant case type power law surely appear subsequent derivation obtain convenient proof apply beta eqs eqs represent term atom law part asymptotic derivation extensively feature mean law proposition obtain surely behavior proposition law define process construction start top illustrate poisson line rate bottom illustrate infinite put minimal necessarily sum eventually case atom one generate see poisson positive henceforth define exactly eq definition principal integer jump observation addition find convenient assume poisson approximation fundamentally tie work able mean count case draw equality power property behavior vary lt lt key laplace give asymptotic integrate integral integration laplace part laplace assume integral thereby part transform link result show count relate law know probability rate law translate relate mean process relate almost behavior count concern feature count almost unbounded statement monotone convergence consequence generate poisson process establish inequality consequence finally consider wish law sure power law poisson wish borel q sufficiently lemma expression type I law imply representation beta process eq x
study runtime example decrease runtime sample ignore extreme bottleneck definition empirical denote paradigm learner minimize learn call return hypothesis throughout improper derive runtime formula boolean mapping follow approximately perform former hardness learn feasibility improper formula rewrite v conjunction problem polynomially cast programming simple however theoretic variable conclude important example know theoretic yield sample reflect reality curve know scale amplitude segment thick node l erm erm exhibit distinguish applicable since assumption work result one let string map polynomial possibly permutation exist treat length permutation formal agnostic learnable size exist polynomial learnable improper whose implie get runtime illustrate node leave define treat pair bit one way example simply hard distribution h contrary efficient free learner hypothesis r trivial soon learner algorithm least pick bit get attempt see work pick event predict inefficient value return see error likely frequently example appear hold pr hypothesis small good one pick least example efficient follow early inefficient rather work implicitly compute product even slightly replace non continuous transfer transfer lipschitz label expect rademacher complexity analysis theoretic erm convex due show erm reveal perform search example identify solution runtime exponentially erm idea hypothesis b l contain respect solve erm list dependence like online learning demonstrate runtime price need paper formalize discuss phenomena dependence discuss restrictive perfect exist involve hypothesis agnostic improper learning half construction phenomenon natural seem introduction base available computationally size open believe outline existence example hold additional happen leverage large efficient surely asset machine application eq result draw bernoulli variable whenever subspace span assumption problem price correspond achieve error round receive arm environment pick learner tell feedback correct correct label frobenius restrict multiclass implement feedback problem multiclass classifier perceptron exploration tradeoff running round require small c round inefficient follow I structure support diagonal thresholding take return index prove perfectly complexity diagonal sort runtime sophisticated sdp scheme clear tradeoff note gap polynomial c definition microsoft research grow understand reduce runtime study example result exponentially growth range every day task genomic recommendation meanwhile world algorithm construct algorithm speak datum reduce situation hardness original computationally problem reduce flip statistical class original might overfitte though keep overfitte check example polynomially example however extra learn hypothesis goal model study construction demonstrate similar phenomenon hypothesis agnostic learn still synthetic continue arise availability upper raise open first study tradeoff runtime sample theoretic complexity latter need learn present efficiently learnable learnable polynomially learn efficiently construction extended decision list large gap focus mostly focus domain crucially provide term technique rely assumption permutation similar sense even however return correct agnostic compute improper predictor hypothesis example consist natural employ carefully decision list
background introduce functional paper part quick hilbert refer reader therein reproduce short hilbert positive throughout hilbert eigenfunction eigenvalue expansion term eigenfunction involve parameter operator illustrate discuss material generate rkh generalization book paper operation bound adjoint mapping order adjoint mapping fa play important positive orthonormal random span eigenfunction eigenfunction smoothness eigenfunction pt reproduce embed lie problem involve time might collect eq fix collection noise vector observation projection inner scenario unified introduce analog analytically convenient study dimensional classical truncation way choice representation time model reproduce property mf rescale yield map vector basis f equivalent truncation remainder shorthand throughout analyze subspace span sample let projection distance orthogonal subspace bound schmidt important form compute matrix special operator thus time hence reproduce truncation k semidefinite throughout definite eigenvalue prove small estimator functional theoretic receive function lie within assess norm approximate measure bad semi norm uniformly radius operator particular second equivalence eq q see background place description representation span goal dimensional subspace span version let orthogonal via z hilbert schmidt notation ny straightforward computation might give approximation objective however account eigenfunction calculation precede lead recall compute trace arise column slack also feasible supplementary claim subspace hilbert interpolation suggest define minimal sense map linearly linearly preserve space aspect qualitative accordingly solve reformulate sdp know equivalent solve moreover lagrangian duality multiplier regularize value path algorithm dyadic one possibly evaluate optimal around standard ordinary pca keep amount variation apply order uniformly space four true signal row observe poor roughly minimum observe particular one considerably optimal oracle value estimate regularization smooth reveal reveal accurate version estimate noise smoothness signal embed noise vary zero smooth increase turn high estimate case begin theorem theorem derive corollary rate z state rate regularity assumption measure vector straightforward upper equation accord signal strength deviation go convenience eigen part involve constant low prevent degeneracy eigenfunction space would define whenever hand go infinity grow subject small satisfying inequality c r sampling sample early property rkh I order kernel integral expect operator decay usual kernel achievable addition condition suppose decay c pair depend small hence one rate minimax reasonable choice although guarantee error critical statement upper nr piece trivial satisfied time model turn consequence achievable truncation operator hilbert decay condition basis truncation base rate consistency closely closely approximation capture define equation bind bind case scale condition namely pt supplementary claim theoretic applicable state abstract form obtain concrete illustrate return case condition need appendix material namely guarantee choose moreover bound polynomial decay verify supplementary truncation sample assume condition interesting become long model truncation satisfied moreover imply follow decay comparison trade basis functional reduce whereas reduce fast regardless minimax achievable standard unit ball range low polynomial truncation corollary similarly truncation optimal become bound eq identity approximate inner product eigenfunction universal kullback leibl relative different condition reproduce kernel eigenvalue q truncation low q calculation time rkhs frequency truncation trivially corollary minimax norm term imply regime begin prove corollary begin heart matrix addition r z define matrix rw algebra eq b establish existence early manifold matrix describe convenient version consequence cs decomposition exist orthogonal matrix theorem work instead version choice properly align align decomposition bind hilbert recall central proof probability shorthand optimal p rearrange sr rs cf supplementary provide concentration well put piece together inequality problem study establish inequality hold constant nontrivial task depend derive law subspace properly align inequality suitably linearity trace separately quantitie equation supplementary material show consequently fix follow lemma supplementary material proof delta recall linearity trace obtain since high bound upper rt supplementary material claim theoretic viewpoint hilbert rkhs sampling operator observation plus consider generalized frequency possible subspace mapping trace condition influence rate convergence subspace work detail cc truncation mn sample rate exhibit sample rate qualitatively interact fast interaction frequency optimal characteristic minimax lemma support grant dms pca generally model map functional lie reproduce subspace span lead model attractive rate minimax functional establish statistic great see book time treat term functional various applicable dimension component one mathematical abstraction reality treat functional continuous arguably practice resource force true sort truncation instance understand justify corrupt noise study affect estimation presence share domain
uncertainty direction encode proxy marginal gaussian determination variance currently wide thousand million arise explicitly store full infeasible gaussian random loop sophisticated develop purpose class variance study variational unless propagation sensitive demonstrate variance computational routine context build f factor follow sample suffer systematic sample suffice capture reliably need application easy variational several fully requirement conventional point gaussian sample draw processor show energy hide response capture induce potential form aspect blind deconvolution unknown kernel evidence partition sensing amount lead application point provide uncertainty estimation overcome capturing consider imply evidence variational mostly focus concave induce potential e variational induce super form super potential quite rich include member super replace potential variational bound evidence derivation demand point map capture difficulty lie exploit concavity vector naturally globally convergent log concave variance upper newly w leave concave minimize inner obtain variational completion loop variational parameter subsequent outer loop bound variational mean field adjust minimize kl shown mean bound sec expectation parameter true match various iterative sequential message pass optimize focus convergent loop bottleneck highlight sec repeatedly variance routine z impossible store explicitly far estimation build monotonically reveal rough structure scale run relatively yield variational prove estimate qualitative follow reliably sample translate uncertainty model enforce constraint use estimator illustrate variance variational fig sample last double variational marginal variance linear iteration runtime variance variance scale space dimensionality htbp monte free energy purpose course note sample reliably extra eq case good reference crucial solve million image deconvolution build glm ie matlab toolbox bound propagation extend implementation future glm ie assume spatially degradation motion blind variant know blind blind deconvolution parameter maximum penalize likelihood determine integrate maximize carry maximization inference complete quadratic computing efficiently suffice exactly estimate poor add extra sparse minima various fundamental domain heuristic blind deconvolution thus software may kernel system perturb typically poorly condition convergence plain gradient design would stationary stationarity fourier employ dft fourier domain beyond conjunction dramatically conjugate gradient fig course conjugate typical bounding attain reach conjugate substantial improvement overhead relative problem aware exploit benefit real camera motion shoot camera still order laplacian induce ks scale employ double describe bound vb propagation ep vb ep fig blind ep reliably vb ep db vb db db ep posterior pointwise blind criterion blind bound leave ground initialization kernel estimate image central area account variance variational allow variational cost stochastic sub random key mcmc systematically bayesian acknowledgment work n education science research foundation suggest sec feedback ie toolbox token department california department cognitive engineering university heavy prior probabilistic viewpoint sparse probable sense shift towards capture latent quantify parameter variance turn constitute main bottleneck variational scale leverage recently map contribution estimate variance much standard fully scalable allow extremely memory conventional prove tv modeling thresholding compress linear random reduce scalable million variable viewpoint compress violate filter response exhibit heavy measurement operator impossible
factorization q mapping mean fouri certain fourier accord formally component arrange order say radius bound mind function need ball condition element equivalent large coefficient qualitatively strongly influence covariate influential subset wish soft empirically suitably g basis attain rate rich density bias rule begin q fourier coefficient form thresholded real value indicator observation square incur thresholded eq impractical criterion unbiased lead estimator q stand improve chen thresholding shrinkage disadvantage thresholde impractical thresholding allow whole coefficient rule approach ds observation usefulness observation purpose hence depth coefficient square magnitude exceed square coefficient leaf exceed internal yet visit tree without explicitly I leave develop shall rely follow proof suggests estimate biased property rather former require scale good balance variance attain minimax rate convergence strategy estimate sample specifically set case case always thresholde sum see expect result describe routine desire update factorization n kx u j I tune estimator risk bound positivity issue orthogonal necessarily happen handle expensive suitably sparse carry approximately sum many logarithmic bind threshold small conservative hold attain logarithmic factor moderate regime minimax risk minimax particular outperform histogram orthogonal attain statement problem estimate statement constant final result time fix constant km mse controlling rate decay case severe reject lot lead complexity computational viewpoint preferable rapidly decay threshold complexity decay practice low tend especially achieve balance mse moderately decay threshold become propose operation explanation possible incoherent compressed testing canonical achieve al group strategy boolean collect tool exist need hoeffding inequality q rademacher u index additional needed need contain countable exist converge pointwise separability ensure sequel class element separate fix element positive consist dimensional statement hold set contain kullback divergence relative entropy exist decompose squared error start necessarily define apply concentration bound list first cauchy schwarz q handle conclude bind eq absolute q argument write inequality inequality eq put eqs second threshold use theoretic low test I estimator pack write shannon observation next upper bounding suppose convexity substitute satisfy lemma conjunction suppose moreover assumption satisfied recursive recall call recursive correct call incorrect recursive q express supremum suitable hence accord see recursive call give recursive call contribute exactly bound call compute scope present low truth comparison strategy enhance regime synthetic mixture hypercube product choose truth shown profile bottom estimate probability sample cc scheme run second retain result five run deviation study thresholde constant twice whose value reject coefficient reliably distinguish zero spurious retain sample theory constant show time second recursive platform complexity keep mind call overhead also call scheme detail lead computation increase polynomially logarithmic considerably recursive become cause expect making later decrease finally number retain number coefficient perform alternative exhaustive threshold see far alternative cc exhaustive search mse time exhaustive exhaustive ten standard therefore optimize related challenge set indirect near bottom remain coefficient subtree sensible number level reduce binary branch trade possibility branch optimize recursive many computer limit recursion restriction recursive typically stack deep queue open branch stack number node free sort latter criterion simulation pruning frequency give hamming measure impose ignore branch resource significant gain branch high sequence level control trade impact technique provide appropriate yield computational mse computation multiple aforementioned bernoulli use ten dimensional component bernoulli average ten string eight expand open queue decrease achieve decrease force datum population effect formalized observation attain rate run polynomial complexity behave real another promising direction investigate form neighborhood coefficient fourier coefficient amount something field govern see sparsity serve correlation whether introduce binary analog something like account localization fourier recursive hypercube factorization property factorization property q leave uv prove case fourier coefficient leibler divergence chi q item combinatorial theory correspondence binary hypercube map eq let fm care purpose state follow lemma follow property associate fourier taking determine sign consider suppose latter consequently q prove item supremum key theorem end copy cauchy schwarz prove unit ball argument bind theorem nf binary density vector bernoulli expansion covariate estimate conventional prohibitive certain underlie moderate flexible trade consider density identically density binary hypercube wish observation arise variety profile yes presence marker recent methodology occurrence medical quantitative science survey panel answer correspond co event intelligence could search engine database store website like component
index draw nash generality see uniform raise nash equilibrium nash polynomially family every two theorem base construction pay ensure nash equilibria close nash equilibria subset let cardinality imagine game whereas complement get player always payoff payoff choose element approximate game nash equilibrium row uniform equilibria formally nash equilibrium row player uniform distribution distribution assertion straightforward player since activity component game averaging imply player payoff equilibrium player strategy payoff element player assign mass distribution argue payoff distance measure nash mix induce mixed assigning mass super disjoint counting argument appendix imply player game run recall anonymous section anonymous game also give anonymous exponent exponent specialize anonymous game define anonymous game index tuple player strategy profile player playing carry flow expect success player among approximation anonymous type sketch appendix tuple probability normalize anonymous game player nash player depend game enforce indistinguishable us player regardless choose play ensure equilibrium every player get payoff get enforce equilibrium challenge include player type player mix player player prescribe nash equilibria construction outline anonymous games family nash equilibrium strategy ensemble anonymous collection strategy equilibrium quantification appendix approximation omit section provide anonymous running course way collection tuple represent moment strategy collection strategy collection recall exist player anonymous payoff nash player turn max argument heart sum indicator sum indicator mean anonymous strategy per player variable replace one change payoff experience indicator nash equilibrium strategy search becomes prune search nash provide rather quantification total variation first collection collection condition sum expectation polynomial moment indicator theorem provide setting moment theory indicator expectation variation indicator quite ess sufficient heart result complement indicator expectation binomial derivative value derivative correspond sign e two sum indicator condition first sum derivative polynomial upon nash equilibrium treat easily polynomial exhaustive search flow anonymous equilibrium integer determine reason nash equilibrium must nash equilibrium player guess pure mix probability handle separately theorem indicator take corollary guess represent strategy player mix strategy mix remark actually find step answer follow could player play equilibrium exploit achieve multiple choice player aggregate player regret multiple aggregate multiple reject mixed profile additive come go equilibrium virtue set regret test nash equilibrium proof assignment strategy player player player player solve trivial dynamic assignment constitute nash player anonymous bit require payoff observation nash discretized value find nash test choice accuracy nash another lose full showing sparse computable sum independent variation cover cover sum indicator ik sparse value kn kn kn q k size indeed collection form collection heavy binomial specify indeed total collection binomial choice choice index precise since sparse permutation need etc index remove aforementioned collection collection tv I I collection give cover produce perform operation moment arise collection operation additive total distance argue total indicator choice claim vector l upper bind possible moment collection indicator k finish proof remain obtain cover produce follow claim time invoke r I hence algorithm produce cover possible moment moment collection indicator binomial add cover nash equilibria simple take anonymous algorithm elaborate moment bernoulli barrier familiar anonymous game many remain nash equilibria game program truly practical interact anonymous nash equilibrium graphical game game alternatively graphical case mit mit berkeley berkeley edu lemma corollary remark games equilibrium either zero polynomial time somewhat surprisingly game equilibrium payoff complete kind sample strategy nash deterministic bring question equilibrium game quasi recent anonymous sense appropriate game fix collection mix exponential prove must exponent player type contrast devise game number strategy set collection player work probabilistic two sum indicator random decrease moment two indicator establish existence cover sum cover nash equilibria nash equilibria equilibrium year progress small corner anonymous strategy different play identity strategy player divide type choose report progress new nontrivial computing equilibrium game entry row interesting randomized nash nonzero quite surprising special easy suggest logarithmic roughly first outline anonymous fix pair mixed strategy look game whether strategy constitute nash definite develop expectation game recently discover anonymous game player game utility player play identity sophisticated case player partition type type playing appropriate anonymous game player identity nash nash equilibrium briefly strategy th mixed nash player every nu nu games player type play type class class row chen find nash hard equilibrium game strategy player contrast class nash show equilibrium mix hence strategy long point simple always nash difference give player payoff payoff nash optimal optimize player payoff class guarantee
statistic respectively typically mathematically commonly represent vertex wherein distinct undirected prominent fashion wide vertex web page direct represent link point page protein network biology vertex undirected edge affinity network vertex represent social great literature focus variety range largely mathematical interest cite short review paper model factor relational hence complex vertex quite geometry modeling phenomena problem quite hundred hundred vertex edge vertex particularly network pair alternatively think setting institute north usa fact three response I network answer despite model knowledge formally pose analogous question ask notably series latter maximum estimate dependency obtain combination might asymptotic characteristic practice interpret scale variance insight arguably real network scale vertex I regime asymptotic corresponding response popular notably derivation utilize tool accordingly serve straightforward illustrative effective trivial answer subtle basic model assumption organize definition main edge give illustrate implication study asymptotic sharing examine extent world find support variant background chapter history practitioner network specify follow statistic despite appeal exponential handle computational concern random graph wherein assume identically arguably small still variant introduce situation practical quickly insight effective network use statistic model vertex edge tendency wherein model network hold draw configuration realization effect realization tie vertex preserve preserve importantly bernoulli bernoulli question parameterization introduce shift adjust realization produce vary configuration typical realization realization like baseline asymptotic configuration would produce shift mean recognize large distinction fundamental implication effective main parameter vertex degree infinity stay tend degree perspective traditional offset asymptotically equivalent enyi alternatively social odd value network able maintain shrink maintain beyond observation effective fisher respectively calculation implication immediately apparent likelihood estimate maximum model finite consistent largely asymptotic constitute consistency estimator argue theorem consistency estimate proof normality usual technique due exponential normality statistic case array require normality vertex triangular array rather array need derivation provide appendix size assume perspective rescaling induce notably subtle fisher analogous similarly asymptotic property analogous previously satisfactory call poisson hand behave tend limit vanish fisher network infer reliable suggest actor contact small suggest exclude varied consistently conservative level tendency interval proportion nominal level prominent rather strong confidence select level simulate study percentage point percent section establish question tie establish sparse detailed beyond initial explore question face point require similar collect ask list precede neighborhood find significant share find construct overlap consist neighborhood baseline correspond realistic expect slope lastly realistic slope around color indicate coefficient clearly magnitude close although possible difference nevertheless enforce tie explanation magnitude substantially argument network close stable tie likely share little two comprise tie neighborhood close assumption violate respective tie lose small small per figure decrease slope tie less adjust decrease increase reduce slope unlikely unobserved contiguous color tie discussion conventional effective sample associate depend case affect meaningful simple already sophisticated say suggest effect arguably type next index likely require complex treatment yet effective sample insight literature unlikely directly method interval network begin gain context begin exponential well present consistency relate contribute lack understand property commonly network particularly package package compute general formulation report estimate unfortunately practitioner seem aware construct theory confidence test formal perspective appear necessary foundation justify practical interval discussion material derivation key result expression main paper begin establish preliminary expression section section offer proof recall model family fisher offset expression order magnitude twice appropriately parameterize parameter model find capture simplify behave consistency normality interesting follow conventional various might technique requirement however behave behave large likelihood amenable demonstrate study behavior pointwise condition lipschitz true easily continuous function compact mean allow standard writing numerator recall proportional average link variable worth note allow towards infinity increase define employ asymptotic suitably normalize standardized variable easy fix bound theorem sum normal note behave numerator mean calculation calculus tend v tend zero establish consistency argument partial show consistency reasoning proof gradient set compact version
purely assumption greatly network influence classic topic economic agent agent major positive seek decision anti game make positive several preference benefit effort depend preference critical sequential vote sequential voting candidate project probability success payoff project succeed receive project benefit voting agent decision subsequent agent make aspect outcome decision combination hand become anti agent seek differ reward nevertheless agent belief uncertain social reward game rational player maximize reward play many different research cognitive storage service selection cloud compute secondary user access share user available storage service reliability availability use service service cloud storage platform receive customer negative product agent make possibility utility access service decision player learn focus design experience device design system learn non approach applicable rational choose action benefit follow design chinese restaurant unbounded chinese restaurant customer restaurant customer restaurant share customer table generally customer customer join process systematic unknown chinese restaurant formulate social restaurant customer sequentially customer request table prefer big big table since table request customer experience key chinese game customer table enhance involve experience moreover table customer receive signal game theoretic chinese restaurant game wireless ii rest provide detailed description chinese restaurant game simultaneous game customer behave perfect sequential game general chinese behavior customer uncertain method response customer game limited knowledge state decision agent agent influence negative characteristic game game draw financial network cognitive simultaneously work effort attempt investigate binary study game decision private unknown game propose attack enough status payoff payoff study stage agent regime change dynamic binary state heterogeneity simplify position game game simplify moreover mostly propose chinese restaurant game game framework study network chinese restaurant table request customer request table table restaurant table restaurant I restaurant exist restaurant decision let state selection game action may mean customer choose table customer utility decrease regarded degradation utility customer number group restaurant restaurant exact may exact may review restaurant relate restaurant information therefore know l l pr subsection describe customer sequentially customer decision later customer customer exclude prior customer belief represent probability receive probability belief bayesian naive updating rule rational bayesian implicitly customer predefine capability information rational customer restriction fix customer rational follow update belief updating learned generally affect like discuss simultaneous customer decision e restaurant time scenario learn request table investigate game understand customer behave simple customer table state indicate table state signal reveal indicate pool customer immediately customer know opponent perfect size customer decision opponent rational customer table opponent depend severe network customer learn table depend severe customer signal customer since customer rational action customer influence describe player chinese restaurant game customer customer recalling stand customer customer rational customer action describe response represent action customer well consider utility player action customer equilibrium popular concept predict game rational customer informally nash equilibrium customer profile none formal equilibrium give nash equilibrium nash necessary simultaneous chinese give customer table current nash equilibrium chinese restaurant perfect equilibrium action player grouping generality customer utility j ix x x nash equilibrium condition nash table nash nash equilibrium customer utility would another table another customer eventually table size customer restaurant table size nash customer table customer equilibrium exchange action customer nash equilibrium nash necessary grouping may state nash equilibrium game perfect inequality hold grouping signal current customer give current grouping table player j sn impossible reach nash equilibrium nash j eq due last customer utility table response customer propose nash satisfy group result grouping grouping customer choose observe customer customer customer candidate customer choose contradict customer customer follow lemma perfect equilibrium customer nash equilibrium grouping propose customer customer equilibrium group customer customer choose another utility become never bad customer choose instead reach contradict n grouping never customer final customer equilibrium thus choose never equilibrium grouping grouping customer grouping achieve moreover show grouping nash equilibrium perfect nash perfect equilibrium sequential game bring advantage decision make decision early large utility equilibrium customer choose table reach equilibrium choose subsequent customer customer show sequential perfect customer advantage get table thus signal uncertainty arrive right table arrive later eventually table collect right decision customer choose later therefore trade choice accurate playing later trade discuss signal model unknown customer table express denote probability customer customer private f customer condition customer uncorrelated restaurant customer make sequentially number mind subsequent decision customer rational learn make decision collect well estimation reveal update updating base update customer reveal expect group customer choose customer customer reveal signal customer see utility key final grouping system belief denote information system state reveal customer belief utility customer close generally impossible impractical recursive approach customer accord ss give ji predict eq customer choose customer include recursive recursive x eq eq x calculate ir response derive customer customer easily verify recursive simulate chinese two customer customer receive begin conditioning system receive give close likely reflect state customer decision sequentially customer game last customer make decision customer number customer table end optimality customer customer customer choose response customer confirm customer increase customer customer mistake choose table customer benefit mistake customer quality previous size ratio table utility choose table odd customer utility simulate customer fig make significantly affect show signal size customer region utility customer large small case choose advantage since signal table even third form belief true less customer decision customer customer choose negative customer likely belief collect customer advantage get chinese nevertheless restaurant game non show customer increase scheme utility quality size customer low customer utility customer large provide equilibrium customer table customer customer correct third customer choose table customer determine size generally state table effect high argument argument customer fig customer utility peak consistent table ratio equilibrium customer decrease customer large utility table size equilibrium customer customer customer size large expect utility ratio next resource pool user act sequentially maximize mean utility scenario scheme customer response resource pool customer average high scheme quality customer quality view customer customer receive signal quality follow table customer choose customer empty table customer dominate good customer opposite receive customer customer equilibrium accord customer utility large equilibrium customer expect share another customer c show advantage customer play less customer scheme large utility customer utility phenomenon customer try identify customer nevertheless observe late customer likely customer illustrative response quality resource pool still play significant
elegant random improve exponent dimensionality use qualitatively insight active thorough empirical batch mode demonstrate impact importance empirically scalability mnist query fast good reflect underlie problem bias process weight differently proceed round probability entire pool sample round round label hypothesis gets weight weight unbiased denote denote guarantee pool round come point uncertain probability mass h hx iy calculate depend current learner possible loss function show conditional use convex loss labeling p tx n j tx tx say h tx hence retain property allow interesting look behaviour square hypothesis return invertible elementary establish integral solve proceed odd substitute ss assume matrix exponentially expert commonly learn interpret prune soft exponential get weight ask answer analyze squared function yield analyze extend logistic exponential assumption excess risk learner invertible finite exist shall boundedness dimensional cube suffice though kernel infinite mapping find learner active excess proceed assume invertible upper bound excess risk motivation square term reduce maximum bind positive semidefinite nothing eigenvalue reduce give upper eigenvalue rank lemma spirit analyze eigenvalue sum bind tool quadratic moment quantity calculate bound establish use bind suppose represent active learner active learner round next quantity excess shall inequality get inequality excess risk assume invertible matrix cauchy use probability bind probability get since get upper probability last provide similarly bind proposition prove follow let theorem tp tm nr get since equation place equation inequality use use inequality apply theorem q substitute lemma tm first square last lemma invertible sample get z enough provide eigenvalue positive definite rearrange q equation fact round square equation inequality equation next column aa subgaussian arbitrary left exponential definition form deviation martingale sum second lemma boundedness upper condition eq datum show lead last exploit hoeffding hence prove add get desire pool employ strategy none unbiased simple strategy point uncertain text classification calculate mostly probabilistic uncertainty close agree label maintain member query boost bag discriminative framework cause reduce one thorough al provable label agnostic unable trivial hinge loss least loss entire member unlike choice implement passive pl variant call learn loss logistic motivated logistic code possible similar uniformly currently pool however implement potential unbiased check helps batch al superior empirical pool primary pool like proceed select pool pool point suggest monotonic optimize point potential dot calculation new query far round subsample value algorithm numerical implement regularize show current importance pool fold perform reduce dimension uci namely axis implementation c error average gap pl see case performance pl always success match reflect performance pl well never pl time huge gap pl query performance random perform via especially round step algorithm problem also solve problem fix budget mnist tend scalability experiment set find budget speedup run dual core active algorithm show scale constraint theoretically prove true view establish tight excess excess high interesting calculate round calculate bin ball randomize pool bin ball equivalent round unlike ball bin round expect bin form subgaussian almost surely q prove et al bernstein symmetric exist q surely dimension version al dimension useful work dimension space
institute mit currently electrical laboratory system mit research interest include engineer electrical engineering receive mit thesis transaction journal ph electrical receive california receive award work software apply technology ph fellowship application communication electrical receive ph degree mit degree electrical stanford move dr work hold wireless california berkeley laboratory university work laboratory research coding signal process security application computer architecture device dr award mit tucker award intel fellowship stanford award department fellowship edu systems institute technology usa email mit edu establish information estimate take sharp reliability decoder appeal low probability union scenario amenable decode average number require random literature fact motivate minimum distance code succeed sparse code ensemble one reliability code attempt area community year minimization code endow hamming herein provide investigate code completion noisy require variety collaborative filter netflix obtain singular generalization completion problem entry arbitrary inner product nuclear matrix also measurement exceed product yield sense observation reconstruction field exactly albeit rank code endow metric concerned check minimization problem besides measurement field code want th string furthermore know bar show linear work work consider herein graph property code linear channel induce assume array rank receive matrix succeed true tend infinity code characterize dense deterministic code code motivated array arise technique unfortunately highly success subspace decode technique pattern work able distance property random code correct derivation similar spirit start rather combine exponent result seminal uniform give multiplicative derive exponent decoder summarize use converse technique require obtain product sense contain procedure min match theoretic lower require hence sufficient condition concern computational recover rank compare exponent min decoder de bind estimate reliability error example upper show fact code comparison achieve theoretic thus much reconstruction main possibility sparse check matrix draw code code derive analog binary block code analysis obtain geometric decoding perform matrix guide strong along et al able seminal work unknown many application information al matrix mirror sufficient additionally analyze inspire derive theoretic derive condition completion argument measurement logarithmic measurement converse match extended analysis conference case measurement compare decoder another author channel symmetric study generator sparse capacity achieve code sparsity make decode amenable technique code decoding pass perfect ellipsoid min code rank analog binary code equip hamming metric minimization finite field various reference table elaborate comparison notational choice describe measurement use reconstruct unknown rank uniformly select random sufficient reliable recovery reliability de extend noisy reliable recovery understand theoretic perspective search section defer rank dense section notational describe notation font font deterministic quantity face respectively deterministic scalar respectively font set event denote element identify integer matrix entry let hamming use mn stack one use throughout definition reader paper definition rank scaling section square ease exposition rank limit assumption converse linear trace sense mass pmf measurement estimate depend e multiplication interested represent precise measurement setup fill capture choose sense general although minimization analyze et whereas unknown general measurement ill pose measurement particular pmf deterministic recovery recover recovery would familiar distinction compress suffice w sense operator matrix rank less rank matrix example random follow tend allow state low one strong recent sequel matrix number matrix q fact present necessary reliably estimate converse necessary object unknown deterministic k nr event q emphasize random follow converse fix rank less equal assume independent rank less recovery reliable I tend degree noiseless section involve elementary inequality k bind convergence statistically matrix validity assumption restrictive practice sense linear reliability recovery problem select decoder denote singleton optimizer analyze singleton equal I error optimization intractable unless sense ar decoder exponent constraints section pmf measurement exploit idea weak fix measurement recover theoretic low rank sharp converse proposition rank decoder converse side min simple albeit rank q equal fact precisely independence uniformity uniform continuity integer depend satisfie loss assume simplification precision argument decoder asymptotically require match proposition rate decay decoder code sense matrix random connection problem theory detail reliability function exponent exist unlike reliability normalization min decoder reliability function upper de inequality convenience event need compute argue latter independence neither logarithm simplify er proof reliability complete appeal min follow may unity lower express bind note yield reliability event essential uniformity independence obtain analogy show capacity inequality exploitation independence error error exponent ensemble linear code large exponent pmf conjecture introduction theoretic perspective compare code literature distance third construction reliability normalize simply quantity instead code random decoding give code singleton eq rank error uniformity randomness code code scheme reliability whereas give rank reliability regime furthermore deal encode matrix amenable low decode exponent general require search propose reduce precede generalize assume deterministic situation minimum generalization min decoder let noisy measurement model choose monotonically measurement increase increase intuition decoder part uncertainty location remark proposition reduce different model technique measurement match converse decoder matrix whereas derivation converse reconstruct unclear also amenable noisy also converse vector assume represent every independently ask also proposition converse converse inspire theorem decoder notion recover proposition code non endow matrix statistically I pmf code theory element belong field adopt approach direct rank codebook satisfy na cn connect directly literature basis j nj discover metric code matrix constraint metric code term correspondence code guarantee self check usual metric element extension field establish correspondence code matrix quantity take deterministic code remark analyze analog index rr turn amenable random generator instead moment mean satisfie namely exponentially compare converse chebyshev immediate eq generator derivation define minimum rank minimum distance rank distance cf pmf let k surely proposition consequence bl code code prescribe decay light code denote minimum minimum go serious implie may hence small matrix code probability say sequence equal denote concentration sequence rank n rank code apply hence n proposition exponentially code contain interval analog substitute definition typical remark code metric derivation require assumption check generator linearly knowledge previous study distance rank property derive inequality decoder succeed typical exceed rank correction derive conclude compare proposition pack sphere pack code n ball ball rh total deduce sphere perfect analog proposition check matrix pmf analyze matrix ham hamming code justify way hence omit allow tight analogy metric code code distance code minimum condition relative linear code remarkably property dense statistically independent fact bound distance ensemble coincide explain decoder match theoretic rank linear recover ask measurement reliably strong recovery measurement uniform min rank probability weak min decoder recover thus say many needed limit increase factor recovery context compress sense finite field derivation precede subsection relative small recovery decode word distance exceed see illustration minimum translate distance code minimum rearrange limit proposition number prescribe recovery decoder recover equal one use follow line disjoint devise procedure complexity decode exhaustive inspire technique mention belong matrix minimum clarity exposition vector rank integer exponentially elementary operation multiplication affect reduce check exactly equal generality read justify nn system partition part equation ax solve decode know write refer coefficient expand solve I employ basis basis linear coefficient terminate increment go second number less since successful equation attempt increment complexity I distinct linear elimination idea dramatically na I solve loss number basis reduce indeed decompose r contain precede enumeration na I different equivalence find equivalence class lagrange non singular inequality consequence equivalence class matrix note corresponding belong class hence previously consideration complexity note symmetry connection introduction table conclude contribution suggest future solve min rank intractable characterize code admit receive polynomial singleton decode however particular line guarantee suggest decode work matrix draw sparse factor strategy hamming loss adopt subspace message pass strategy cause assumption additive value channel field also achieve however uniformly fig fundamental limit code metric construction analogy channel code low code probability recovery conclude analysis complete remain table minimization field combinatorial demonstrate subgraph number clique ellipsoid semidefinite fact minimization characterize information limit recover field give noiseless noisy measurement even sense decode dense code theoretic succeed herein interest analyze require motivate sense matrix fix property sense isometry joint use programming necessary sufficient guarantee et sense random linear lp decode lp compress sense reach sense new analogously whether derive thereby provide reverse grow literature design tractable interesting decode code interest analyze possible tradeoff reliable low analog finite solve optimization acknowledgement suggestion improve acknowledge discussion thank liu fig joint precisely definition inner subset equality condition
practically useful n respectively main procedure complicate give long family replace respectively multinomial approximated degree jj around accord light approximate eq update analogy update accord type q compute size fortunately tune definite answer tune concern number nonlinear shape initial fitting case filter informative likelihood simulation logistic nan matrix regression nan imply covariance make support simplicity well theory stochastic observation increase importance sampling budget particle plain smc specifically adaptation update adapt proposal fast enough less twice bad initial iteration run typically iteration concern exact optimum slowly let framework stochastic em constant step mention smc successfully filter k unobserve space transition sometimes initial conditionally particular density respect development straightforwardly optimal record recursion filtering perfectly g q omit brevity time associate approximate follow design adaptively describe gaussian toy optimal available vanishe expert value hide state filter separate initial algorithm observation belong family expert expect adapt accordingly importance particle produce zero spread confirm look proportion particle sort decrease proportion mass number adaptation plot row propose visually normalise adapt shift row show adaptation support thus span importance balancing drop near nan proportion unchanged look evolution fit result couple serve lack close htbp focus away single adaptive particle filter bootstrap simulate record adaptive filter proposal entail filter symmetric rotation shannon two filter shannon improvement surprising light consistent particle infinity relative effective sample chi exact briefly kernel stop partially sequence mutually gaussian state observation leave censor nan display row kernel mass third un normalise kernel reference depend relatively line particle adjustment multipli weight adapt lead adaptation un normalise middle bottom leave normalised match save sample un normalise widely match non finally drop family large illustrate gaussian student achieve allow kernel kernel half purpose thank concern case highly relevant adjustment logical smc design open rely algorithm proposal instrumental fit version apply relatively approximation every evolve among nonlinearity adaptation imply limited markovian possibility adjustment multiplier thank comment value jj jacobian mapping consequently thus conversely complete figure filter approximate integrated logistic enough strongly skewed mixture leibl divergence target instrumental single optimisation illustrate successfully nonlinear leibler shannon entropy decade smc simulation rare liu flexibility method efficiently step composite particle markov dynamic datum assimilation material normalise dimension equation smc density recursively weight f measurable function state main former particle specifically mutation particle transition particle update correspond measure ratio eliminate importance particle accord factor introduce adjustment multiplier weight particle likely contribute state target locate choice adjustment affect let particle dynamic closely choice adjustment multipli yield perfectly weight call adjustment express close one refer approximation deal proposal approximate approximated nonlinear always e g multimodal cause severe center student go log unimodal nevertheless approach optimisation particle thus recent state art meet success implicitly sequential sampling minimize choice definition maximal completely single carry minimal coefficient variation smc instrumental shannon instrumental specifically use cloud update resample view standard proportional tend tend asymptotic addition shannon variation give sound theoretical shannon measuring suggest shannon entropy purpose adaptation matter adaptive design smc adaptation adjustment kernel henceforth implement auxiliary enough hand way simple integrated let weight choose proposal allow partitioning region kernel whose student proposal closely appear learn community large em procedure smc decrease minimum gain already minimal computational overhead organize notation throughout adaptation mixture treat optimisation paper result several unless th trace c transpose quantity define equation r smc sample notational argument wish move particle adjust obtain weighted yield introduce normalise q partition ideally update drawing particle independently firstly close secondly form normalization cope aim take importance consist drawing position importance resp adjustment introduction importance absolutely proposal multipli transition resample schedule give methodology simple framework describe foundation smc development recall space method replace auxiliary particle induce term discrepancy induce index term quantity family instrumental distribution member mixture expert take approximate already closely resemble weight partition smooth assign region eventually dominate region weight sometimes resort mixture let without partition transition cost broad class flexibility ease state space resp statistic method proposal illustrate multidimensional multidimensional mind fulfil discuss optimisation algorithm paper recursive parameter key smc em gain simulation describe scalar sum unity treat auxiliary specifically convenient state integrate satisfied student eq sample addition statistic mapping note additive subproblem assume problem fundamental case integrate regression hierarchical plain possibly deep expert overhead possibly overhead reduce flexibility somewhat expert ji definite
vector suitably collection well list order gene diagonal via monotone transformation diagonal list list obtain position value well suited list desirable weight depend exchangeability carry exchangeability gene give list sided exchangeability score e diagonal list gene actually contain consequently affect diagonal possibility contain present exchangeable list position exchangeability gene position change general support similarity kind expert biological e g concern list positive gene biological global weight influence may probability list relevant list microarray gene symbol probe two datum gene patient good cancer outcome contain gene patient subject main gene list list use gene experiment far rank usefulness exchangeability apply set bootstrappe account modify accord snr compare patient gene positively outcome place deviation outcome gene extend list list way list vector describe accord contribution list position snr eq comment subsampling time keep snr vector side mean exchangeability gene define list magnitude matrix exchangeable position list two extreme end final gene list snr median subsample top create aggregated list product subsample ranking product list extend list describe step positive data exchangeability gene list set five step among result set depict variation underlying list among stability ranking overlap five appendix top figure plot list obtain clear exchangeability list list variation exchangeability list indicate correlation bottom row figure list figure stability note gene share feature discrimination patient poor outcome correlation expression gene estimate exchangeability next distance rank adjusted contain exchangeability exchangeability score position construct bootstrappe compute extended list distance extended list variation list vector distance compute extension much extend vector estimate incorporate comparison diabetes extend list imply dissimilarity set dissimilarity also diabetes appendix f histogram stability ranking desirable interest ranking assign gene position rank extremely therefore ranking obtain examine rank gene list discriminate two patient fold validation performance classifier compute five describe gene level center standardized value train standardized level gene classifier classification receiver operating split classification bottom gene gene ranking rank always ability top gene list considerably observe expense decrease biological extend median aggregated univariate multivariate genetic ranking criterion gain biological knowledge ranking highly unstable list much framework variable list order pair visualization g multidimensional collection list stable list list several list exchangeability concept quantify redundancy gene incorporate list way gene list choice dissimilarity new used gene response htb panel right label randomly aggregated ranking coincide indicate ranking aggregation original ranking variation list interestingly part list stable thorough position gene subsample modify set gene ranking rank collect position ranking place position provide used exchangeability distance exchangeability j bs j overlap exchangeable obtain eq consider sensitive sample exchangeability two estimate integrate part negative difference integral note differentially express sided exchangeability distance position length sided exchangeability v exchangeability score datum therefore repeatedly uniformly form exchangeability distribute take exchangeability score main article motivation treat positively poor snr somewhat adapt exchangeability fast high exchangeable gene top list decrease trade effect choose rank construct case similar general present measurement information example reference list contain resemble document field retrieval weighting could position list gene often position influence vector position part give ranking part expression gene exchangeability plot article ranking give overlap gene set aspect since row overlap gene top list overlap indicate exchangeability relevant ranking absence two gene list list top rank htb study stability distance figure ranking independently list extend ranking find important list exchangeability dot list gene position list extend position contribution cr measure exchangeability coefficient relationship expect exchangeability highlight relationship simulate rank use last exchangeable contrast also exchangeability relate response subsample keep group pair exchangeability correlation average realization exchangeability gene group consist group moreover highly exchangeable last detect mm average simulation simulate synthetic matrix sample sample relate group mutually situation compare two sample exchangeability score vector time group part average exchangeability take variable exchangeability score mm mm represent exchangeability positive exchangeability variable pair value provide brief propose list compare list order list compare order review dissimilarity measure list essential part short checked simple commonly compute overlap gene differentially gene check significance simplicity method make software package gene spirit base diagram score recently relationship gene experiment check gene bottom modify kolmogorov statistic combine statistic average gene ranking compare list bottom list adjust list high stability middle permutation propose permutation metric somewhat possibility module rank within module matter method compare list rank account formulate list choose suitably note association universal explicitly compare list similarity score cosine similarity coefficient geometric describe quantify overlap gene gene list positively correlate list exchangeability account set reverse role draw let give list compare experiment list gene gene correlation metric exponent control choose position list gene wish ranking finally list matrix create list q list maximum deviation repeatedly label calculation estimate two rank q vector also feature module gene argue module penalize difference within module practice put module order list value list rank q compare ranking similarly preliminary overlap list gene contribute overlap list accordingly replace desirable one pt set frequently list interest biological often complicated instability sampling variation may partly redundancy imply exchangeable exchangeability functional account variation exchangeability representation list support list ranking incorporate data microarray cancer patient robust sampling biological microarray throughput pattern copy output throughput consist quantitative trait order response gene association exceed study gene subset interpret understand process hypothesis inherent interpretability list change redundancy gene thereby depend list substantial gene overlap apparent instability small ranking extract experimental exchangeability redundancy general incorporate exchangeability list gene microarray list quantify list gene gene mean conventional contrast method tailor specific
present characterize correctly identify discuss associate group approach breast recovery block machine signal process attractive domain interpretation data moreover inference dimension amount traditionally approach involve penalty prove theoretically practically penalization estimator model correlate convergence refer covariate jointly lasso set define restriction model group author encode recovery subsequently notion natural generalization recover display first group overlap support formalize propose construction support stable diverse pattern tree greedy penalty formulation group norm group support encoding idea support combined regression call notion learn problem regularize study notion strong recovery recovery typically always exactly group sufficient consistent group weight group disjoint question potentially application problem cancer identify molecular signature gene pose group biological connect reliably empirically explore application simulate contribution extend preliminary publish group predefine support asymptotic regularize estimation length play overlap address error pathway gene structure latent group work several formulation correspond section notion section toy illustrate consistency discuss present variant covariate finally experiment artificial gain well real breast gene article c g usually say article usually pm pf p convention whose ball convex equivalently lie sparse usually adjust minimize loss level group group neither decomposition remove group latent partition covariate belong group covariate group select precise effect differentiable covariate leave full nonzero lasso select overlap however entirely illustrate overlap third neither one formally extensively study risk general assumption surely q support complement equivalently induce group covariate introduce feasible assume least g illustrate solution component satisfy precisely group immediately therefore sparse solution likely interestingly reformulate cost penalize reference formulation group overlap group equal standard overlap figure unit shape circular consider coincide appear propose group nonzero covariate nonzero reduce penalty enforce impose covariate belong penalty intuitive classical empirical risk remain article investigate detail group theoretically empirically idea component separately appear multi decompose regularize norm share latent group coefficient associate robust decompose trace norm sparse regularize decomposition relate idea interesting length induce length set penalty latent group norm note support consider consider case function relaxation submodular naturally note theoretical analysis study paper propose view complementary useful prove section valid formula subdifferential penalty context optimization satisfy proper convex classical base penalty norm homogeneity decomposition g consider form dual norm equality due convolution convex p g plug effort variational rewrite solution optimization optimize lead I g formulation positive case respective express reach deduce problem g solution incorporate rewrite I h I see admit convex hull suggest visually ball hull horizontal disk define g g eq show hull terminology union convex subdifferential sum conclude g deduce g regularizer penalize lasso covariate belong group implication regularizer stacking restriction note occur group stack successive loss operation consider particular consider restriction eliminate reformulate group lasso overlap expand overlap trick hand extend consider endowed definite kernel section implementation implementation section show section consequence lasso restriction overlap present mkl disjoint concept present consider gram function p kernel multiple find minimize kernel reproduce kernel reproducing show precisely form let sense objective solution equivalence overlap turning introduce problem show mechanism original introduce one extension overlap combine function input mkl use orthogonal typically correspond paper mkl formulation binary mkl reformulate view structured mkl interaction obviously derivation priori lose rbf relationship covariate expect non within non algorithmic structure applies present explicitly computer memory access require large alternatively efficient algorithm proximal proximal operator via dual associated norm regularizer natural ask answer latter suggest express formalize informally notion characterize induced decomposition formally strong decomposition g term dual variable support follow immediately support definition support variational constraint group weakly constraint notion resp support resp group determine j g another variational formulation j definition consider illustrate situation strong entire abuse notation write explicit decomposition correspondence relatively include note support group note fall back overlap decomposition unique unit group necessarily cover support group convexity length illustrate w w dual strong represent line separate triangular colored support weak strong group coincide color group adjacent result never motivate uniqueness obtain vector variable true particular sparsity induce regularizer coincide interested result group lasso disjoint support complicate several exactly express might notion concept situation hard consequence notion characterize study generate finite hard imply consistency however require study notion continuity end particular correspondence correspondence element space notion continuity space correspondence say open resp correspondence singleton value correspondence notion main hypothesis denote g clarity eq h tend subproblem standard argument restrict tend rest construct condition probability whose computation gradient enough variance get correspondence contain show contradiction condition b hold condition consistency outside support converge high imply support show nonetheless support solution union ensure situation necessary sufficient usual disjoint favorable j mutual incoherence overlap motivation g difficult outside set result generalize convergence bound consider version sense show dimensional neighborhood optimum focus group recovery relax give ball bounding image paper complementary recovery compress sense understanding associate motivation different solution test context definite kernel context context overlap significantly arguably consider group indeed notion support norm accord guide ask support show useful attempt characterize prefer small scenario discuss relevant false informally fact contain small enter cover large enter change dual group unless g g natural weight exactly general consequence lemma choose latter group group cb previous c w b prove behind geometric imply intersection redundant associate unit norm redundant exist unit ball g gd h impose redundant pi g g like sufficient redundancy might restricted family soon group group without size become special necessary redundant redundant find g quasi weight dominate dominate include suggest without large group active correspond possibility get ball impossible select support would union detail bottom set possible g intersect characterize discuss require single group group result weight dominate entirely assume dominate constant satisfy eq duality inequality dual corresponding follow group could trivially gap small select give critical dominate equivalently possible trade surprising interpretation penalization expense support support put simply support support variance reduce note consequence pattern analysis propose rip take generally subgaussian thresholding mapping support sufficiently assume absolutely retrieve support provide redundant never g g g select show pose consistency write support never weight select positive negative outside select condition see possible ensure group hand usual easy probability arbitrarily uniformly accord large small possible fail guarantee theorem summary control select uniformly satisfied furthermore c k want incorrectly select incorrectly base similarly choose noticed analysis ignore overlap incorrectly select address sufficiently large nonetheless allow positive ask contain element one could negative previously element pattern kkt none select k kp cc c chernoff group interpret group number put thing differently negative correspond certainly process active group another discount group select contain enough depend long tail two aim specific group solely motivated need criterion guide fine analysis scale dedicated collection require definite recommendation note view weight adopt relate vertex connect wish connected graph clique subgraph alternatively experiment subgraphs length formal choose control connect typically connect overlap group priori subsequently assess synthetic match support datum cover group union group offset set express group latent support b ht mm variable axis represent gray penalty band report frequently support regularization pattern small path hand time third root method path replicate test average fourth replicate help recover decrease connect simple successive index graph correspond protocol experiment group sub report choice cc axis level gray axis blue band support improve result consecutive group exact boundary influence penalize extreme group extreme weight root size small covering effect weight scheme form create nest covariate level setting compare weight scheme replication extreme regime way select small possible entire path regime ideally would theoretical ol obtain ol hold lead simply return case across correspond corresponding size recovery recovery path pattern covariate quantity reach weight right bottom blue band belong last table illustrate effect weighting result correspond recover term selection fourth uniform encourage lastly suggest performance mse point create selection spurious regime figure illustrate behavior expect large extreme active intermediate group active allow except yield adequate choice lead large path hard regime fourth behavior lead pattern reason fine pattern precise term mse reach optimum grid fraction covariate dimension show suggest helpful use large reasoning size mse ds ds min ds motivation possibility microarray use prior gene modify various mechanism biological gene involved amount interaction empirical biological database involve small set hope gene likely involve estimator practical implication expression small gene select number noisy addition select functional lead increase interpretability signature goal predefine gene set various database canonical pathway gene restrict gene indeed keep pathway poor break section combination group biological breast dataset cancer gene least pathway unbalanced use loss weighting example validation specificity regression pathway keep training practice microarray result gene keep select cross validation experiment noisy notice change lot split often sure cause choice choice accuracy observe improvement use weighted lead consistent improvement outperform
dataset ard hull term gp ard contrast ignore ignore additive irrelevant order example additive relevant r r ard add learn ard irrelevant variance variation subset repeat kernel however necessarily axis align combine rotation axis align process sum gaussian covariance process converse hold positive along exp put mass complicated hypothesis unable difference marginal try taylor remark relative degree matter huge subset cardinality specify sum kernel future likelihood cc department engineering university gaussian process low gps generalize generalize additive gp hyperparameter see expressive efficient increase regression example logistic generalize easy interpret extension smoothing spline add fit increase end allow simultaneously se gp much flexibility input introduce process allow interaction interaction kernel parameterization kernel constitute powerful interaction significantly efficacy major advantage interpretability implement recently call learning explore interaction suffer fact validation use necessary train dataset outperform se solve classification kind capture determine difficulty specify gaussian represent medium sized impact efficacy ccccc kernel st order kernel draw draw gp st order nd figure compare gp model high order gp order gp range depend house approximated price part small house capture unseen precise dimension define additive dimension assign interaction covariance base kernel full necessary specify additive hyperparameter kernel variance order control interaction heart nh vary widely come st come se gp specify degree model decomposable discover suitably flexible advantage allow order additive na sum quickly become intractable term th let recursive polynomial formulae computing trick additive base merely remove polynomial trick evaluating se evaluate gram additive interaction allow invert gram matrix typically order interaction computationally simply limit dimension recommend approximate order hyperparameter experiment support spline allow active closely relate smoothing spline spline along plus spline triplet weight number exponentially order practice ss via penalize fix sparsity hyperparameter procedure method separate individually allow automatic determination gaussian mat ern scale et kernel emphasize locality local argue care method solely require st interaction interaction rd exp additive additive structure distant space example strong similar dimension provide geometric comparison kernel additive non problem gp demonstrate gps data shape area peak training gp green function come first gps discover suited deal collection additive refer additive base gp first order gp gp ard additive base set hyperparameter fit mean inference propagation table split specify concrete nh gp square exp additive l concrete nh gp gp breast exp heart exp bold along bad sometimes model large experiment se perform order well
precision procedure qualitative procedure systematically htb change binary cancer micro jointly frequent dna profile dna run treat cancer group individual stage tumor smooth whole segmentation constant detect segment theorem propose detect multidimensional statistic generalize procedure knowledge observation free efficient programming simulation particularly keyword statistic segmentation estimation series contiguous stationary segment appropriate analyze time driving jump known segmentation arise range eeg processing square adequate instance presence criterion optimize recursion without require prior knowledge suitable first distance relevant rank analysis approach limit point way multiple group propose statistic amenable dynamic optimize term dynamic test change point criterion estimating point instance penalty add location adopt reference therein original approach norm propose rely slope approach preferable bic calibrate paper homogeneity section procedure derive describe observe transpose testing group convention rank j rewrite cumulative f denote continuous course direct maximization exponential maximization dynamic algorithm eq optimization start use procedure number rarely focus paper number principle change two trend rapidly increase hence least square regression fit residual minimal treat significant non alternative dash line performance assess arise report dimensional piecewise predefine four feature well simultaneously moderately db ratio jump amplitude assess replication correctly estimate point use tolerance whether correctly conclusion qualitatively ht programming multiple detection compute intra scatter add indeed approach slightly
ml knn ml knn knn knn knn k knn multi label several correlation improve correlation mapping paper novel method label sample subspace structure decompose wherein row consist matrix residual space correspond estimate via group coefficient concentrate sample belong evaluate real aim wherein belong label relevance early lp transform label several task label task lp treat unique label share label although bp lp variant multi multiple problem exclusive thus necessary discriminative correlation lead imbalance label decompose multi problem demonstrate exploit label group multi structure imply randomly final prediction thresholding relevance parent binary several subset build hierarchy classifier subset cc predict feature predict group method separate I correction result subspace formulate extension knn dimensionality feature preprocesse multi linear multi always decompose formulate problem correlation study label learn structured decomposition sparsity assign label via training estimate subspace stage matrix correspond nonzero space subspace explain decomposition low group cause specific linearly subspace coefficient concentrate label able method build mapping label decompose sparse multi label increase imbalance compare different several sample decompose residual explain reveal linear I wherein coefficient thus multi subspace representation correspond label belong wherein multi rank wherein stage group multi thresholding label preserve mapping information encode explore label matrix correspond matrix bound indicate cause monotonically iteratively value complete represent characterize label residual main time require impractical multiplication decomposition accelerate letter projection w I wherein matrix include multiplication require initially distribution firstly calculate adaptive algorithm summarize stage acceleration htb kn kl py ny l ir I introduce group representation stage decompose component sparse space component label lie subspace group wherein define corresponding coefficient nonzero concentrate belong accord solve problem integer final prediction vector thresholding group sparse although via select coefficient threshold guarantee summarize htb evaluate dataset annotation scene music categorization web compare knn five evaluation evaluate effectiveness cpu experiment server core ghz intel processors gb ram prediction five metric ham score prediction give label wherein one rate operation exclusive accuracy scale medical music select yahoo website table sample vector yahoo yahoo education yahoo yahoo show cpu knn table matlab train classifier parameter knn knn yahoo hamming score cpu second ml knn education ml knn ml knn regularization uncorrelated subspace
c mrfs cross scale coefficient across texture moreover global illumination classification model learn powerful mechanism classification affine space pca handwritten digit recognition oppose convolution learn algorithm lie group rotation intra variability operator clutter responsible intra variability define linear network wavelet modulus locally signal affine compute art handwritten texture descriptor classification appropriate include complex form invariant transformation rotation elastic manifold kernel operator address space high property operator lipschitz relatively log invariance linearization operator invariance interference class domain penalize rotation concentrate translation invariance carry difficulty application review cascade wavelet transform modulus operator affine training describe art digit database software build transform begin wavelet mapping coefficient modulus operator operator operator lipschitz continuous wavelet orientation angle large frequency function operator q complex wavelet orient complex wavelet numerical gaussian finite q derivative amplitude frequency fine scale translation improve frequency frequency modulus modulus eq concentrated may frequency modulus wavelet concentrate frequency xx interference combination depend exact location wavelet modulus modulus wavelet transform thank towards coefficient wavelet modulus locally q convolution carry wavelet j insensitive reduce phase modulus provide co occurrence scale corner edges texture one frequency fine wavelet average procedure eq co occurrence family angle satisfy verify invariant key x eq term signal locally linearize lipschitz class computation source material illumination variability build affine path large affine first covariance affine model empirical empirical mild covariance belong dimensionality learn affine computational estimate affine dominate calculation eigenvector thin operation signal let select affine family embed affine l ik classification class centroid affine space highly discriminative dimensional far adjust discrimination dimension approximation adjust dimension penalization penalization factor parameter optimize intra class variability penalization thus handwritten recognition texture discrimination illumination wavelet length mnist digit provide good classification compare currently table compare optimize coefficient find optimal compatible classifier yield deep apply pca validation linearization approximation affine digit intra class function compare space belong intra fast decay affine
discrepancy name context call approach root statistic exploratory datum choose simplicity include regular histogram string estimator discrepancy principle approach also penalize deconvolution discrepancy exist weak rather old integrable density decay slowly inconsistent extend exact version quite asymptotic even sample size asymptotic mostly previously suggest wide size last conclude existence distance kolmogorov define continuous depend usual easy note probability sign dirac sn show surely condition analogous statement prove find pp surely n n df show solution previously sample explicitly underlie surely already function monotone bandwidth principle unique suggest subsequently seem sample show realization mentioned numerically possibility discrepancy usually criterion know formula kolmogorov function function law iterate logarithm see discrepancy principle need obtain behavior estimator similar theorem hence part h tight page continuity interval identically around continuous consistency bandwidth bandwidth go rather let denote old exponent follow q lebesgue borel set integrable old continuous exponent similar old exponent next consistency estimator discrepancy old sufficiently exponent go slowly hence density fulfil yield chapter imply fx dx surely almost consistency threshold consistency rather inconsistent rough vanishe quickly iid draw compactly n f ab compact almost surely surely example density discrepancy lead consistent inconsistent estimate uniformly function iff fulfil w trivially case elementary since compactly directly consider threshold version discrepancy propose order need generalization nonnegative part second df df give small surely discrepancy intuitive case slow principle law iterate logarithm introduce correct nonnegative propose go chapter version high almost surely triangle theorem order form choose risk depend accord interpret threshold suitable density discrepancy principle guarantee asymptotically optimal sketch asymptotically yield depend invariant rescale kernel translate rescale standard small choice lie extremely severe sample size lead version principle work mainly conduct mention large estimator include discrepancy included find suggest extensive describe good whether discrepancy principle perform discrepancy principle specific applicable method quantile kolmogorov ks propose fulfil method l nr formula extensive simulation variant standard brevity largely size density package density figure opt variant typical sample bandwidth fairly bandwidth k obviously large small fourth nr asymptotically arithmetic density size table density ks ks nr cv lr ks nr ks ks nr lr risk although risk quantile kolmogorov ks ks kolmogorov statistic beneficial risk mainly nr although asymptotically choose bandwidth poor seem capture multimodal density bandwidth e surely density density e bandwidth estimate large discrepancy nr distribution density select lead perform worse discrepancy except nr small fulfil discrepancy principle consider inconsistent state bandwidth cv lead inconsistent applicable discrepancy relatively principle choice perform lr consistency guarantee density generally simulation consider much largely base iterated discrepancy easy branch mathematic rarely estimate density infinite peak work surprisingly discrepancy principle size much choose normal nr poorly take recommend ph thesis tu university collaborative nonlinear author support discussion lemma corollary foundation department science university de investigate discrepancy
variable consider base width variable evolve environment influence trait influence fitness evolve convergent regressor regression suggest hand suggest fit display evolutionary along available www c regression trait dynamic trait accord sde generalize study generalize yield research focus evolve markov highly correlate system adaptation treat trait meaningful importance maintain algorithmic predictor herein optimum sde imply predictor appropriate sde computation ease presentation sde diffusion random process initial solution e x let conditional sake pair specie sde x jt j jt j trait time e successive increment take isometry equal j acknowledgement anonymous associate anonymous comment allow manuscript substantially partially national science award roc second author work foundation award trust fellowship national mathematical synthesis science department security nsf national foundation ef pt studying trait relationship develop herein adaptive estimate novel square algorithm implement process comparative comparative group relate specie comparative evolution specie evolutionary history view evolutionary specie often incorporate list study reference describe evolutionary relationship specie evolve statistical dependency tree trait trait brain paper consider response trait toward optimum precisely trait stochastic differential sde eq measure adaptation toward mean change evolutionary dynamic responsible trait toward indirect force optimum correlate evolution trait brownian motion bm optimum bm ability central tendency dependent evolutionary predictor response find curve relationship trait value develop identifie rely free univariate make computation knowledge methodology set evolution adopt adapt base overall pattern paper describe provide implementation summarize suggest direction topic foundation let trait sde change predictor predictor variable appropriate evolutionary trait evolve evolutionary trait optimum evolutionary curve trait regression tb specie whose pair trait correspond branch evolutionary root specie evolve herein see evolutionary sake unless otherwise result trait deduce common eqs estimate thus need residual lemma trait covariance eq adopt variance regressor vector method observe value specie represent trait value regressor entry equal coefficient optimum mean trait incorporate process rely predictor enjoy close branch length accord depend optimum therefore estimate adaptation trait
cost apply map near neighbor centroid centroid high preserve separation shift generate centroid minimal deviation optimally denoise one level hard low noise method cut optimum temperature flat maximum like report mutual gibbs distribution expectation easily transformation calculation integration mutual around except modification analog mutual term transformation depend auxiliary truncate value decomposition close select svd principled cut spectrum convert code optimal simulation theoretic determine state selection decomposition exploratory thereby unitary essentially induce rotation quite rather interested justified underlie cutoff define truncate svd method approximation coding approximation capacity enable compute capacity channel originally discrete problem select rank truncate thereby investigate challenge space method indicate svd remainder derive svd problem principle page description optimization di satisfying temperature minimize play central give interpretation hypothesis approximately define code receiver via transformation asymptotically vanish channel achievable encode effect lead high patch overlap decode possibly approximation capacity select maximal going list quantity truncate frobenius optimizer convenience frobenius svd instance relation basis linear basis rank cost minimize encode mutual eq originally space binary string sum first volume hypothesis infinitely receiver distinguish union reason capacity analytical calculation transformation r weight u sum np condition temperature distance solution temperature suggest high constraint temperature infinitely htb discussion infinitely infinitely negative infinitely transformation volume practical criterion svd integral finite space hypercube isotropic identity way grid experimentally investigate study influence integration capacity mixture isotropic lead capacity sphere standard finding fig n none could possibly codebook indistinguishable capacity contrary capacity analytical solution infinitely serve htb transformation mutual grid increase increment preserve transformation density impose large dataset
spatial model investigate generate instead randomly add lattice generate add link proportional q connect degree link fig obviously adjust spatial network link e super link connect therefore tail long heterogeneity large heterogeneity confirm indeed heterogeneity suggest heterogeneity real modify spatial perfectly preserve see fig length identical reflect general conduct affect optimize process report always become shift small dimensional imply exponent previous homogeneous distribution heterogeneity shift value however optimal enhanced introduce heterogeneity heterogeneity favorable degree average short however shortest overcome effect degree heterogeneity keep average independent introduce heterogeneity spatial law shift far decrease heterogeneity explain law distribution exponent system heterogeneity explanation wide length empirical heterogeneity short explain heterogeneity range sense understand design thank zhang chi helpful suggestion work support science spatial range long constraint short spatial network propose spatial constraint result heterogeneity exponent shift short length synchronization reproduce play internet phone network grid past decade system model locate plane distance generally reproduce spatial associate link introduce network link link site dimensional choose distance length reach exponent control number formation link length short author optimize aspect availability customer company carry traffic constraint observe degree network total system many work reveal network propose degree heterogeneity heterogeneity affect exponent link shift short heterogeneity degree heterogeneity place phenomenon exponent briefly describe locate lattice near neighbor pair site receive site e separate regular long control average length long correspond vice versa path network minimize certain optimal accordingly exponent b usa average receive proportional therefore space equivalently nod
model use slide window window far union separate clarity mining relational weighting temporal view active current step relational treat additionally relational instance temporal predict attribute weight recent highly decay rapidly define ed w historical long lie exponential historical inverse w could e traditional appropriate temporal learn temporal since diverse simple use validation learning extend naive bayes heterogeneous relational subgraph model topic topics nn ga nn rl nn ga design mirror conventional assume attribute independence label calculation link calculation incorporate temporal attribute instance heterogeneous algorithm search feature average mode dynamically relational create split use standard zero conditional computed sum link attribute standard attribute weight select appropriately augment traditionally prediction vote propose ensemble method ensemble information relational vary link temporal ensemble propose ensemble five temporal ensemble sample discrete strategy perform construct temporal temporal weighting additionally bias toward past strategy time link method ensemble transform temporal space temporal weighting learn parameter temporal ensemble noise temporal dimension node observe past link temporal learn distant temporal randomly relational likely due diversity correctly predict instance additionally temporal assign dataset attribute I communication email communication extract www list period email user report contain comment user size text message dynamically individual discover centrality cluster prediction task pt conv tool tool window module ex count email email topic eigenvector count topic email count topic citation database citation information research paper extract automatically web seven machine paper ai reference addition work author temporal relational evaluate relational area describe model weight union window otherwise information dt relational ignore temporal window besides vary classification evaluate different relational vs also different centrality team demonstrate temporal representation discover attribute temporal outperform ignore dynamic significant improvement traditional relational ensemble vary apply mining discover link temporal topic feature correspond evolutionary classification effectiveness discover vary incorporate pattern task brevity compete non dt auc relational model show case appropriately assess prune primitive temporal relational relational information use model various explore relational relational compare primitive relational classifier dt dynamics relational interestingly improve weight learner suboptimal dt see improve indicate focus attribute explore small representative set framework representation choose correlate class influence link attribute learn explore appropriate show attempt apart superiority outperform necessity temporal optimize select strict representation link model worse significantly scalable include search exponential link relational representation depend g citation temporal construct robust reduce relational increase temporal bias temporal relational representation aid mining evolutionary improve explore selective e motivation link decay rate learn temporal ability selective temporal attribute find selective temporal simple task temporal relational classification vary relational relevant relational repeatedly almost temporal relational ensemble traditional ensemble outperform temporal information link sophisticated temporal ensemble instance investigate ensemble use temporal wide ensemble ensemble traditional technique ensemble clear lot ensemble aim reduce significantly provide information utility information increase significantly information compare ensemble auc randomization primitive apart accurately attribute team centrality topic primitive temporal representation help minimum temporal relational relational complex find team localize change frequently unlikely temporal relatively percent temporal ensemble ensemble pattern indicate temporal notice change project frequently could responsible increase accuracy importantly significant model performance temporal example randomization attribute additionally randomization identify two significant randomization attribute thereby value association step due attribute change ensemble standard traditional ensemble figure traditional ensemble step even rely attribute relational past present use future behavior attribute ensemble topic useful indicate topic way understand among vary technique relational discover pattern relational weight attribute vary temporal insight temporal discover temporal consider link distant successively window link attribute node attribute successively past successively include recent attribute conversely successively auc increase increase drop temporal also quickly since paper publish distant generally show justify auc ability predict instance future might always noise unlikely past relate determine past behavior anomaly decrease back stability modify temporal classifier model network whereas stability lead structure unstable tree gradually evolve ai relatively stable addition also find perform amount hypothesis complementary advantage especially temporal justification method select representation ai ml use yet insight measure year ai ml task figure ai cite majority year year paper paper indicate transition paper become researcher factor paper influential omit brevity ai lag interestingly probability begin indicate past behavior use communication discover communication interested classifier version latent estimate gibbs representation justify complexity latent topic network investigate topic communication evolution use discover pattern explore relational c cc topic c topic code file os type mail fix release build day amp pm module home file support module check main thing good exception van ms list topic find positive related sentiment exception social appear van various representation classification necessity relational representation link attribute see relational perform indicate meaningful appropriately exploit remove brevity temporal relational outperform clearly annotate effective communication effective moreover time correlate ensemble mining insight network effectiveness scalability flexibility ensemble mine temporal acknowledgment research nsf contract number research make support air office national science fellowship reproduce purpose notation view
nash correspond equilibrium game see wireless sharing decentralize equilibrium game distribution far none player know strategy actual nash article information whether actor competition present principle ne finally access compose amount bandwidth token accounting protocol exchange service token spectrum datum transmission type spc spc spc resource sharing request actually spc communication kind allocate bandwidth spc thus allocate introduce bandwidth need bandwidth allocate free allocate however allocate token accounting protocol exchange service token represent depend whether green connection token bandwidth token indeed green due fail ht describe spc customer spc share connection green green determine indicate sensitivity spc tolerance degree towards risk spc sensitive green spc big say spc spc customer characterize parameter want spc resource tolerance indicate tolerance sensitive degradation price able pay connection sensitive amount spc spc request utility depend request request spc decision allocate spc bandwidth split competition receive bandwidth spc spc utility request spc competition spc rise focus competition spc competition access could model spc could motivated fact environment regular connection need spc access could game thank run software good strategy article game denote spc green part would played stable player mention spc spc share regard request request bandwidth amount elastic delay file connection price file transfer transfer file denote file spc amount bandwidth require file spc bandwidth minimal request amount green user signal strength spc request bandwidth limited since request spc connection could besides bandwidth amount request avoid amount aim interval request actually discretization focus request accord profile depend sensitivity degradation fix accord ht sensitive great propose cost pay maximum element permit green connection couple amount parameter highlight minimum bandwidth green request spc spc bandwidth allocate request accord request spc triple amount spc resp mean connection spc receive connection spc spc eq answer respect request spc equal formally spc bandwidth availability green bandwidth spc outcome consider spc connection maximized compete formulation describe player player strategy define pure corresponding couple integer quantify game since several allocation decision spc utility maximize spc gain spc allocation represent triple represent connection spc express cost connection utility nash equilibrium nash potential admit least pure nash equilibrium response pure nash good response correspond profile let profile pure nash equilibrium stop player change response new profile strategy mean spc end spc bandwidth game restrict admit one equilibrium want whether know converge pure nash equilibrium maximize step request use spc spc compute repeat profile converge prove consider game pure equilibrium sufficiently value nash decentralize learn nash equilibrium incomplete principle follow player discrete update strategy local represent gain spc gain technique else u correspond respectively player margin percentage step mention restrict pure nash equilibrium article try exist nash nash characterize spc file category specifically simulation present take consideration extremely gain sensitive spc want file communication article simulation strategy simulation strategy notation analytically compute present fill highlight nash payoff rise verify pure detect strategy converge expect expect expect remark strategy match pure vary checked pure nash equilibrium keep vary h c iteration fact consider ht ht nash eq gain even strategy per still pure nash equilibrium nash equilibrium gain pure reach stable game make represent rapidly fix respectively connection maximize number propose duration connection connection mean duration software update slot q focus per pure nash analytically permit game restrict pure nash equilibrium profile learn pure equilibrium strategy strategy necessary user give minimum amount file competition model game spc
tradeoff term monotonically decrease argument standard point newly assign never increase cluster pay penalty still decrease finite clustering perhaps objective past criterion aic penalize mean motivation constructive monotonic connection dp finally datum log choice sampler cluster formation cluster text extension dp arise dp top define prior mean mean share appropriately hierarchical reader detail hdp inference dp straightforwardly hdp outline datum mean share local cluster dp I hdp index define base mixture dirichlet base yield follow analogous global process prior extend employ hdp summarize derivation determine hdp require threshold work global association set assign create cluster close start global penalty distance plus sum local whose specification hard hdp initialize step else g dp mean cluster cluster intuitive minimize k whenever create appropriately cluster across monotonically minimize theorem monotonically local convergence two objective hard relaxed large eigenvector relaxed indicator suitably process clustering mean relaxed dp matrix stacking whose correspond matrix optimization note relaxation orthonormal standard argument cluster eigenvector relax cluster cluster common thus relaxation mean dp great possible develop dp mean normalize briefly review straightforward map j note express space expand c c unnecessary explicit assign cluster near implicit hdp point weight objective k function vertex edge adjacency disjoint collection cluster cut normalize seek minimize cut find cluster minimize minimize v c equivalently prove normalize mean normalize ratio let extend w k c objective optimize construction utilize distance cluster perform penalize normalize set dp k car balance scale breast brief goal enjoy many unlike scalability k ground output application match cluster ground truth parameter potential way clarity utilize desired iteratively find distance distance among repeat distance utilize hdp replace sum gibbs covariance fix inverse prior yield though bayesian consider two place via validation key terminate within run contrast figure clustering first learn use validation tune cluster poor validation sample converge contrast dp reach additionally dp benefit among gibbs comparable accuracy uci labels truth cluster datum set gibb yield run gibbs run ground truth table dp achieve datum scalability vision datum patch second versus dp fully obtain gibbs infeasible demonstrate highlight baseline ground choose uniformly covariance set generate gaussian per individually share result score across set baseline mean datum dp mean whole obtain score yield hdp form cluster mean dp mean individually share via hdp versus dp scalable retain benefit bayesian model several future basic mean relaxation iii nonparametric process process generalization comparison helpful proposition corollary offer cluster hierarchical utilize multiple flexibility method viewpoint inspire mean gaussian hard result monotonically elegant like analysis multiple argument dirichlet extension relaxation thresholded cut statistic impact field learn instance dirichlet document dirichlet result gain simple preferred bag collection motivation algorithm implement work variety attempt design scalable viewpoint connection mean mixture gaussians means view maximization matrix mixture show dirichlet nonparametric argument simple hard except form point cluster centroid monotonically include take hierarchical model hierarchical dirichlet set take hdp novel cluster result cluster across cluster turn mean local cluster global cluster additional extension mean arise relaxation compute compute eigenvector highlight connection bayesian graph early algorithm monotonic unlike formulation cluster conclude result underlie chain indicator proceed point perform indicator cluster z ic normalizing start new cluster gibbs point currently assign often g g infinite particularly dp define covariance mean prior compute closed calculation z assign z happen obtain assignment
use measure rather minor since always valid valid valid old existence univariate find kernel consider assumption density function exponent level density kernel extend assumption give correspond observe theorem give depend lemma case term dominate assume star shape generally star shape corollary equation risk note hard estimating level cut logarithm also assume contour old low smoothness plug although construction apply cut interest illustrate prediction practice bandwidth drive density intuitively estimator near kernel quite estimation guarantee contour bandwidth candidate idea applicable validity small preserve validity correction input level construct satisfie use q use conservative coverage much big level sized subsample finite validity loss negligible although excess minimize symmetric small loss difference detailed require illustration well present bandwidth sample select table coverage lebesgue repetition coverage regardless decrease cut tune excess loss support corollary agree figure realization dot show sample splitting blue curve inner set respectively grey clear three capture close panel require parameter practice estimate prediction picture hull construction ideal region least fast depth excess axis stable bandwidth especially moderate contour dense allow moderate phenomenon interest construct combine idea statistically level method compute near without regularity validity algorithm popular device comparison conceptually stable future aspect bandwidth selector well study procedure smoothness possible develop deal nonparametric n yy yy level estimate view process nest empirical vc kernel suitable formally define easy nest empirical theory eq eq triangle result exist constant conclude part eq subsection neighborhood exponent enough q allow switch q applying empirical definition inequality suppose event exponent condition event enough consequence proof observe therefore suppose event condition right side depend example nsf support dms air force fa university e investigate region establish region regularity condition approximation simplify bandwidth selection theoretical demonstrate prediction want prediction product pc ny name prediction region beyond anomaly item fall prediction previous indicate sample investigation mass concentrate choose give rate lebesgue contour formulate region thorough introduction book criterion efficiency region prediction region formulate notion validity however evaluate probability work prediction hand free efficiency closeness provide contour unknown benchmark set prediction estimation difference loss eq region minimal propose validity constant cost check linear rate formula smoothness behavior near contour special combine idea prediction estimator sample validity nature region analytical form carefully tune value efficiency approximated density level always region validity finite validity computational cost refine rate convergence tuning bandwidth density drive demonstrate construct region method general construct prediction exchangeability prediction equivalent block value coverage prove order completely nonparametric rank n pp finite coverage usual focus aspect discuss subsection augment region algorithm example investigate histogram mixture plot middle middle three estimator bandwidth lead valid lebesgue bottom plot bandwidth whose corresponding minimal prediction region give closely characterization level lemma also region introduce density denote cdf version sample respectively eq outer set proof accord much validity next investigate efficiency region subsection efficiency term convergence level value mass exactly continuous equation equivalent contour outer plug ny nh regularity condition consistent refine modified condition
swap swap example like uniformly graph similarity enyi swap metropolis toward empirical qualitatively show value eqn expect world dirichlet scalar vector compute sort repeat replicate average normalize nonzero eigenvalue replicate procedure average result behavior qualitatively quite prior shape shape separate become concentrated around swap top eigenvalue quickly merge eventually go infinity become regular relative frobenius show swap regularize sdp regularize construct laplacian sample wishart eqn laplacian edge implicitly sdp observe criterion relative appendix present criterion spectral error intermediate value plot replicate replicate implicit regularization improves regularize sdp correspond stronger improve strong intermediate level figure improve wide level illustrate obtain level figure similar explicitly case implicit regularization parameter illustrate depend illustrate optimal proportion converge high value like agreement interpretation quick laplacian exactly regularize present instead result make implicit extension suggest extend characterize implicitly eigenvector statistical application though illustrate become increasingly analysis department mathematics stanford university stanford quick nontrivial eigenvector certain regularize program provide manner regularize often ridge lasso respectively interpret gaussian laplace regularize sdp estimate conversely imply run base nontrivial eigenvector eigenvector correspond exact heuristic time output sense recently formalize context graph top nontrivial optimize equivalently optimization program objective program semi nontrivial interest usefulness graph image segmentation semi etc ask diffusion run heat quick nontrivial eigenvector algorithmic tradeoff box solver optimize interesting regularization sdp rather diffusion sdp constraint analogous regularize regression regression prior laplace respectively detail whereby interpret population laplacian drive random define inverse population posteriori estimate prior assumption regularize sdp heuristic say sdp interpret regularize regularize computed sdp solver laplacian heuristic nontrivial eigenvector laplacian background section describe section describe implication importantly light certain decision brief appendix define set symmetric case combinatorial laplacian semidefinite one often eigenvector eigenvalue define normalize version laplacian laplacian graph laplacian laplacian compute black also walk state walk diffusion base negative evolve dynamic three dynamic following evolve equation evolve seed evolve term add penalty interpret incorporate prior observation constant depend encode computed maximize equivalently value minimize regression respectively ridge impose interpret impose problem normalize view laplacian induce observe formalism follow observation specific another laplacian node population degree degree laplacian matrix analogous wishart trait accurately one however laplacian appendix provide justification precise denote support eqn analogous eqn play object function eqn eqn eqn linear program note sdp particular program except prior estimate sdp appendix prior heat prior invariance subset semidefinite set eqn support orthogonality sense empirical eigenvalue laplacian equivalently normalize scale factor simplex exchangeable permutation neutral degenerate possibility dirichlet must eq one implication prior nearly
interactive mechanism modification multiplicative interactive object section private interactive online mistake analogy analogously multiplicative weight normally maintain distribution element element universe update query element ever need update key pick universe let element initially hash mapping update ever implement run neither result depend interactive polynomially compute inner product project project vector normally represent however limit representation also become standard solution vast differential exhaustive et answer arbitrary sensitivity interactive mechanism make however answer query database interactive answer sized family counting improve al improve differential result arbitrary query comparable multiplicative generalize reduce release interactive unfortunately discuss private bound suggest private release query substantially run guarantee bad query give release mechanism give size discretized set far efficient sized simple relax notion long bad case efficient algorithm sphere respect point algorithm attribute run et bound expectation give average private boost although require I algorithm error complementary error guarantee require database interactive algorithm mistake implement multiplicative analogously set thought universe involve multiplicative small universe give universe difference select universe run adaptively whereas subset linear randomly database contrast bad utility query projection family independence previously use stream limited streaming element universe mapping universe define database think sometimes evaluate linear normalize qx universe polynomial quickly query list universe class even many accurately linear interactive one time output capable answer interaction mechanism interactive mechanism query answer next interactive mechanism let query interactive abstract qr q interactive mechanism mechanism adaptively stream query query polynomial size require define notion neighboring database database individual I differential randomize act database range differentially neighboring database interactive output choose adversary interact adversary differential privacy laplace center fundamental privacy sensitivity laplace preserve neighboring database procedure preserve privacy understand individual privacy prove composition universe ss tx tx class sparse small weight analysis use must database sequence define hash table argue never run argue apply infinite ever attempt consider private potential state number tx x x tx si expression argue every drop least begin know drop x x tx tx tx tx tx update q never report observe variable ed therefore last follow recall choose proof update theorem differentially mechanism interactive set run query sparse bind prove recall small application algorithm lead universe conjunction release multiplicative weight follow differentially interactive release set time roughly moreover per query super polynomially class interactive release algorithm privacy polynomially sized class constant interactive release query together database vector obtain multiply require projection use hash function family use independence prove limited wise independent also preserve inner product integer random matrix vector ax ax ax ax ax identical ax compose output kt wise independent hash tu hash wise purpose integer finite select represent function hash last bit wise independent hash mapping write u therefore composition preserve magnitude matrix consequence chernoff limited independence random query independent entry projection recall query I qx dominate rademacher equivalently write mb mb b markov wise independent dt dt follow chernoff plug recall union query query prove sparse wise entry denote q also use laplace prove I scalar prove logarithmic projection implicitly random using introduce laplace analyze source together first probability conditioning event occur complete briefly mention super polynomially conjunction say conjunction sparse size differentially private interactive release polynomially run accurate super polynomially sized interactive super polynomially sized also achieve interactive interactive class query release run polynomial unfortunately fast release random projection seem powerful tool query comparable pt fact pt accurately answer differential privacy class relax guarantee zero polynomially universe interactive interactive technique sparse universe runtime mechanism universe universe universe specify database record think infinite task statistical query database provably preserve individual record use become accurately answer privacy date come type accurately arbitrary family query database unfortunately size universe database impractical technique query rich class sensitivity answer axis align query vc conjunction offer choose query open privacy release know exponential polynomial restrict universe typically database polynomially universe sparse query still entire universe vc dimension rarely ask analyst medical study might analyst disease sparse analyst knowledge overlap participant analyst database beyond merely universe information privacy preserve matter way analyst algorithm accurate preserve differential act adaptively choose stream arrive interactive shot output datum structure encode interactive query answer sequence take
compare four consider range compare reflect bottom panel full observation full mle substantial fold standard case low usually perform well cell use cell useful right real practice variance ratio substantial whether substantial improvement leave fraction interestingly standard hashing b bit hashing poorly hash bit hash scenario improvement b bit involve contingency prohibitive sum achieve improvement consider permutation provide formula convenience assume large verify q q q ratio without std standard binomial really multinomial define probability event solely however combine ensure accurate ratio even hash mainly focus estimation estimation application use text column characterize tool user understand context contain column describe compression coordinate reduce message pass small estimate recent bit hash storing bit prove bit storage bit target encouraging result improvement application page adequate substantially increase permutation similar work svm hash good estimator hash apply three q convenience presentation intersection unbiased provide variance delta straightforward skip lemma less obtain resort likelihood mle equation unbiased result theory variance estimator panel reduce variance improvement fold magnitude bottom confirm well know may right panel well pair occurrence real web page contain present verify pair mle two magnitude unbiased mle simulation bias mle vanish hash low bit b available form form low practice convention estimate hashing tool multinomial estimation cell e summary conduct permutation achieve improvement review basic probability
distribution count different context indeed invariant reason favor assume permutation note associate order table new tn n tn metropolis probability sufficient positivity instrumental employ stay feasible candidate proposal l un see positivity condition propose candidate option candidate cell count recall free permutation cell equal count cell count th mn j mn c modify table generate order table return free return belong mn mn cell using eliminate cell attempt describe local move union various order translate mn mn mn mn candidate table instrumental distribution mn n mn hasting need n n effort result chain irreducible positivity discrete performance sample jump move small less move around specify tailor arbitrary table produce proposal whose uniquely determined particular bound constraint contain table sn tn sn necessary sn computation substitute sn reduce upper perform binary search index u jj b r table never n algorithm return sequentially guess calculate si j si ss ss cells line tn return equal purpose sufficient repeatedly iteration proportional keeping free result successful different generate move explore make adapted table generate call dynamic basis example zero eight table effectiveness describe section compete unable generate basis algebraic importance sis chen calculate cb henceforth implement package table structural zero produce burn chain cb sis generate sample run replicate define hasting proposal distribution monte carlo mean batch p estimate sample batch carlo error value batch iteration throughout batch process mac intel report compute c sis chen al sample sis implement solve optimizer code replicate material individual group business self teacher employ survey horizontal structural ten education level similar manner freedom interaction subtract zero correspond zero al must increase marginal table level freedom interaction lead exact section sample agreement second million iteration table fix e markov chain dynamic two show p increment calculate sample table dot represent mean incremental estimate solid represent way datum table classification eight relate economic page yes child yes education school yes education yes cell mean observation table contain observation applicability relate limit due marginal two interaction observe interaction degree value markov markov cb cb exact cb monte markov cb error p recent chen report version sis use sis chen chain replicate run second markov million second h count vary fast varying total value show exact number iteration increment sample dot incremental represent number markov basis sample exact instrumental key tm candidate would c instrumental increase obtain effective determine lead chain dynamic good running finding properly adapt table procedure integral part dynamic markov basis need complete practical determination one step question study basis algebraic spirit list relate basis material computer material contain file run base markov basis importance sampling description file txt acknowledgment grant sciences university thank author comment generate markov basis contingency cell marginal low zero framework find markov table move structural zero eight code article keyword contingency chain structural sampling contingency perform arise nuisance minimal sufficient validity nan approximations raise cell large structural e g occurrence one way importance chen chen chen chain thompson al literature table concept behind understand pairwise n tn tn general computationally solve simple markov move equal connect primitive extend log basis basis determination table cell arise limitation inference lower induce theoretical basis include unfortunately quite carry principled assessment algebraic dedicate software far paper dedicate basis preliminary complete start perform basis contain three repository useful move simple decomposable avoid basis advance instead connect current applicability markov basis example handle approach present set basis table algorithm chen et applicability zero conclude variable contingency observe ki ci km cell order array ni ni ci cn cross c kn un bounds specify cell structural zero express take addition constraint satisfy minimal consistent odd observe induce bound li programming constraint exact count cell cell determination need comprise integer array order equality system e assume determination complete analysis constant constant perform volume arise conditioning multinomial term straightforward computationally infeasible induce markov front expensive walk impractical reference table basis contain number move difficult handle algebra necessarily need entire chain basis generate ahead consist connect neighbor union neighbor two algorithm always valid reciprocal cell include h uv u l strictly neighbor e move one correspondence table expensive value give increase array equality correspond stack along determine reduce row gauss number system efficient programming small translate determination bound represent vector version linear cell cell n determination use bound system line algebra gain determination cell calculate application certain combination integer linear constraint associate sample cell make sample positive key sis calculate estimate quantity exact various desire table distribution sis describe chen chen many completely numerical involve way table table contribution chen previously cell value version sis eight table argument example situation sis distribution importance sampling procedure section c method table marginal limit applicability index cell free I imply
outside preliminary notation map identity associate composition restriction matrix operation complement form column rest unchanged na ni ai n q way projective system act remove row edge obtain vertex label edge n projective projective restriction map restriction map projective system restriction permutation map I permutation monotone monotone uniquely determined subset monotone set say subset subset symbol subset distinction become restriction ga km usual composition collection class edge whereby without correspond clarity restriction respectively projective relationship collection elementary category category c must satisfy correspond composition must represent object need make object implicitly restriction category object give element object define monotone cover n preserve aa show infinitely infinitely exchangeable project subset look subset either restriction intend notion subsample subsampling depend way make chinese restaurant crp permutation modeling represent among population observe social reasonable intuitive sampling subsequently edge restriction accurately reflect removal remove without node subsequently u radius less depend choice removal nx nx necessary infinitely action ar r collection exchangeable mean every b exchangeable suppose finitely cardinality cardinality permutation restriction ne ne ne ne consistency restriction reduce eq restriction element infinite endowed implication start exchangeability negative b infinite exchangeability direct corollary exchangeability doubly sequence non negative exist space graph denote vertex induce hausdorff hausdorff know throughout triple vertex adjacent argument os account comprise edge graph information implication miss link absence know monotone set ccc five lack involve affect cluster involve description overlap subset onto also helpful nature naturally certain statistic individual network undirecte let collection equivalence relation every network implication parse link social among individual clique due unit sample assume point imagine infer observe image monotone cover collection monotone exchangeability g element population presence within reasonable inference cluster context cluster sample restriction newly classify already exchangeability gauss allow restriction realization poisson observe complete nothing
decomposition therefore nx n jx conclude proof hold q precisely proposition linearity true interesting reference reference product compactly department university also useful theorem remark remark width nsf grant dms statistic al united adjust langevin mala move incorporate mala start stationarity suitably chain mala imply dimensional compare walk mala previous target applicability correlation rescale mala weakly hilbert scalar show decomposition suitably resemble eul discretization martingale continuous numerous arise affect quantify cost part develop mcmc arise application product gain class hasting propose accept proposal brownian langevin proposal langevin quantify increment proposal rwm adjust langevin mala aim analyze markov determine optimal increment precise suggest existence accept may large proposal large move large move still order satisfy dimension question concern choice suitable required stationarity proportional diffusion research along paper concern rwm mala direction converge determine parameter rwm proposal mala require target rwm mala quantify efficiency gain mala rwm employ inform feature optimal maximize rwm mala analyse huge concrete tune proposal rwm mala algorithms acceptance probability practitioner criterion given ask extensive see distribution subsequent beyond slightly variance diffusion limit contain diffusion employ considerably complex form limit diffusion perspective motivated nonparametric condition separable space inverse additive noise absolutely first measure subspace reference normalize interested mcmc finite approximation project onto eigenfunction brownian operator rwm apply dimensional weakly furthermore target scaling exponent speed limit I remarkably case fundamental method dimensional measure develop limit numerical derive limit wide hasting local consider mala context field model behave measure contribution output mala suitably apply dimensional approximation limit diffusion acceptance regard remarkable langevin form add demonstrate develop numerical obtain technical estimate conclude state develop fashion theorem measure induce structure finite enable mala scaling probability follow notation sequence expectation sequence constant notation real satisfy notation let separable hilbert trace family hilbert refer function represent via often move expansion see allow variable product norm rr rx rx alternate space identity r k b rd rd orthonormal say trace j include every belong furthermore induce alternate measure condition formula valid eigenvalue existence impose direction comprise functional however identification identity avoid first derivative operator exist constant ensure trace full measure full derivative x x h h refer interested approximation end span eigenfunction notice subspace approximations reference introduce law random normalization support nx equal schmidt u strictly repeatedly give langevin leave similarly define informally sequel lemma show quantify various repeatedly sequel consequence proof satisfy functional define satisfy sm satisfy order formula sm estimate constant lemma lipschitz similarly globally mala langevin diffusion motion operator mala proposal euler proposal n time euler discretization derive reject ensure average grow detail choose diffusion stationarity lie markov chain evolves implement acceptance coordinate form langevin proposal acceptance define eq expect acceptance stand also bernoulli success may bernoulli key behind quantity behave like summary markov describe project mala lebesgue start stationarity solution informally scale mala proposals exponent exposition simple explanatory dominant inclusion carry n proof lemma evolve stationarity n n show acceptance acceptance diffusion summary choose probability infinity exponent move jump expect adopt efficiency exponent analyze describe mala nx acceptance mala rigorously prove corollary accept rigorous theorem show rescale hasting mala mala converge weakly follow result accelerate stationarity explore explore stationarity maximize choose acceptance one mala algorithm acceptance first implication mala explore invariant implication law speed invariant fast practitioner tune acceptance express mean probability speed function maximize precisely product acceptance outline drift martingale statement paper pointing proceed full subsequent simple scalar scalar markov weakly acceleration scalar variance success sequence converge acceleration diffusion original idea arise instead work hilbert value coordinate present precede account variable autocorrelation bernoulli variable markov mala start stationarity invariance principle mean acceptance depend randomness quantity approximate approximation lem lemma bernoulli limit mala drift martingale constant define ensure array drift decomposition read exploit behavior version approximation resemble euler prove martingale difference q process brownian strong brownian outline continuous additive define closeness see eq continuity continuous x infinity prove approximation chain evolve space drift eq go infinity rescaled diffusion follow general diffusion chain separable chain suppose stationarity martingale converge converge globally lipschitz rescale converge weakly dt brownian motion independent define map equation accomplish nz n ax nz nu nu nu nu nu last nu z nu n first check satisfied sequence mala key lemma weakly state satisfy nx need satisfied establish asymptotic chain quantitative every infinity verify let n proof independently consequently j lemma jump globally follow normalize uniformly functional measure converge weak contain prove need function normalization uniformly bound convergence surely converge surely prove sure result state exponent repeatedly sense principal come function converge law z rigorously generality quantity expand quantity n ny pt n n purely change involve simplify constitute q expand remainder taylor x x lem ny put together state globally nz ny simplify demonstrate definition tx x nx nx give equation b n quantity lead term quantity eq
proximity fact composite valuable regularizer problem product induce every shorthand p proximity operator convex proximity operator exist recall subdifferential subdifferential nonempty compact subdifferential gradient establish relationship proximity subdifferential proceed positive define fact immediately namely norm search simple soft thresholding otherwise last large example recall fact theory refer exist mapping call define well fact fail state tool fix mapping I appear fall class every penalty prescribe penalty e form overlap apply formula separately proximity operator compute function z n di fuse consider immediately fall intuition fuse vector case choose incidence consider composition norm specifically absolute permutation xx rr form singular order correspond proposition proximity base von call fan uv right vector von sometimes von trace fan state system diag suppose contrary obtain th component positive contradiction form task nuclear consider gx r assumption convex occur build proximal use compute proximity special case proximity operator develop apply minimizer unique minimizer subdifferential inclusion formula combine inclusion term proximity formulate convex minimizer conclude add subtract fix conversely hold proposition b spectral eigenvalue iterate without need use provide proximity mapping bb part proximal corollary motivate proximal approach lipschitz idea behind therein approximation point simple iteration accelerate nesterov specific property objective optimal achieve certain interior iterate reweighte square apply successfully nonsmooth hessian cost system accelerate involve desirable accelerated see restricted computation proximity method exhibit far empirically accelerate influenced nesterov update achieve recursive fista computing operator sequence combine compute efficiency nonsmooth problem demonstrate art case exactly computable overlap proximity know hierarchy acyclic graph order improve moreover report incidence penalty aware composite apply variation lasso build nesterov software advantage proximity regardless linear scalability methodology simulation case square run first consider synthetic set simple embed generate uniform randomly nonzero group group assign discussion choose coefficient group normalize like zero noise run datum correct zero due objective reach efficiency iteration different indistinguishable conclusion cpu show depend almost grow higher comparable proximity insensitive require cpu grow remove show nesterov match tree structure lasso berkeley segmentation extract dictionary place branching practically indistinguishable synthetic draw explicitly connect cluster dimensionality accelerate decay overlap monotone decay graph make progress close long future accelerate optimum verify fast time matrix quick addition incidence total without datum nesterov identical favor solve class nonsmooth give nonsmooth term composition example cover regression term nonsmooth feature deal rich regularizer efficient exist demonstrate specific fused group handle composite accelerate yet theoretically suggest simulation room acceleration case method include range learn linearly composite problem wish useful discussion support air force grant fa grant h nsf international european cm
apply determinant newly covariance datum aggregate entropy within gp integral covariance eq whereas scalar maximum datum define solution next candidate adaptively domain overlap one high dimension large sampling computationally feasible monotonicity determinant follow counterpart collection approximation computationally costly computationally variance lead rough widely heuristic minimum discuss point collection proposition nonlinear dynamical evolve action control state nonempty evolve scalar often observable mapping nonlinear system relationship possibly simplification observation noise model variant respective control nonlinear discrete partially observable input nonlinear maker maker maker limited goal describe dimensional reference model dual control formulate system output output variance function x good quantifying observe adaptive control maker start zero little put emphasis quantify section dynamic shannon information maximize base discrete dynamic relationship u goal control system reference let system control objective follow side unlike static close approximate reason reasonably dynamic costly purpose utilize methodology approximate objective adopt candidate solution sum utilize visual iterative account newly receive start exploration action reference estimate gp associate approximate strategy solve couple make regard present firstly turn control unknown system step depict figure note mapping htp htp htp unknown map affect controller relationship obviously hard control initially increased gradually summarize greedy accurately process long twice control less keep mind identify control htp htp htp cart classic system case position cart nonlinear state period position cart cart angle angular standard available step look strategy control cart result provide benchmark htp htp cart control use ahead controller act external dynamic choose obtain show satisfactory within htp htp book provide valuable insight relationship discuss however focus book problem play important introduction machine literature gps get increasingly popular characteristic book comprehensive treatment gps additional area active neither discuss quantification build shannon theory use gp discuss collection discuss objective measurement subsequent active gp heuristic measure old attract interest research community dual focus adopt dynamic reinforcement application identification dual present address focus limit quantify serve quantification allow joint two illustrative cart develop static way influence control action identify select static constraint control present mainly future research investigation exploitation trade elaborate weight person theory constitute control decision priori posteriori identification exploitation select action provide point identification control quantify serve art regression quantification identification control objective illustrate map position control cart make limited obtain accurate often infeasible time fast change controller collect underlie dual obtain control controller controller must controller perturbation differ amount information controller controller aim stationarity prohibitive perturbation action observation provide single identify discrete continuous perspective process underlying equation time training adopt allow quantification goal explicitly combine objective pose multi control control iteratively objective resource approach uncertainty additive likewise variance additive observe provide incorporate simplify provide supervise gp regression rely gp describe iteration observation determine optimization take objective information measurement objective issue infinitely collection incorporate many concept implicitly build upon information learn specifically field acquire control adopt bayesian use process iterative capturing meta control despite heavily utilize different aspect image lack expectation inherently information play datum develop strategy obtain scalability worth note problem seem social economic information variety fundamentally decentralize resource allocation decision change quickly characteristic decision nod another security management costly another biological operate system summarize present utilize control regression gaussian modeling observe learn specific gp provide follow spirit trade one hand try minimize try estimate unobserved end fit follow predictive unobserved balancing prior capture basis observe gaussian regression preference
site merging course run mean current estimate run strong suitably unconditional site suitable site lattice magnitude large cluster site cluster size site probability idea essentially inside another simulation list function grid run header change empirical cluster different estimator produce subsequent run run run estimator generate neighborhood generate unconditional manually simulation n run pp generate use binomial input success p run run remark update six neighborhood provide tt tt run scheme permutation belong run maximal edge tt edge edge cluster neighborhood order connect l l calculate either construct neighborhood cluster cluster edge tt tt lattice corner cn neighbors n ci b neighbor ci cn interior ci generate unconditional cluster integer generate determine single modify furthermore simulation manually site probability inside estimate simulation site giving position site give site xy subset corner c c c single est sim p estimate est run run modified remark update step neighborhood site size outside edge site site belong list list maximal tt label order magnitude remark moment triangular lattice random connect neighborhood edge else l merge remain tt tt size tt status store tt status active store add updating step special neighborhood complement vs complement list list single subset tt neighbor edge label order edge remark always label connect neighborhood edge edge calculate maximum cluster merge edge cluster large tt size tt size l exceed site inside size output n list inner site outer site binomial probability bin site determine entry les indicator k bin would like di discussion mm remark phone phone author develop algorithm shape presence boundary object condition interior paper method write digital image noisy image paper efficient quick detection object noisy use indeed look first question ask question diverse automate moreover picture procedure picture surprisingly majority image statistic skip stage reconstruction object permit nonsmooth mild interior detect automate analysis cancer material regular precisely shape advantageous object testing approach density assume continuous tail connection object seem spatial consider triangular periodic linear pixel top exponentially object detection build datum stop view machine learn valuable field machine indeed unsupervise even rate convergence describe algorithmic nonparametric processing also research language programming implement substantially free language advance already build type describe store processing image implementation algorithm explain image method behind introduce suitable modification allow fine image subsection implementation modify image processing package convert sequel store image grey entry vector intensity pixel convert display vector mat mat mat mat mat mat mat mat follow code generate lattice n neighbor n ci ci cn neighbor neighbor n analysis find thresholded remarkable algorithm linear find linear respect pixel depth construction span explore order encode neighborhood neighboring site belong site store site explore current site unlabele get store site site new proceed site unlabeled neighbor seed point label site span tree whose display root please slow much generate span span tree span r pos rank pos l pos I else pos pos pos current something nothing backtrack pos current else return cluster image nan find object test extremely finite site minor change make cover well let probability site
many detection exist within number change mean change situation change parametric simplifying perform efficient contrast book describe many often detect change rank statistic method compare empirical paper source presence zhang produce offer detect require side contrast figure well point real problem extensively review poor rise essence page consider offline match length likelihood approach language word length word change point count homogeneity text review variation uniformity appearance use calculation length adopt stre finite simplicity write subsequence string define prove length consistently true length markov surely though complete suggest class however non fast convergence perform plug estimator insight shannon stationary ergodic state ergodic finite alphabet string length typical mean expect matter formal dependency involve involve return take possible directly e px normality condition corollary extend condition simpler analyse partition match avoid parse block see cover understand result iid even source partition fix block long return source iid similar prove typically ergodic concatenation parameterization independent sequence length change concern contrast common algorithms quantity help set match position length position among detect tend tend direct look suggests simply normalize martingale address expect find hope easy rate relative stationary ergodic relative rate role analogy argument probability md string set typical string graph generate n theorem prove consistent theorem build understand behaviour establish martingale tool martingale consider develop control inequality mention process write nx kolmogorov kolmogorov brownian fact supremum principle sense dimensional appendix prove consistency martingale illustrate behave length model see look affect analysis language figure form english switch author natural english two language language write english distinguish suggest english author typically consider text already distinguish show text marked vertical change estimator idea information work variety source related toy direction hope establish likely return dependency exist time distinct regard towards behave case expect perhaps suggest decaying suggest toy decaying case fast toy scenario multiple point change believe real come direct theory work issue point stream spirit burn believe act proxy analogous figure n control envelope fluctuation see toy fluctuation fluctuation fluctuation overall first technical regard operation enyi binomial autoregressive discrete ar recursively take u j nd j martingale definition prove induction bin bin bin u e write argument since birth death standard martingale jensen inequality since bin n j n equation deduce martingale z c c k n j bin n j lr deduce variance lr c martingale characterization essentially version stay minimum minimum figure htbp change function stay function martingale bin explain see equation mean leave change point part similarly equation mean concave right two follow exact curve lr lr rl accord small limit proof assume curve either interval know mean standard conditioning decompose term mean nz mean term function mean coefficient decrease similarly equation since add contribution example lr rl together deduce take tend zero tend adapt slightly loss generality equation interval empty suggest empty deduce empty otherwise divide use bn divide union interval q make mistake imply p bin work grant mathematic thank pt pt conjecture condition pt alphabet detect change property motivate idea length information novel parametric enumeration form example show chain distribution require avoid consistency relate toy establish martingale argument string symbol alphabet source piecewise source section offer perspective substantial consider match length length elsewhere string class process length describe position choose place create direct link formal model change believe
freedom desirable node jointly schedule wireless computationally intensive multiple become extremely challenging study available report wireless author power effort schedule early paper direction technique apply energy generation wireless solution node spread wireless wireless nature ideally every coverage allow receive limit effectively form graph note interference continue irrespective transmission towards network call wireless provide late transmission via intermediate connect loop flow source flow depend availability transmission consumption choose slot restrict node instant point transmission form transmission network one though final throughput gain capacity involve activate simultaneously define activate power link power scheduling scheduling become gain centralized setup channel notation channel link channel gaussian calculate area spend retain power capacity calculation whenever statement setup general modify exploit objective maximum statement contain link average power form statement refer versus sign value refer represent effective rate choose flow mode mode link power refers call spend mode correspond power spend slot represent availability rest pure system statement applicable mode interference vector capacity capacity interference channel mode mode spend one problem user sum noise interference assume matrix network clearly know optimization link iterative scheme link simultaneously continue link sum monotonically link capacity link spend active perform namely capacity optimum capacity interference instant namely link channel link initialize li constraint loop one choice see mode turn million note ignore calculation require get need solve network generation claim node thing variation network even network large basic solution start choose consider choose new improve master mode scheduling master attain improve iterate converge suboptimal solution suboptimal problem optimum search evaluate infeasible avoid heuristic sub optimality example heuristic ready implement denote among link mode solve solution optimal obtain next depend initially discuss early well convex chapter multiple optimal solve problem similar claim performance node ht number channel number gaussian random source rate flow avg availability unit approximately optimally get solution problem unit achieved give fairly amount heuristic solution unit optimum ht direct source flow nod heuristic multiple initial point see follow solve achievable user unit unit network
imputation offer assess imputation use validate code categorical surprising imputation validate imputation cross validate grey significance pair investigate first already categorical yield rare present protein array status gene find prediction mass experiment novel search child product systematic heart heart adjustment health life assess observation comparison perform imputation categorical generally amount imputation ill nearly dependent e binary category statistical successful category variable number nearly introduction lead removed variable make implementation statement bar bar grey black three order magnitude level pair encode significance child result run imputation imputation simulation lot behave scenario estimate imputation minor role compare tb three simulation assess compare imputation previous show far fast however run considerably require code categorical variable knn tree comparative tree high tree forest tree strongly increase note imputation number equal node tree run perform always miss number tree simulation imputation basically multivariate consist require distributional aspect come biological near multivariate imputation imputation quality need subsequent represent potential include relation dimensional greatly excellent uci repository molecular thank medical centre thank child center child rich provide thank anonymous constructive comment science program biology disease disease motivation modern acquisition throughput technology depend imputation offer however majority imputation variable type handle separately ignore relation cope several imputation evaluate imputation tree constitute multiple imputation build forest able imputation without need biological introduce range handle comparative method relation imputation error additionally exhibit attractive cope availability article journal bioinformatics rd version miss fully biological enhanced measurement technique field provide multivariate greatly exceed type arise technical diagnostic opinion additionally often contain structure parameter parametric exclude imputation restrict appearance field imputation base combine categorical idea book assume detail imputation unlike restriction logical domain record motivation imputation structural aspect forest mixed type allow interactive nonlinear effect address miss imputation train rf miss proceeding iteratively completion thresholde condition dimension complex structure rf suit apply furthermore rf allow bag compare imputation lasso perform proportion competitive use source error computationally attractive error imputation average deviation hence need pn absolute imputation imputation method ten categorical type imputation weight euclidean choice large effect imputation implement imputation mean imputation knn mis good original paper standardize constitute gene apply vary scale center imputation matrix imputation cross standardize require different regression comparative categorical algorithm imputation contrast specification default setup mainly simple miss imputation imputation assess pool choose procedure imputation control transform error imputation categorical variable comparison code categorical categorical deviation apply validate variable back categorical completely rate versus follow publicly include patient pd health remove shape energy patient follow level time give perform even well data dimension
supervised fitting cross validation type monte report integral associated estimate integration number sample additional computational bias true sample sample distribution fitting optimize condition severe degradation slightly trade nominal wide range rarely never optimize operate interested summation generate expensive evaluate standard quadrature mc powerful mc technique drawback expense prohibitive question technique exist lower combine mc statistical technique traditional method brief monte stack introduce technique call stack monte carlo stacking error problem problem show reduce return source truth set many bias estimate bias euler equation run deterministic return run lead square must error mc refer extremely mc second mc converge correct answer decrease carlo kind concern fluctuation monte estimate fluctuation carlo generation help sample alternate eq sampling measurable use mc several million accurately importance unlikely occur reduce sample location monte leave location method sample generation algorithm unit hypercube technique seek supervise fitting make integral polynomial fourier expansion compute quadrature fit piecewise polynomial induce correct answer increase exhibit fitting impossible make difficult know additionally datum exhibit overfitte datum inaccurate location result fit evaluate address issue validation several fit fitting stacking stack sophisticated winner strategy amongst stack view version nonparametric technique therefore argue generalization use stack single reader stack apply stack gain netflix competition combine prediction stacking partition instead correct stack close comparison stack stack monte incorporate avoid introduce poor carlo function necessarily perfect fit equation instead term fit properly mc optimal deviation intuitively correspondingly low perspective alone add thus incorporate emphasize fit poor one fit nearly ignore heuristic ignore fit inferior extension however set introduce fitting change depend hold accord special approximately calculation fitting modification option fit integrable expand calculate replace significantly computationally attempt concern five fold point training minimize inverse leave calculate monte give fit alone either first example analytic examine set obtain sample test fold cross unless plot versus green sample blue polynomial form fig ten number point high outperform carlo range perform additionally see validation bias generate fig fit whereas outperform example fitting fit upon carlo fit accurately contour generate hypercube fourier expansion low individually ref program synthesis conceptual future certain frame eight probabilistic effect technology represent burn apply order easily follow eq generate take standard find twice apply respectively x exact trend two sample magnitude calculate calculate cost sample expect error uncertainty quantification test response surface noise signature output recently propagation tool robustness pressure signature minimal specifically fidelity field roll angle uncertainty pressure like true space fit great fit increase despite fit still strength additional cost number introduce stack monte reduce carlo unbiased thus monte able effectively use extremely generic generate mc also monte furthermore discrete result regime examine explore application fidelity incorporate form fit curse cost entire evaluation assumption
ambient bold symbol capital product denote strong convexity recall private tt step problem utility algorithm generic differentially private guarantee framework convert private generic privacy sub private differ one privacy formally define sensitivity th mention early requirement degree adapt private formally sensitivity decay linearly convexity typically satisfy output magnitude learn private algorithm along concentration bind privacy guarantee framework convert private guarantee bound tb algorithm assumption change sequence noise output lemma c projection I function differential see measurable addition see apply eq good definition differential privacy composition argument differential privacy order needs add intuitively large incoming lead arbitrarily bad exploit iterate algorithm union probability differentially private technical lemma stage preserve privacy entry observation privacy sensitivity differentially private proof noise let z tb hoeffding get depend private private algorithm see output diameter lipschitz time add chebyshev inequality function sequence private diameter two private strongly lipschitz show restrict practically exposition privacy obtain update eq elementary algebra theorem algorithm see also differentially private variant private regret key observation behind differentially differential obtain add structure preserve propose sum add see lead well next iterate function tx trees structure differentially l b guarantee privacy state sequence rl regret lipschitz schwarz bind private fact eigenvalue bound ta observation use problem differential provide goal output time privacy treat concatenation sum partial change na noise privacy get generalization provide description label root label node add noise node thick node b depict change partial denote leaf string follow level label right concatenation string label two summation vector root j rooted leave tree root tree correspond leave root output add perturb node summation node level illustration code ts ds root ss ss minimum form string string q privacy utility partial sum entry consider ratio inequality equation combine formal utility guarantee proof inspire technique tw dt output differentially tw vector tree select sum differentially private sequence differ noise independently hence thus lemma ratio use differentially deterministic hence differential reasonably broad drop albeit perturbation modification analysis sub fit private spread practice inefficient completeness technique analyze differential privacy framework differentially online algorithm good show good bound offline well differentially large class offline offline later practical describe offline one observe minimize formally consider specify data minimization provide via execute incur comparison produce private noise output detailed presentation framework underlie privacy bind differentially private prove privacy need perturbation sensitivity output produce execute dataset entry triangle maximum continuity lemma guarantee utility rewrite incur derive convexity utility bind number dimensionality inequality lipschitz equality follow vector least lipschitz continuity bind rhs plug exist offline method differentially framework wide namely however significant advantage pt learning theorem utility loss popular svm require lipschitz minimize fix constraint bind bind dimensionality believe primarily usage add practical iterative guarantee differential privacy extend force propose differentially framework offline compare wise optimum algorithm approximation privacy two practical practically regression online offline online regret provably preserve guarantee specifically fashion minimized problem variety finance apply differentially private guarantee logarithmic dataset year dataset fix generate dimensionality contain offline use ridge non private normalize number incur synthetic note log close requirement weak pt ccc iteration incur level privacy plot scale privacy meaningful provide privacy especially low logistic regression square logistic private logistic point purpose private show average algorithm privacy reasonable private point reduce underlying sub maintaining show bounding sensitivity consider algorithm private differentially private differential special cost logarithmic regret guarantee show private differentially private algorithm offline learn offline obtain error regret open optimal similarly privacy logarithmic research direction differentially private various course lemma claim conjecture microsoft com university edu edu preserve learn continuously change preserve privacy challenge subsequent etc learn customer behavior customer etc practical use privacy measure critical order sensitivity I arrive regret goodness utility two convert privacy good popular variant regret linear differentially private offline offline art modern privacy customer generic scenario engine ad serve ad relevance past search two key query goodness ad engine try guess history available ad get online engine several reasonably pose severe query ad click guess past query thus engine correct guess relevance privacy etc concern learn one generic privacy preserve formal notion programming notion interesting lot progress however without privacy contrast g click relevant ad produce roughly analyze output produce privacy hard differentially private handle typical programming several theoretical algorithm convex algorithm incur exist private offline notable notion private online specifically two differential privacy whose privacy record privacy current set count differential typical arise practice contrast consider practical namely handle offline result particular expert differentially answer count offline datum entry total answer start subsequently answer
consider due analytically latent consider simplicity vector length grid note however dimensional large infeasible extremely slow mix recent gaussian hyperparameter slice general mix gaussian element burden impose typically use drive heuristic recalling devise accordingly evenly knot local bin spline compute cholesky spline autocorrelation choose truncation level computational consideration gibbs inversion operation sampler step computation dimensional draw distribution draw non model apply methodology handle choice diagonal towards shrinkage towards shrinkage redundant straightforward gibb present infer adaptive covariance diagonal become distant nearly band limit upon invert band limit efficiently versus naive issue maintaining analyze covariance regression compete alternative covariance nonparametric replicate discrete length additional additional precision draw sim covariance ccc sim residual sim residual covariance sigma set prior parameter section evenly space determined round experimentally sampler insensitive initialization lower dimensional one analyze mix initialization form cholesky initially take spline location sample iterate couple iv drive newly sample indistinguishable present burn run discard true component display noise covariance density mu sigma jj sigma mu jj sigma sigma ij covariance high interval show row represent mean represent error bad magnitude see able capture component worst contain density interval interval small band representative capture observation standard build instead benefit exactly mean regression model wishart analyze dataset element decide whether remove chose predictor region result remove compare miss likewise sample mcmc regression base actual model result gibbs discard clearly indicate regression predictive improve naive depict constraint fact lead improved kullback leibler element base true predictive predictive kl regression regression replicate dimensional choose evenly spaced knot sx diagonal maximum value covariance show ccc est beta analogous nonparametric wishart mean replicate large near replicate exactly truncation replicate gibbs thin examine display aggregated replicate average covariance range norm indicate good commonly volatility financial wishart discrete account slowly ty specify discount observation maintain wishart construction integral run backward frobenius depict wishart approximately twice furthermore towards error accumulated high degree force cc x nonparametric covariance wishart replicate analogous apply nonparametric capturing spatio temporal united states surveillance grow although increase public surveillance include rapid distant spread drive medium coverage surveillance control health www site us surveillance key report population week plot show report six state available http www media fs severe dr david united human predict million n united ccc vs trial possible nonparametric google trend dataset thick trend united york covariance line scale united state show show plot indicate period event shoot severe aid rapid researcher search query predictive methodology search logit transform query examine rank variation perform query explanatory region base result rate closely track actual advantage datum http www google publish ip address fine aggregate reporting finally google trend methodology search search activity raw search query track quickly various medium week week vector element consist google health human service block observation omit reporting end year number respectively exploratory figure plot period state move window aggregate f event due dimension simply observation dimensionality ability correlation cc california allow temporal correlation important predict function define rate week capture x ix model google line particle jointly univariate joint lose individually predict rate select simply analyze pattern substantial miss solely google trend ability introduce methodology cc trial trial california california trial map trial map map map trial map trial plot capture compare figure clearly demonstrate temporal google entire dataset length hyperparameter examine truncation level run chain discard burn chain examine every initialize parameter assess perform diagnostic index york california south event spatially correspond state factor period imply correlation shrinkage component trend follow google estimate national york plot notice display key mild distant area distant york california highly expect south fairly state infer south finding display specific note dimensionality solely level unable crucial redundancy report form covariance miss dataset also ability method cite observation observation readily handle limited matlab take machine four intel core ghz processors gb bayesian nonparametric potentially formulation collection herein induce dependent methodology tractable dimensional simulate google trend interesting direct propose fall address limitation collection observation predictor collection cope framework hierarchical propose building complicate additionally continuous response employ probit lead probit current include augmentation impose covariance defer another consider possible approach avoid specify discuss draw process large become infeasible scale large predictive process directly use integrated approximation dramatically affect hyperparameter address one thus allow formulation datum image voxel spatio dependency represent covariance imagine replace element spline hierarchical allow variability subject interesting future domain acknowledgement author helpful discussion covariance theorem every cover repeat second continuity sure almost sure continuity shrinkage prior surely enough wishart x surely let pairwise surely almost xx inequality sup ij diag prior imply recall e probability symmetric semidefinite wishart conclude combine positivity proof ns sure exist almost convergent nk last equality e lemma recall matrix distribute imply iterate moment location eq wishart conditionally n fourth moment slice really covariance derive accord zero drop derivation notational summing index cross arise share process moment derive eq dictionary conditioning rewrite dictionary conditional take definition although literature regression vary relatively little focus develop allow induce covariance dependent loading loading unknown induce highly flexible tractable conjugate miss theoretical nonparametric capacity grow within collection portfolio likewise temperature etc record spatio variation imagine arbitrary potentially multivariate setting although literature univariate multivariate I capture correlation typical inference address decomposition precision propose predictor row problematic exist alternatively submatrix coincide submatrix additionally involve assume definite factor still flexibility approach rank dramatically increase parameterization volatility discrete volatility volatility linear return suffer curse typically limited dataset dimension volatility assume autoregressive process approach recently latent conditionally wishart precision limitation wishart process challenge theory description intra relationship review dynamic linear conjugacy cite volatility ability dependency often lead additionally value within typically herein volatility modeling covariance wishart marginally inverse spatial dependency arise computation scale accommodate spatio temporal upon factor symmetric positive without xx establish decomposition explore vary predictor factor relate model related conditionally work
belong skew term geometry r entropy divergence family multinomial beta study unified family divergence family generic furthermore entropy calculate exponential distribution among shannon entropy entropy divergence exponential family communication field know entropy measure amount accord close expression shannon entropy theory seek code underlie language since practice true unknown observer rather entropy unknown leibl divergence orient notational convention rewrite enyi entropy modify one axiom characterize average enyi single prove l fx classical illustrate rule later discrete entropy q since enyi tend shannon enyi entropy keep shannon decrease close entropy report multivariate technical motivated multi generalization shannon derive entropy entropy non tend shannon shannon enyi entropies entropy monotonic function admit density denote natural convex since f p fx ff characterize member later say observation sense show factorization regularity condition class statistic family lebesgue distribution order family multinomial exponential family canonical decomposition prove exponential family follow exponential r enyi entropies particular standard center closed entropy shannon entropy enyi entropy shannon fx illustrate generic let start family exponential successive canonical decomposition family r enyi entropy use rule converge shannon univariate exponential natural kx enyi shannon eq tend calculus complexity base may mean calculus result enyi tend shannon entropy h tx conclude probability define divergence geometry also rewrite ar case coefficient hellinger family measure enyi divergence follow skew jensen gap let member exponential enyi exponential skew enyi divergence bregman divergence exponential direct appendix natural log member enyi
round possible trivial rich fact minimax allow contain corruption regret low proposition appear minimax bound key corruption learn task follow regret map restrict mirror imply vanish corruption dependent class parametrize matrix may simple parametrization corruption define affect determined possibly parameterization q drop subscript simplify introduce throughout parametrization problem start corruption cast within base power weight feature absolute sign correspond round base loss want enough magnitude parametrization allow strategy correlation elaborate induce classification weight thus pick monotonically discriminative sign hypothesis restrict op pp z scenario impose soft corruption pattern nature corruption process observe define weight array example sensor wireless array sensor fail situation sensor measurement surrogate neighboring feature neighbor say miss ki put neighboring sensor specify graph predict feature vice versa entry would add constrain diagonal mirror frobenius norm function could simplify presentation shorthand incur define exhibit regret theoretical recall setup imputation mapping eq q retain observe feature predict encode framework expectation product like quick inspection result optimize hypothesis play framework restrict relaxation relaxation natural restrict simple existence follow denote consider rr overall interested q imputation optimize hypothesis bound imputation optimum easily find jointly take saddle concave n k drop convex relate replace relaxation piece define eq strictly semi definite convex defer lack arise relax involve ensure relaxation relax correspond point solution saddle problem equivalent proposition behind replace rademacher justify predict like class purpose give hypothesis rademacher possible triple lie optimization problem note hypothesis correspond relaxed deal reasonably dimension thereby still provable rademacher inner outer sample bound brief sketch defer hypothesis prediction use lemma straight bind come us control gap risk follow sample term covariance vector online show rademacher relaxed ridge use baseline miss zero gradient ridge evaluate several uci dataset dataset artificial pixel central remainder corruption corruption base deviation pattern tune corruption present use type force additional choice regularization task entry correlation coefficient entry pixel allow corruption imputation perform relatively poorly corruption imputation expect provide imputation rich corruption use perform least imputation offer significant vs method omit top corruption fraction indicate corruption across sdp solver instead find well regression see improvement least well imputation corruption continue perform corruption figure display improvement cccc corruption table corrupt significantly imputation able outperform imputation naturally miss perform well cccc regression corruption bottom naturally occur introduce I motivate construction theoretical empirically result imputation strategy accumulate task index contain distinct associate classification different show total regret von duality distribution clear optimize set alternatively vector individually minimax regret accumulate measure complete order regret convexity last regularizer finally old yield step non negativity substitute simplify yield theorem formulate problem rewrite ridge problem maximization find regression component hadamard contain dot product instance fractional problem eq due quadratic substitute result show precisely long necessarily semi fractional add additional add make replace complement positive semi complete cauchy schwarz follow rademacher term cauchy schwarz separate remainder second term expand apply expectation schwarz cauchy schwarz supremum use follow complete analyze wish bind separate depend cauchy add summation inequality divide prove corruption corruption corruption corruption choose induce corruption instance choose tune induce show corruption sum available corruption average trial fold test applicable trial random corruption pattern across update optimize fix writing give equivalent optimization respect respectively equality couple optimize subject norm work case deal solve minimax q eq positivity constraint matrix optimization concave von minimax max observe minimize maximize direction normalize appropriately I n leave constrain equation imputation view natural know even corrupt hard alone miss expectation start simple imputation strategy regret setup weight equation derivative loss first control natural
pairwise yield kronecker tensor aim formally product edge generate gaussian kernel generate approximate type armed arrive kernel assume universal like product pairwise well product theorem say suitable rkhs contain close relation likely concern outperform kronecker domain symmetry relational formally indicator return element recently alternative pairwise kernel limitation generalize pair graph impose lead direction connect repeat measurement reciprocal symmetric briefly summarize learn reciprocal let start definition binary hold relation every multi unobserved opposite transform reciprocal reciprocal relation represent preference win direction interpret interpret edge ordinal ordinary interestingly framework reciprocal give proof immediate domain least product obtain represent reciprocal relation relation every relation specific relation constitute application symmetric relation relation relation call preserve arise domain special setting turn easily incorporate mapping mathematical kronecker look obtain kernel reciprocal kernel protein bioinformatics latter enforce every edge twice edge method kernel represent impose remark well flexible represent space symmetric every every reciprocal version reveal kronecker product kernel reciprocal symmetric require kronecker relation lot play competition name show symmetric relation despite application characterize relation relatively preference modeling argue reciprocal preference certain rational human decision make well play etc extensively theory reciprocal relation traditionally increase reciprocal relation call v special case moderate relation notion put minimum form numerically fuzzy term learn reciprocal relation reciprocal call reciprocal rank reciprocal acyclic graph rank reciprocal relation figure yield interestingly rank reciprocal relation simplify joint simplify r respectively reciprocal follow decade reciprocal notion powerful reciprocal relation reciprocal property relation rather applicability need root euclidean space euclidean ranking hold transpose euclidean basically see multidimensional restrictive one embed realize rank object reciprocal nevertheless type sometimes situation euclidean occur existence ranking relation metric square symmetric reciprocal synthetic standard reciprocal kernel kronecker product pairwise kronecker pairwise reciprocal kronecker product kernel illustrate kernel learn family base measure consider measure family include coefficient coefficient member first coefficient coefficient originally correspond conversely member analyse give member generate statistically bernoulli mention feature feature last vice illustrate score experiment set validation hyperparameter sample replacement relation predict missing relation experiment set set unseen present result test use pair compare multiple testing rbf node mse training explicit grid conduct regularization table outperform relation method demonstrate easy relation generalize node enforce outperform symmetric kronecker clearly exception experiment successful probably enforce apart difference pairwise case experiment conclude knowledge really help predictive pc hardware window learn correspond number occurrence document experiment second connect node use size start node except gradient mse fail decrease rely early experiment kronecker pairwise kernel exist aim achieve around show succeed kernel lead prior symmetry relation error large small instance enforce symmetry training performance ordinary reciprocal kronecker metric explanation decade imagine resource dominate limit define fitness advantage draw disease etc specie specie dominate relation reciprocal simulation specie simulate specie thus specie represent limit specie dominate function specie training validation determine validation testing factor try probability specie ordinary reciprocal pairwise kernel regularization grid obtain repeat sign significance conservative difference statistically rise worse prediction relation ordinary pairwise reciprocal kronecker extend approach handle relation kronecker feature propose feature pair object constitute addition relation learn incorporate property experimental synthetic world really help improve recent development fuzzy look grant metric closed universal rkhs dense every input accordingly hence universal approximate metric separate vanish compact space write accord write dense accord confirm separate vanishe point consequently arbitrary rkh belong rkhs obtain replace observe connection immediately arbitrary write single proposition drive bioinformatics retrieval relation recently investigate relation standard preference framework introduce extend consider exist relation model reciprocal relation establish important link development fuzzy theory usefulness demonstrate relational modeling forecasting winner computer protein interact protein document mining friend site example application machine mining develop relation specific learning scenario summarize object variable usually consist infer unknown relation predictive unseen predictive advanced new assessment like relational preference learn learn relation edge relation aim preference relation also development fuzzy theory article elaborate connection contribution typical real ordinal relation relation simplicity learn relation possible binary example infer protein interaction bioinformatic world relational furthermore recent logic literature algorithms mathematical relation impose domain knowledge learn relation even already domain example social notion person reciprocal neither like person person b approach one model distinguished mean loss object setting restriction arrive classifier relation restriction mathematical framework fuzzy theory use notion relation account real scaling relation somewhat middle actual matter provide learn different domain satisfy distinguish symmetric domain similarity require triangle similarity learn undirected reciprocal bioinformatics preference relation win probability gene formal rescaling relation convert reciprocal symmetric property preference preference learn reciprocal relation interpret relation application symmetry relation see constraint framework object study edge weight edge node indicate
approach early ard share four penalization solution ard dictionary approximately ard ard except middle bottom decomposition ard divergence al show generative statistical relevant audio explain equivalent power investigate decomposition short second long condition sequence play pair combination subsequent temporal analysis window ms overlap two frequency temporal log signal sound string sound truth run ard initialization return low nmf multiplicative ard nmf initialization nmf solution fit use matlab implementation good run factorization wiener reconstruct produce component inversion decomposition nmf ard produce ard nmf al order fig display ten first component produce ard nmf axis two comparable histogram standard nmf ard al histogram fig ard nmf confirm relative relevance upon weight drop component individual component sound string sound contrast tendency piece split component leave component visual reconstruct split distance contrast flexibility ard nmf choose desire ard fig et component resemble closely attack merge note weight remark ard ard nmf retrieve choice decomposition point nmf desire experience correct kl kl leave ard plot yield inferior ard nmf cost apply stock section price comprise american g stock price rd th trading day stock price representative stock price company display leave plot capability set entry indicate miss integer perform nmf incomplete estimate multiplying infer basis activation normalize stock datum estimate n miss stock well ard nmf termination literature column normalize unity compute relevance weight initialization order right observe trend increase infer addition ard components ard ard penalization method dense prediction rather fig large observe ard perform uniformly standard nmf miss stock price kl nmf ard modeling know choose ard kl nmf though retain ard component retain ard nmf nmf stock fit assumption well ard well comparison return ten run clearly inferior demonstrate predict analogue analogue across ard calculate choose flexibility ard nmf tie prior exploit ambiguity induce term accounting present monotonic result multiplicative complexity update preserve initialization efficiency approach validate propose method offer flexibility exist deal prior moment rule recommend work fully seek variational carlo handle principled concern tensor acknowledgement like acknowledge work discussion share also thank improve theorem support project factorization paris france mail fr matrix nmf divergence family include leibl important fidelity automatic determination dictionary tie parameter minimization posteriori drive course prune spurious component efficacy robustness perform synthetic stock prediction factorization model order selection entry nonnegative factorization find factorization dimension dimension early reference nmf work seminal contribution lee become technique diverse process financial usually minimization q nonnegative separable scalar parametrize square kullback kl divergence divergence short detailed crucial conversely overfitte seek find solution fidelity information bic set scale linearly bic assume relevance determination ard pca computationally monotonicity auxiliary directly update leverage technique decomposition correct order produce fairly limit reference monte mcmc strategy nmf evidence high reversible jump computationally intensive another reference close principle irrelevant component zero qualitative significant author conference publication firstly paper parameterize show flexibility quality image secondly herein decrease local whereas mm bayesian detail ard nmf extensive numerical efficacy ard denote matrix matrix denote thus consider nmf review originally introduce later definition limit divergence another square euclidean parameter control observation either learn cross map parametrize respect greater separable function kn kn kn kn kn h kn kn kn kn kn kn kn kn kn kn kn kn standard nmf recover mm building everywhere function whenever replace iterate mm minimum attain key thus auxiliary approximate objective decompose sum construction consist jensen denote result generally previous iterate minimization thus follow exponent common drive drive prior constraint observation prior integrate element set activation integrate analytically prior activation prior x x assign note tie together column vice relevant norm fidelity common row finally impose relevance relevance every related note familiar family mean mass pmf form pmf vary generally gaussian coincide coincide shape b aa base log ratio divergence scalar cost whenever negative form posteriori statistical half observe monotonically third force tend serve pruning kn describe determination weight update previous k attain strictly choose ard ard division dispersion tradeoff fidelity term knowledge validation case optimize literature estimation factorization fix kn audio power multiplicative model upon choose principled focus selection sample write shape parameter v real law large number element exponential inverse gamma expression empirical ard calculation defer supplementary material sometimes fall moment observe ard ard informative section confirm small conclusion robust draw automatic pruning map develop independently extension array column coefficient exponential prior square euclidean kl divergence major optimize solve author change multiplicative along treat update occur row may nonnegative pose parametrization model tie converge reach automatic projective al projective seek nonnegative span fit ard originally describe et al additive half relevance describe adapt relevance nmf multiplicative noise address employ nonparametric set large hyperparameter place assign sparsity enforce factor contrast herein unify nmf decrease section music decomposition stock price demonstrate decomposition model nonnegative generate sample weight inverse gamma sample half depend ard reason define noisy matrix kl noise db noisy poisson element datum db gaussian observation db number choose set iterate converge limit ard dispersion fair initialization
solely exchange exchange play source network bias network contribute arrive generalize class adaptation far ahead original strategy allow source noisy step diffusion two adaptation signal network reveal network mean analysis lead far choose improve steady combination metropolis context consensus suffer degradation since ignore noise profile node combination outline far ahead mobile move neighborhood evolve even critical adaptive track variation profile order cope dynamic environment issue vi letter denote letter plain deterministic radius argument form argument stack column noise consist scalar successive use dependence scalar measurement form denote phenomenon study hybrid adaptive minimization problem several diffusion type node problem adaptive manner learn time review combine second small size whenever node broad diffusion diffusion selection example recover identity node evaluate estimate rely measurement left vector mean receive neighbor describe weight introduce quantity step involve share information node generally subject quantization likewise share subject perturbation objective perturbation diffusion weight enhance happen link additive fig observe information comparison reference noise without li compare ki k four source appear appear measurement independent zero covariance lk I lk lk uv lk white spatially independent covariance noise perturb diffusion continue symbol avoid eq denote node matrix worth respectively scalar zero notation ki easy lk substituting get recursion compare compare block q arrive follow presence exchange consist nm w exchange adaptation exchange weight recursion ready strategy exchange assumption verify assumption matrix th submatrix driving term driving term convergence whenever see bind size steady namely argument jensen induce hermitian ii iii combination determine solely regression note perfectly compare dynamic range mean filter analysis tractable diffusion exchange derive excess square analyzing error flow global error positive hermitian choose notation stand quadratic denote link ignore approximation side rhs third term error independent rhs express fourth rhs relation depend w relate linear ahead verify guarantee select conclude diffusion stable mean small size convergence exchange variance steady iw notation proceed evaluate first rhs q use identity rhs give likewise rhs approximate eq ignore steady rhs matrix vector representation mc v ki r lk lk di lk I lk semi hermitian arrive steady state positive hermitian steady say select evaluate q link bias section examine simplify sharing sharing within neighborhood expression recall expression optimize impact arrive expression eq source performance b I obtain expectation steady give term summation source contribute combination denote q minimize stochastic rely well also fan hermitian positive semi scalar norm numerator motivate separate node noise covariance rule particular scalar minimize expression manner note eq minimize bind network algorithm variance combination product neighbor r l lk propose rule estimate close allow estimate realization lk lk ki lk factor arrive adaptive diffusion benefit step vary analyze able adopt walk stationarity weight vector change accord zero initial recursion verify continue steady stationarity steady lead network define early hold environment consider network topology adopt size fig randomly generate fig white link top link collect collect k mml examine simplify namely share rule metropolis degree node iii rule rule fig also result see relative rule diffusion achieve steady uniform relative diffusion algorithm exchange assume change along circular fig dynamic express generate trace randomly db noise variance simulate fig algorithm denote average estimate fig horizontal axis along red trajectory blue trajectory along trajectory exhibit ability high investigate source exchange environment link bias mean noisy exchange derive motivate choice help also scenario change simulation illustrate finding
use even obtain strictly speak equation I really alone believe error implication therefore converge limit importantly order apply boundary normalization order laplacian happen point explore boundary laplacian spaced panel scale interior laplacian scale factor panel log would boundary boundary gradient normal normal direction half negative slope compute pick near plot expectation rigorous square compare decrease edge going see reflect intuitive nn go pde boundary boundary laplace hope regular boundary laplacian partial x connect element point boundary axis fy fy fy interior g along direction inside pde fx h fact point condition pde add point normal h h fx become fx boundary use implement method pde laplacian grids laplacian laplacian suggest eigenfunction satisfy condition density outside normal need boundary positive integer eigenfunction unnormalize laplacian see eigenfunction x boundary condition laplacian density eigenfunction panel eigenfunction eigenfunction boundary eigenfunction fact second eigenfunction correspond sign symmetric normalize eigenfunction correspondence eigenfunction eigenvector ix therefore xx boundary along normal direction asymmetric towards test graph nn graph study cluster panel shift less fix notice nn graph eigenfunction near boundary reflect near neighbor asymmetric graph decrease regularizer limit study point compare laplacian regularizer boundary follow thin quadratic manifold control distance near replace constant independent integral integral become use conclude term fact test several density value laplacian ratio figure ex ex test suggest maximum report table result theoretical suggest pattern decrease numerical precision h degenerate unbounded boundary behavior happen bring e dominate satisfy boundary safe product essentially short importance regularizer reproduce hilbert rkhs reproduce kernel unit uniform expansion green laplacian reproduce orthogonal nan expression hand without finite green reproduce expansion kernel approximate obtain scale quite due boundary eigenfunction engineering algorithm graph data attention certain continuous operator research topic exist interest evaluate boundary considerable paper analysis discuss convergence implication volume manifold scaling elsewhere manifold manifold implication manifold laplacian well amount analyze theoretical aspect manifold different bandwidth operator appropriate understanding object light property guide selection practical versus unnormalized suggest preferable practical algorithm converge fix suggest iterate superior laplacian limit cluster infinity kernel bandwidth study operator arguably significant coming since explicitly interest simple pixel gray scale image zero boundary image motion configuration human body sensor limit range boundary process boundary graph boundary operator direction normal boundary choose bound laplacian near boundary interior fix function appropriately scale derivative likely correspond influence laplacian ignore boundary laplacian confirm provide way laplacian regularizer application bound norm would minimizer boundary confirm related investigation different unnormalized graph px boundary seem view illustration boundary theory boundary effect reproduce dimensional manifold take boundary laplacian regular grid pde proceed compact riemannian satisfy necessary implication discuss review away define instead geodesic hand discuss discrete normalize weight function depend three discrete degree degree unnormalize two define accordingly l nd include unseen l fx notion apply sample become close close result interior need boundary converge weighted limit find dimensional manifold small neighborhood equivalent boundary neighborhood key fact near argument notation rest use without subscript subscript sufficiently unnormalized laplacian random laplacian normalize eq expectation limit help result sufficiently distance thin different study manifold point small comparable tangent whole tangent half ball radius center coordinate fix local coordinate
riemannian survey function robust tracking step compute gradient gradient short geodesic illustrate idea geodesic seem would robust lagrangian previous corrupted order geodesic formula equation equation respect correspond respect gradient equation easy rank compute geodesic singular orthonormal left set singular step approach fact lagrangian exploit leverage success estimate subspace help dual gradient augment lagrangian step estimate admm recover sufficient admm fortunately iteration subspace update achieve practical experiment show produce corrupted outlier take depend tradeoff step track subspace predefine stochastic literature proven however identify subspace change obviously change rule shrink dynamic adapt change need track steady adaptive rule produce empirically achieve fast adaptation gradient e take slightly step otherwise mean subspace slightly along step besides sign step inner proper equation update inner consecutive gradient min max f control control estimate identify subspace precisely change dramatically take adapt subspace undesirable conservative update increase limited large subspace change subspace change drastically accelerate step though demonstrate much level change prescribe always equation variable far calculate level select let exponentially linearly new follow discuss hand level new quickly change really dramatically idea sec e sec application prominent separation foreground background surveillance imagine stack column several frame fact subspace separate foreground background track suited application task challenge video static background move foreground video third simulate camera examine dynamic video background use foreground background static foreground select train frame randomly video real may video confident select pixel set experiment separate foreground simply subsample frame resolution cycle frames stationary subspace subspace second foreground separation stream fashion deal frame perform separate second time separation high video effectively video background foreground separation separate separation quality experiment admm track resolution separate virtual virtual c alg eqn alg alg alg virtual alg full virtual video change dataset frame adjust track unlike run algorithm pixel pixel separation numerical result matlab virtual virtual virtual camera choose camera width resolution virtual adapt frame virtual camera track frame separation pixel second camera camera frame pixel track pixel separation computation second frame robust subspace low incomplete datum question constrain global great minimization alternate triple estimate without useful situation fast know business explore one promising separate background surveillance video may wind result kind movement movement foreground acknowledgment pure mathematic internet together suggestion college university com usa edu chinese university edu algorithm subspace robust stationary corrupted completion foreground perform high quality separation move popular achieve per keyword alternate track surveillance long application communication localization medical imaging leverage reject reliably collect modern standard setup difference massive ever people example estimate surveillance city netflix rating million thousand movie day amazon com second survey whole equally example indirect really sensor response inconsistent corrupted fast tracking challenge outlier use manifold fix operate amenable streaming contribution work propose online subspace tracking algorithm adaptive tracking combine augment solve via discuss dual optimization triple use admm inherently successfully highly successfully recover outlier significant state art completion component analysis algorithm nature separate video surveillance frame matlab motivate robust subspace tracking give subspace tracking familiar robust introduce subspace robust detail discuss critical implementation limitation compare several world video surveillance conclude future direction build problematic corruption noise detect computer like would need anomaly order anomaly subspace datum often anomaly rely tuning heuristic principled seek sense combination residual outlier without outlier sensor network collaborative video surveillance monitoring experience failure signal move foreground surveillance camera include recent seek whose datum majority computation perform pca slow consequently identification develop robust emphasize effectively low dynamic literature tracking previous consider vector object exactly incremental descent algorithm minimize vector fit subspace variable operation outlier early survey track subspace come literature signal process vast literature qr context fast incremental svd modification thin work tracking aim large useful direction arrival estimation introduce target receive array follow music music tradeoff array time track incremental thus make suitable follow improvement analysis algorithm conduct ambient operate along algorithm require conjugate descent tracking oppose careful give nice compare none subsection address issue robustness miss address problem fraction extend handle outlier relate differ may outlier great subject investigation netflix matrix entry rank complete prove norm minimization recover incomplete low algorithmic singular thresholding column fast completion order evolve subspace orthonormal span equivalent outli entry may white
receive simply order statistic n x order fall histogram suppose x hold probability z thus technique enjoy wise bound regime upper linear scheme besides histogram differentially sensitivity analog technique release sensitivity enjoy dp allow sensitivity demonstrate dp q logistic g hx x give diameter note draw first require give except consideration valid give empirical cdf independent close quantile cdf inequality small give apply statement probability rearrange take achieve quantile eq namely hence concentrate probability stem triangle inequality demonstrate mean polynomial examine choice statistic eq cdf evaluation analog behavior quantile differential privacy exploration boundary differential privacy conceptually reasonable differential privacy whether adversary really access histogram cell reveal adversary application two dp synthetic histogram example histogram bin well privacy disjoint ht example histogram empty technique relax differential differential result arbitrary suggest privacy replace relaxation privacy gain deep guarantee release extend bin allow besides relaxation report reference thm proposition thm privacy privacy database effect release add analog property procedure histogram show histogram accurate histogram differential privacy show analog global release privacy dp computer science privacy give strong mathematically rigorous disadvantage guarantee come statistical propose notion differential privacy privacy differential could concern ordinary differential time privacy loss great explore version propose privacy tradeoff introduce ordinary differentially technique context histogram introduce concept identify differentially differentially histogram low relaxation privacy enjoy nonzero thus histogram relaxation analog composition database example input consist database column individual differ coordinate neighboring satisfie privacy measurable intuition dp space privacy differential strong guarantee essentially effect differential key therein much simple differential noise call interactive goal database user query one way release private database release arbitrarily histogram histogram focus mechanism e value rather mild partition space lattice take simplex histogram observation bin essence element laplace rate histogram norm valid histogram satisfie correspond subset differential privacy private histogram although set restrict space histogram demonstrate least hypercube demonstrate present risk point result differential upper move give result relaxation I tend fast differential privacy subscript whenever improve risk uniformly rate question problematic large remain open technique property would wise differential privacy admit release mechanism keep histogram namely cell contain privacy view random draw certainly denote strongly restrict invariant permutation strongly affect replace instead affect draw random privacy randomize differentially private analog differential algorithm differentially decrease inverse dp relaxation quite namely privacy meet take randomized algorithm strict relaxation
detect avoid detection probability require calculate marginal multidimensional whole estimate integral computationally easy one complicate multidimensional integral may difficult problem typical need assess reliably estimate integral statistically density assess metropolis hasting determine integral newton despite well known simultaneously density hasting receive integral test posterior mixture calculate accurately criterion size posterior whenever make assumption regard derive exist harmonic property metropolis require simultaneous give detailed summary also performance case undesirable integral demonstrate collection relative number probability calculate integral likelihood truncated estimate integral select select parameter posterior importance conversely asymptotically estimate convergence therefore integral eq impact slow also key ensure rapid practice certain unimodal likelihood reveal posterior need unimodal symmetric reliable reflect skewness asymptotically mixture various scenario prior parameter increase measurement reliable estimate estimate assess converge estimate accordance definition scale simplicity converge convergence practical except aic converge use integral integral radial velocity limit amplitude reference amplitude short hyperparameter density multiplicative proportional correspond model integral insufficient detection term note choose lead choice simple corresponding parameter becomes set prior decrease prevent call improper way unit invertible system way transform prior constant superposition radial variance become variance chance posterior always use convenient prevent undesirable define use retain statistical allow result radial velocity choose density artificial build still hold free statistical reason dimension space effectively amount turn decrease integral integral simple reliably receive posterior series different velocity noise suitable assess nearby report period day cat epoch enable reliably bayes close close provide slightly great support rather accurate estimate use receive reliable estimate integral assess lc epoch reliable integral estimate aic odd well lc estimate aic demonstrate four artificial radial determine number parameter conclusion use improper conclusion artificial epoch epoch later hour amplitude zero describe standard every simulate measurement set table contain factor signal model signal direct bf approximately bayes detection periodic weak detection bf bf bf see could detect chain converge clear maxima none probable rest converge clear periodic broad factor estimate integral unit prior factor signal clearly undesirable side effect prior yet bayes factor turn term enable detection weak coefficient density consequently prior integral generally challenge integral limit statistical method integral truncate calculated draw posterior density deriving field restrict problem certain dimension reveal know choose star know enable angular namely anomaly aic reasonably accurate well rapidly somewhat biased complicated possibly test current use parameter suitable value essence yield converge rapidly test probability small practically respect odd great though bayes select possible cause large cost low receive correspondingly effect deal unity make support european author acknowledge significant improvement assess model eq essential calculate posterior model sample dropping notation marginal value marginal estimate integral choice posterior denote eq practice undesirable property instance require likelihood information broader correspond dominate except simple harmonic extremely usage limit large impact sum extremely slow reason approximate marginal integral representative
stop iteratively square due instability penalize nontrivial simulated multivariate normal zero draw ridge along double non differentiable induce tail conditionally make choose full keeping penalize likelihood initially compute thereby zero grid use show iteratively fail small algorithm numerically penalize distinct logistic interact poorly way coordinate wise gibbs situation multimodal favorable signal moreover tractable univariate thresholding analytically narrow handle double penalize square differ subtle checking maximized confirm iteratively least yield optimum understand circumstance recommendation use logistic beyond scope penalize quantile fit corresponding gaussian scale quantile th value sufficient normal distribution percentile whose three model traditional package along regression double pareto penalty error loss new th percentile straight significant difference double pareto double systematically sparse pareto error goal conditionally representation mixture implementation maximization together acceleration many variant improvement collapse choice explore option fact relevant posterior argue predict future well predictive generalizing version symmetric prior quantity p original along discussion formula typically chain scheme identification logistic active anonymous associate comment generalize special view pseudo generalize yield th quantile maximum machine represent convolution lead distribute marginal improper necessary mixture still apply multinomial logistic modification indicator ik follow write conditional independent indicator coefficient product regularize multinomial update eq sign divide function equivalently obtain identity derive expression classification collect z recall row mixture since rest corollary example variance mixture regularization generalize prior regression expectation maximization wider demonstrate include quantile acceleration augmentation normal regularize objective multinomial predictor unknown log phrase minimize unnormalized negative consequence penalty discrete generalized miss problem unify many include negative outcome support machine robust key derivative sufficient expectation conditional variable estimator generalize disadvantage maximization slowly fraction information large basic exception substantial gain quasi acceleration combine good maximization newton robustness super convergence research correspond example include bridge estimator machine jeffreys pareto augmentation decomposition p working code estimate although may specify user notably study specific include support logit thing estimator refer issue machine inverse approach mean representation likelihood normal density combination mixture likelihood choice fact avoid deal distribution mode hyperparameter mix exponential integral identity density use involve represent expression lead identity improper limit density loss first pseudo support third regression canonical improper multinomial category exploit guarantee penalty check presence mode return compute mode gaussian mixture conditional row exploit variance maximization quasi newton remainder conditional remainder numerically newton like omit formula iteration increase density explanation acceleration offer mixture I wish regression assume outcome code represent update estimate stationary I row resemble iteratively square due subtle difference iteratively least matrix entry rapidly numerical failure nearly initialize solution illustrate run pure goal standard different draw factor loading standard nearly orthogonal acceleration conjugate different hypercube latter case
indicate proportion outlier ph bold noise x outlier ph bold r input level noise x bold indicate r r proportion c outlier ph value r proportion proportion train cpu cpu proportion outlier train cpu ph train train test cpu ph outlier cm r r set dim size ph ph remark corollary edu school filter building replace half start examine fitting contaminate criterion prevent unnecessary removal training challenge non optimization derivative free outlier correctly nonlinear global robust approximation task science processing application rarely provide reflect nature quite efficiently sophisticated huber basis wrong phenomenon typically occurrence routine range need eliminate examine notable cell least infinite phenomenon popular discard half discard small asymptotic residual something maximum likelihood estimator outlier residual stand much devoted regression paper deal see estimator affect chen et square robust backpropagation anneal fitting regression minima ann training square criterion high minima applicability traditional fast backpropagation article fitting fit combine ann removal outlier fine clean backpropagation second improved prevent unnecessary removal undesirable prevent unnecessary impose removal introduce definition several optimization section ann exist present comparative study briefly introduce origin global discussion intercept explanatory identically goal determine term absolute deviation mention sensitive outlier affect estimator small proportion sample size overcome ols estimator introduce least deviation contaminate datum detect account high follow discard build residual evaluate partition square residual small residual advantage heuristic method evolutionary semidefinite method mention achieve contaminate outlier easily detect either eliminate closely case interest mention minima instance estimator inner minimizer permutation distinct determine efficiency efficiency ols estimator fully ol efficiency method efficient high efficiency preserve initial provide robust cutoff beyond absolute residual exceeds remove adaptive weight ols step normally initial combine efficiency final ls nonlinear nonlinear vary model data output hide respectively input bias term represent ol residual backpropagation weighting method combine multiple minima criterion outlier demonstrate unlike regression shift towards attempt use huber criterion estimator residual growth large effect quasi term quasi huber either priori estimator scale latter estimation estimation mean evaluate robust criterion ann ann discard treat build wrong several introduce allow criterion mention criterion need discarded mention numerically sufficiently ann decide design removal subsequent backpropagation approach fully efficient ol ann mention objective clean remove large backpropagation objective ann clean consuming multiple iii ann execute use ann study negligible cpu implementation language library translation complexity achieve competitive cpu calculation graphic gpu c valuable traditional cluster core gb ram execute thousand limitation identical parallel parallel execute primitive calculation gpu sorting gpu summation parallel gpu artificial datum set paper artificial set example consider take segment indicate take subset outlier outlier divide subset center random variable coordinate combination euclidean norm explanatory segment robustness ann investigate deal model explanatory early explanatory variable numerous article robustness ann investigate follow explanatory investigate deal explanatory early one function surface datum standardize network publicly consumption regarded value use water consumption set water consumption real world take difference value optimisation fix cpu start package ann default datum training subset contain noiseless domain function ann average cpu procedure depend table rmse backpropagation bold apparent ann view row train figure reflect large rmse consistent method practically filter remove contaminate clean outlier repeat across datum set furthermore mean cpu backpropagation long backpropagation increase explain rough gpu core quality cpu appears contain rmse cpu time world consider backpropagation good comparable backpropagation absence reach artificial look figure correct robust note origin lose undesirable criterion bottom give
shift instead notation general assumption operator robust nr nr proof state uniform function measurable b define separable q universal uniform totally say ba refer next list need bound instead lipschitz define b follow statement em closely theorem instead n enough ns qualitatively robust continuous case get imply due weak even topology kernel space qualitatively neighborhood measure ns ns assertion step step continuity svm functional bootstrap svm qualitatively compact totally bound continuous shifted assumption compact metric lipschitz map triangle definition shift definition continuity combination functional continuous respect topology obviously separable therefore qualitatively every bootstrap qualitatively b complete proof axiom mm van complete robustness shift nonparametric unknown known borel separable qualitative robustness special show vector stable functional svm generalization proof mention borel algebra borel metric borel algebra borel complete borel bn value n ns nz nz nz ns nz ns nz nz bootstrap bootstrap value value call bootstrap approximation sequence transformation q valid continuous neighborhood nz qualitatively neighborhood ns n ns
distance recover subspace base subspace clearly linear occur image identify highlight conduct show scene hyperspectral paper estimate towards bayesian draw operate conventional mmse manifold mmse approach minimize distance manifold formulate entail along rank unconstraine coordinate q derive give make k since ty previous equation stand wise vector q k ks scheme suitably like university comment point also lead kn db kn prior prior snr snr snr department france university subspace operate manifold usual adequate metric manifold alternative propose carry minimize distance estimate consider metric eigenvector carry illustrative include von obtained analytically implement monte provide hyperspectral processing signal span evolve consist mode estimation play central recover resort kn kp estimate arrival multiple snr swap occur subspace circumstance bayesian enable estimation investigate approach herein assign minimize yield illustrate new performance assess conventional pure material hyperspectral conventional minimum wish range stand operate subspace mmse systematic mmse minimize euclidean I distance natural natural distance two subspace orthonormal svd tp unitary seem adequate mmse span intuitively appeal face minimize square angle argue subspace mention estimation author hence parameterization since subspace span eigenvector word notational convenience case derive approximate monte carlo constitute schmidt low difficult approach mixed need jointly circumstance coincide differ mmse note meaningful unitary relevant investigate occur final comment unitary range although address grow set manifold excellent reference interest geometry attempt herein illustrate conventional condition first step derivation consist distribution accept von trace matrix arbitrary function argument respectively observe thus view manifold manifold span orthonormal matrix possible namely proportional proportional cosine angle sum cosine angle close purpose display fraction figure identical increase additionally close distribution angle exhibit show figure density practice condition assume vector assume knowledge condition condition recognize p see therefore form projection maximum posteriori estimator contrast use refer von fisher np pp although knowledge exist burn unitary successively unit nk therefore draw interestingly arithmetic set differ truly minimize require compute prior posterior distribution mode close matrix gibbs distribution condition zero eq orthonormal noise follow eq express eigenvalue representative information th signal second facilitate derivation condition priori distribute interval choose low resp posterior thus problematic gibbs draw p p conditional recognize distribution generate accept reject straightforward show condition still need initial illustrate monte matrix signal ratio burn sampler estimator mmse priori figure evaluate conclusion draw estimator well prior snr enable subspace prior mmse poorly since average remark mmse well restrictive conclusion precisely uniform db make well propose subspace decade receive great interest purpose monitor mapping concern analyze decompose signature retrieve signature band obvious physical consideration kind proportion satisfy positivity constraint hyperspectral induce abundance pixel subspace simplex vertex strategy hyperspectral literature generally identify well linearity context occur consequence hyperspectral image capital image notably assume pixel hadamard reduce fan generate signature library assume interaction interaction display fig resp white represent low linearity image increase left corner note contain pixel linearly belong conversely local pixel live linearly subspace determine detail span refined estimation develop matrix suppose subspace contain computed form expression e evaluate projection state
coordinate coordinate update gradient coordinate calculation operation cycle dense per qp number outer cycle ie across column update associate converge characterize precise estimate warm start accurate moderate experimental glasso inversion note operate explicitly compute keep retain glasso optimization upon consequence panel precision produce update square plot right precision produce glasso minimal value life problem might viewpoint approximate sparse glasso purpose glasso maintain update entry principal return figure typical glasso operate optimize regularize require dp glasso working maintain glasso estimate track simple rank update describe unlike glasso positive precision column iteration advance compute warm guess solution dual warm necessarily address states row glasso maintain warm glasso column maintain work glasso partition th remain condition furthermore w box qp since box e qp combine violate row update glasso maintain matrix encounter counter warm tuple option converge easy numerical choosing example encounter briefly describe iid sample glasso take warm glasso fail warm note sufficient update glasso eigen surprising w light need guarantee iterate remain pd glasso converge establish pd warm p glasso glasso ensure warm glasso glasso every glasso dp consider update update diagonal entry satisfie update pd glasso coordinate simple lemma arise regularize box valid program respective quadratic pd glasso glasso converge definite warm start due primal glasso require initialization construct dp glasso require half update example glasso series play problem start dp glasso wish early glasso starting start dp glasso diagonal trivial algorithm glasso glasso warm start describe example micro example model concentration zero follow generate iid entry set one spaced article glasso algorithm coordinate glasso initialization wise version glasso warm suggest converge due fine grid glasso diagonal path wise glasso warm start glasso comparison glasso dp glasso glasso block dual include dp comparison glasso glasso update glasso box dp glasso report qp glasso constrain computation cpu ghz glasso operate glasso operate dual example base successive q across row tolerance primal glasso inversion comparison compute expensive experience base precision glasso matrix glasso present grid show eight evident glasso warm algorithm large typically sparsity become slow end viewpoint warm glasso probably far warm glasso design glasso warm section base plot path plot vertical dot estimate precision population minimize green blue dp glasso seem prominent latter table present type time level glasso return moderate glasso across entire path high precision glasso glasso see warm winner among decrease winner glasso warm starts glasso warm start see dp warm starts glasso warm report comparison example primal warm winner observe primal warm fast warm type difference matrix htp warm start dp glasso warm start grid panel covariance vertical line correspond htp solution dual warm primal primal warm dual warm glasso warm second combination glasso warm start across example htp solution average warm primal warm path glasso table dp glasso warm start consistently experiment process filter gene gene thresholde microarray threshold gene large form gene pool subset warm warm warm accuracy warm perform htp warm warm explore apparent glasso explain leverage glasso dual problem dual equivalent course maintain definite inverse tight essential inverse glasso solution sequence warm glasso former maintain operation dp glasso glasso solve update qp property end maintain start glasso package implement dp glasso make thank group stanford discussion comment presentation experiment warm glasso take seed seed gaussian matrix denote absolute solve glasso warm start eigen produce glasso clearly undesirable glasso seed generator iid gaussian matrix solve warm glasso fail converge example row covariance eigen compare glasso glasso follow set eigen matrix definite manner value scale perform primal primal warm warm result show result overall glasso warm explanation though dp dominate dp glasso warm start dp glasso warm scale index acknowledgement discussion statistic stanford department stanford university graphical undirected zeros glasso popular allow one efficiently glasso converge warm start explain outperform glasso study glasso solve likelihood p glasso dp operate coordinate sub conclude realization definite covariance task unknown especially ordinary mle poorly behave regularization framework estimate precision correspond algorithm graphical regularize negative q sample amount shrinkage glasso analyze propose suitable implicitly insight use absolute frobenius denote normal sub write equation solution component wise sign glasso use partitioning way partition form glasso hold get plug glasso operate read fix stationary see glasso solve rest except current coordinate sparsity easy update move onto next converge glasso outline successive glasso keep initialize cycle repeatedly implicitly solve warm round save row convert produce glasso iteration value zero successive glasso produce monotone dual value curve produce glasso confirm plot block behavior lead problem correspond equation qp constrain qp kkt optimality stationarity kkt box qp substitution relate result theory dual qp denote primal relationship duality qp qp glasso column unlike require glasso ascent constrain dual box glasso optimize lagrange dd
generator hold configuration plan fixed run heuristic planning plan heuristic planning learning thus serve slow policy improvement approximately improve learn step reasonable curve never policy started show high episode improve rate episode present find trajectory time trajectory algorithm correspond path behavior planning planning regard curve present find start step drastically episode reach optimum per episode find path show several trajectory heuristic planning episode algorithm successive episode strategy bad trajectory use trajectory extremely learn curve evident comment short instance help close focus visit reason exhaustive work long although planning contrary policy distribution branch however propose algorithm planning planning planning episode around episode suboptimal significantly reach episode case identical policy significant planning find policy deal save resource achieve good reach length quite interesting plan novel reinforcement planning well planning module incorporate ability make take heuristic strategy apply three conclude heuristic planning therefore role since scenario inform sampling problem work planning environment software open support c heuristic play play trajectory system key batch resource option nevertheless rely complete search scenario application decision paper plan finding select branch likely outcome branch proposal excellent suggest analogy role human behavior business activity make may rapidly change system importance business design adapt rapidly change work decision make rational rational recent heuristic year issue sensitive learn variety application total benefit effective call engineering decision seek achieve environment action influence thereby choice take indirect delay consequence planning decision I entity simulator role play game game quickly although establish paper game select target point entity interact avoid trajectory automatic carry different search literature nevertheless computationally especially environment decision rely exhaustive planning strategy heavily precise complete environment game scenario apply suited incremental mechanism action mechanism application strategy literature exhaustive cite author use sensor able use application reinforcement learning activity even education application policy student fit survey search branch branch branch selective previous novel planning module bad express action particular example distance goal architecture grid environment treat markov sequential decision work inform sequential problem go experimental devoted conclusion far go briefly combine heuristic solve agent homogeneous planning method plan current heuristic search encounter tree alternative node root tree max stop state action rest conventional heuristic save value design result algorithm agent reward reinforcement learn view asynchronous dynamic dp reinforcement act optimally markovian domain consequence without receive domain similarly td try consequence immediate one discount straightforward greedy choice parameter exploit value agent assume planning take produce interact approach planning path cause another function terminal execute r model ss r terminal plan new gain intensive available computational resource easily reinforcement planning experience model directly sometimes indirect involved plan next action reinforcement planning learning plan act rl planning rl step world record next reward sample pair conceptually plan rl agent propose planning strategy incorporate advantage shortest like environment g heuristic search branch branch successful selective incorporate heuristic make search reinforcement contrary consist environment bad priori bad reward lead priori trajectory simulate interesting behavior hypothesis human brain orient learn experience trajectory use strategy jump law thing analogous heuristic system grow system behave activity connection agent base aspect behavior relate motivation intrinsic publication
kernel see latent output strength entry old connection connection form network structure bayesian dynamic write conditioning understand output correlation couple condition covariance influence distance point point gps view amplitude mixture kernel entirely square motion kernel switch addition signal correlation volatility model gps influence account small factorization instance case dataset marginal dataset introduce signal node magnitude result fitting bayesian weight constant switch switch scale fix value predict value different approach mcmc w kernel compose function q diagonal since way order matrix weight use bayes incorporate gp volatility volatility volatility likelihood p p volatility minimal procedure component significantly change bayes use bayes mix poorly correlation node mix impose elliptical recent mcmc correlate joint costly numerically unstable highly w joint construct construct infinite generally heavy assess long take fit posterior kullback leibler variational computational nj nj f use deterministic pass vb form q f nj nj vb available ard however require take obtain message descent lower see straightforward contribution mainly take block covariance operation regression function evaluation volatility normally invert numerically unstable allow scale complexity vb dominate iteration function give covariance weight calculate weight mean per iteration vb reach overall account compare multi complexity volatility also new model compare variational vb chain inference keep comparison date task set account account change correlation volatility gp co krige output gene account dependent specifically volatility prediction make generalised wishart estimate suited exponential function function measure gene expression level hour hour replica gene activate protein focus least influence level factor output training replica replica average testing dataset alternative scale sparse accounting volatility volatility standardize small used replica replica training testing create similarly vb comparable outperform likely high indeed may mcmc mcmc level toolbox typical second km region access location wish predict standard correlated enhance location enhance function weight vb output correlation heavy figure generally around structure result learn spatially vary beneficial able predict dataset show three dimension learn marker use absolute mae predict experiment mae krige unclear log transform dimension unit skew include mark give mae gp cost accounting correlation moreover scale method run method intractable volatility dependent making incorporate generalise wishart correlation output volatility prediction ahead forecast historical understand financial follow exactly predict five process especially become benchmark assess make ahead forecast forecast predict compare generalise wishart original wishart table vb competitive even historical learnt value less meaningful encourage ahead forecast likelihood historical prediction covariance fully volatility outperform vb volatility perhaps suggest r vb mcmc vb mcmc mcmc mae vb vb co gp historical mse forecast mcmc empirical mcmc empirical gaussian process interpretable structure extension gaussian output heavy predictive scale scalable procedure empirical several benchmark structure easy gradually periodic hope thank valuable gene briefly process regression introduction define etc exponential display long trend length interpret learn useful determining past forecast covariance input covariance introduce velocity motion correspond time ar process markovian independent mat ern eq time model special recover many periodic period gibbs allow length scale datum noisy note gaussian integrating input regression predictive marginal amplitude noise section online book website length reduce mode posterior weight vb explicitly constrain representation find extension improve multivariate volatility weight propagation vb need straightforward analytically fit q
prevent class order versus technique successfully multi extend classification build expensive perform test member observation test classify entry extend binary label sign real return one positive value take sign vector entry indicate event I label prediction sensible since confident label observation affinity bag ensemble repository ensemble bag forest grow five label predict label use rule compare bag bag classification predict ensemble bag majority figure vary level predict easy easier distinguish ensemble suit versus consistent strong ensemble neither boost bag ensemble figure apparent rule ensemble apparent ensemble solver easier large percent correctly difficult classify l rule bag vote breast w htb approximate coefficient solve constrain advance magnitude fraction absolute direction change expense importance avoid prevent coefficient keep basic rule figure rule yield complex deeply subtle within depend close ensemble use tree complex far ensemble variable large tree tree tree variance rate general distribute change tree ensemble variety tree bottom also rule rate allow result method substantially use less ensemble chance phase subsample experiment take decrease subsample terminal lack difference enough use tree term build term less linear regression ensemble nearly would complex nonlinear significant predictive variable control many thresholded descent figure include range capability threshold decrease use experiment package return elastic net within regression elastic wise method nan cycle partial soft update modification also allow decrease exponentially meet increment warm prevent cause train inherent interaction subset clear solution prevent overfitte capture enough characteristic decrease generate experiment use coefficient method change find false twice negative probably negative rate experiment use iteration iteration minimize operation experience scale figure accuracy solution increase high coefficient experiment weighted sum leave control weighting minimizing importance become thus problem choose multiply extract central exploit property reach shrinkage spee many zero choose solution fix initialize expand contract user variant generate show figure roughly accurate solution generate solver thresholde dramatically behavior weight square reasonably build selection difficulty risk build different plot simple come tree use build capability coefficient figure successful correctly classify build attribute importance correctly important save expense indicate coefficient magnitude rule sift look consider rule order large table repetition attribute repetition vote influential repetition vote high rank form attribute available rule repetition indicate magnitude rule large contribution attribute look different order receive example rule order rule try set rule top rule decompose rule repetition influential attribute attribute repetition available rule repetition test repeat figure compare rate attribute number attribute model successfully rank attribute essence extra attribute act attribute accurate capability well fewer allow extra exclude datum collect save store vote influential try influential least small attribute train figure htb namely boost bag variety class ensemble extension method highlight ensemble well tree decision tree ensemble return ensemble relatively number like class build would separation make training rule control correctly capable produce finding difference coefficient method consider ideally solver would way important method indicate magnitude share similarity trivial reflect return return return rule method identify attribute contain method benefit return weighted rule rule rule important rule technique attribute error building model set traditional ensemble decision bagging provide weight relationship certain tree complex deep tree limit affect fall correlation simplify post processing capture rule need regression correlation exist computational ensemble matlab write reduce training portion coefficient grow ensemble substantial unnecessary quickly method machine computational ensemble solver programming implement language acknowledgement suggestion appendix method detail fast negative take coefficient negative constant minimize trivial second term square want derivative substitute gradient function rearrange switching update iteration coefficient take everything act predict scalar use gradient simple move proportional large negative gradient absolute length fully update affect update coefficient function update follow stepsize exceed step th substitute add turn subtract indicator turn negative david berkeley national laboratory road popular accurately predict class learner offer capability offer little insight ensemble accurately advantage indicate predict rule give multi bag tool classify high automate grow powerful successfully disadvantage interpret ideally like generate quick give insight understand attribute response dependency ensemble method fast base learner capture capture bag test refined dataset neither bag boost strong modify extend bagging boost attractive combine importance regression quick collective make learner build form grow node combine ensemble utility modify coefficient test wide ensemble class classification repository dataset ensemble look dimension large th predict class future unlabele belong binary two classify risk map predict label function develop problem consider risk train expect set combination learner coefficient risk minimize take solution many solution interpretable like possible prevent influential use impact control receive deal enable coefficient brief gradient tree th pseudo entry subsample memory iteration build note intermediate rule training pseudo unable multidimensional regression termination behavior capture control dependency build learner past build model need approximate construct distinguished approximate
parent east child anchor west east distance child circle child em em child parent child parent child node child parent parent child rectangle em parent child node edge parent parent anchor east anchor grow east em child distance node edge minimum child anchor east em child child right anchor east grow scale shape node write gamble final chance us branch decision strategy subtree compare gamble move branch cd consider cost cost insufficient decide cost option create unclear follow policy take predict involve normal decision easy check induction apply six gamble tree work know agree invoke relate terminology method provide theorem state coincide method yield strategy entirely nature take arc reduce form decision unlikely specify eliminate subset tree form reward entirely normal form set decision tree gamble early non empty event let follow equivalent satisfied fairly straightforward notation prove convenient normal gamble chance gamble gamble formalize method start find gamble gamble solution normal form formally definition strategy say calculate check whether optimal gamble xt tu tu tu tu operator popular reason wish even trivial decision tree least path child lot gamble impractical particularly avoid implementation backward property adapt traditional informally difference focus represent generalization gamble backward require clarity prefer rigorous induction operator move tree subtree set form remove retain gamble decision normal move next node root reach normal node call gamble empty gamble gamble unless optimality gamble affect non element attribute gamble call irrelevant state non gamble non gamble contain together check family empty gamble path appear frequently investigation axiom satisfy backward gamble let equivalent label satisfie section coherent proof strong let non gamble bx b satisfy empty gamble bx ax z immediate proposition consistent immediate proposition corollary decision fail property gamble contain envelope cc gamble clearly dominate dominate interval backward induction easily see consistent tt backward induction theorem property induction fail serious inferior backward induction gamble example subject decide provide site yield pay site optimistic minimum size west grow east level child circle node distance parent parent near edge parent edge parent child rectangle parent circle level child edge parent child node node child edge parent edge draw draw circle em parent child parent parent parent child parent child child node edge parent near node parent low table et incoherent correct ccc form decision uniquely identify gamble gamble notation baseline anchor grow transform node circle em child parent node parent node parent em child node edge child em node parent child distance right parent east child anchor shape circle pos child pos child edge parent parent east anchor east shape child node x x child em east anchor west east transform shape x depict backward chance node report follow outcome good marginal great x xt x eliminate dominate dominate calculation maximal maximal corollary differ al detail trivial give probability gamble backward induction gamble except computational benefit form knowledge assign coherent low set function plausible maximize solve apply list backward induction help solving differ cast result coherent fail interval surprisingly low property fail fail precise property conditioning independence want satisfy argue solution specified reach one acceptable subject mind advance carry function cause undesirable example subtree moreover always computation burden secondly gamble stage eventually even perhaps surprisingly normal exactly tree intermediate since coherent investigate uncertainty present gamble order path empty gamble empty I family empty gamble property property definition empty gamble therefore gamble gamble event empty gamble empty gamble finally satisfy property gamble property empty empty gamble lemma solution typical choice large tree find backward induction choice yield work backward induction instance classical decision maximize probability limited maker may handle include amount probability problem expect utility suggest systematically systematically decision criterion admit solution decision problem induction main contribution backward induction form option uncertain consequence option base subject seek represent maximize advance choose maximize utility specification fortunately induction utility previously reach backward coincide probability form criterion strategy generalize backward induction summarize replace move strategy go de idea differ many expand low find reason first mention feasible induction eliminate strategy hence far efficient secondly might paper present formally define characterize discuss large conclude low section informally event reward root chance node leave grow happen last either also cost weather leave outcome predict figure size em parent east anchor east draw circle child em child em child child level child node edge parent parent parent child rectangle em em child node node child parent draw parent node parent edge node child node level edge parent node node edge child em child parent distance em child parent child parent edge child draw circle em child child parent node parent edge level child em node edge parent edge node child draw distance child node child edge parent edge parent assess utility next solution demonstrate backward allow identify instance state element subset call capital arcs chance function uncertain reward state yield reward whose sum finite express close assumption whole expectation empty mean probability obtaining follow
decision incorporate validation answer systematically find address prediction reproduce model validation inform calibration rely understand limitation maker quantify failure satisfie maker requirement framework quantity assess become necessary activity decision computer state validation accurately capture behavior uncertainty effect quantify correctly approach partition next calibration calibrate predict small discrepancy improve principle incorporate goal detail specifically examine situation value available scenario impractical still want incomplete directly example situation computational aim characteristic wind nuclear assess environmental impact failure predictive systematic call validation suggest hierarchy validate phase update pyramid ability use high maintain author split experimental validation pyramid experiment calibration rigorous propose partition datum calibration average split reference instead way disjoint calibration choose analyze split choose term split optimal find validity replicate assess reproduce quantity interest information must produce satisfactory second fundamental issue validation set challenge model confident prediction concept mind satisfy inform calibration reproduce data validation quantity able answer work receive intensity intensity decision reliable explicitly step algorithm section reduction iv short h demonstrate quantity interest updating prediction make update newly assess predictive capability drive require metric calibration decision may perform suitable tolerance quantify ideally metric provide easy metric percent nominal measure determine tolerance procedure flexible choice threshold choice application inference probabilistic input function briefly capable reproduce work mutually suitable demonstrate inform inverse metric analyst system develop partition split admissible inverse problem impossible area improvement approach subject future treat calibration obtain depend could evaluate calibration detail visualize metric model measure consider split threshold ability reproduce must change perhaps next optimal highlighted figure assess ability predict partition threshold fail threshold use make prediction tolerance decision maker conclude valid demonstrate may perform metric maker partition stress procedure extremely advantage allow range specific reduction assess prediction solve markov chain monte sample equip hybrid briefly next present result camera gate camera count gate gate linear regime predict raw receive gate range micro second quantity interest processing gate require data nonlinearity gate introduce gate width incorrectly rate camera respectively calibrate problem algorithm mention unit horizontal cutoff tail cdf datum include correspond percent characterize value gate include calibration validation use datum tolerance gate leave set fact point provide gate keep single unit inverse prior update compute result requirement find plot show predict thus look represent first split form receive micro second might optimal split contain low gate close calibration
ny sf sf tf sf consider I random ny ny z apply keep notation use without b kt k h k n kt h b h kt k h h h r k h rely finish proof ny computing functions unconditional characteristic nn z independence bootstrap consider value account mapping theorem argument recognize show case q follow similar pick account eq hand know eq q argument similar least last identity argument arbitrarily similar manner brevity omit proceed write similarly q strong prove vi proceed prove consider square envelope maximal inequality consideration subsequence display happen surely compact k inequality hold subsequence mean different constructing interval smooth asymptotic argue bootstrap confidence bootstrap remarkable study procedure show consistent also condition procedure external influence encounter every science relation finite point simplicity compound crucially among nuisance page build bootstrap nuisance generally scheme build employ illustrate issue arise smooth nonparametric regression therein bootstrap asymptotically valid inverting bootstrap bootstrap use form estimate slightly suggest complete validity go cite evidence suggest nonparametric bootstrappe residual bootstrap build certain process suggest absence weak limit addition smooth approximation residual finite superiority smoothed procedure triangular validate achieve first argument version develop conclusion broad function specify work nonparametric procedure ci neighborhood change jump stage rely approximate ci immediately organize scheme notion section prove generalize constitute section residual bootstrap show consequence general point proof sequence parameter convention suppose concave leave third piecewise w interval maximizer small stand small maximizer nm property square estimator page n two process way distribution brief review function usually root break simple generate nh nr say weakly consistent weakly strongly choice mostly essential smoothness expect perform phenomenon root x follow property bootstrap draw bootstrap scheme bootstrapping widely residual n predictor response z smoothed density successful scheme choose nonparametric n j n j bootstrap usual bootstrap regular number z n available satisfactory solution vary framework convergence prove scheme procedure give theoretical evidence scheme widely page typical see failure situation investigate estimation situation give rise asymptotic compound poisson asymptotic distribution estimator independent dominate procedure usual problem crucially depend generalize page triangular array array k throughout operator constitute bivariate nm estimator k nm convergence consistency result estimator assumption iv regularity property like fact hold independent easily converge uniformly calculus aid theorem page v proposition sufficient measure triangular achieve would highlight satisfy suitable assumption z n vi q converge two independent sided compound continuous limit element compact terminology leave limit topology topology q endow metric become subspace page value refer reader difference prove easy iv find tight tight compact rectangle sequence process weakly mind definition z continuous poisson give mapping theorem lemma weak weak jump piecewise continuous integer jump compact see apply first show define h square notation eq argue scheme residual definition note I state sequel strongly refer second let interval z still n n n side probability translate proposition k prove estimator conditionally surely bootstrap vc hold also lemma recall process compact rectangle iv bootstrap scheme follow lemma evident situation know hence therefore think convergence measure conditional distribution say mean measure characteristic fail limit hold converge probability n two ny measurable random moment ny aid able main compact neither weak probability weak enough pick observe e h converge weakly probability weakly eq converge probability note imply hence e limit probability latter happen iv nh characteristic probability weak weak make unlikely rigorous depend note conduct depend argument illustrate bootstrap approach unconditional centering process kk n k small inconsistent random sequence I random addition mutually jump I e nh bootstrap analytic compute approximate limit two distribution immediately random resampling arise bootstrappe fix inconsistent speaking resampling fail basic residual compute bootstrap conditionally surely strong argument bootstrap scheme introduce give start center empirical variation almost smoothed bootstrap result prove converge compact r nm define paragraph minimizer seem complete statement limit prove bootstrap scheme bootstrapping achieve proposition regularity bootstrap smoothed want fulfil procedure estimator construct choice kernel see bootstrap random next scheme turn hold probability scheme framework number condition remark subsequence p pair size take approximate bootstrap provide coverage length scheme make estimator kernel density kernel reference bootstrap refer fix bootstrapping residual scheme c coverage avg length coverage avg smoothed avg coverage avg c coverage avg length coverage avg length coverage length smoothed c scheme coverage length avg smooth fdr outperform coverage increase length fdr bootstrap procedure inconsistent obtain scheme single smoothed right panel top histogram indistinguishable guarantee efficiency choose tuning smoothed also require smooth insensitive bandwidth certainly asymptotic right bottom bottom broad change two unknown twice w least problem least bootstrap bootstrap produce build interval contain empirical replicate w iw iy j j j smoothed procedure must difficult method adapt case additive form smooth like helpful comment state characterize kk third let w next maximizer small topology converge topology open ball radius maximizer belong contain rectangle function observe jump every pure jump functional statement jump suppose jump h k express jump jump topology wise continuity page integer see large thus finitely jump happen finitely jump correspond jump jk supremum concavity unique supremum maximum write statement topology lie c proof imply second inequality sequence consequently imply n account lemma aid proof vc subgraph envelope vc index satisfy envelope sample probability notation lemma page z vc index class iv z also replace z z z z z z ny iii ny inequalities ny n ny z z n ny b similar iv lemma prove rearrange term nz z n decompose n n unique maximizer page theorem prove nm take iv make inequality consider constant expression st rd note add display get note expansion admit n lemma enough precede consecutive expansion see take define bind maximal imply see inequality z n expression n completely tight j ne imply since function z vector tight process agree number choose least jump jump form vi happen exact z j tight fourth tight uniformly show distribution end consider linear grouping rewrite imply note ne n z imply follow argument ne vi write characteristic put arbitrarily device prove
track node individually internal environment external include noticed believe indicate also world consider point choose extent share political b change political role choose em measure influential reflect opposite initialize assign validation degree learn influence sort influential area research criterion include since comparison ranking impossible study influential influential list compatibility matrix em diagonal intuition share learn one direct seem political year comparison year voting record year turn highly regard historical trend political increase strong every unbalanced entire majority comparison step agree show increase window measure quantitative highlight correctly eight term present co evolve mixed membership blockmodel model able reproduce hide able detect record different world explicitly pair result complexity network approach introduce greatly enhance efficiency foundation grant grant nf grant fa dynamical structure model co evolve network specifically membership blockmodel probability observe two membership vector membership evolve network change variational real social important scalable modeling world network inherently dynamical system attribute network change often mechanism node illustrative social influence mean interact evolution neighbor properly characterize selection subject extensive suggest continuous time agent base network evolution agent characterize depend well local evolve markovian choose select maximize serious missing attribute observable network grain suitable dynamical model blockmodel extensively blockmodel block node assignment influence relationship account role co evolve blockmodel mixed membership modeling suggest impose assume membership drive parametrize membership aggregate mean trajectory separately describe track whole shift membership experience change political experimental membership node form connect process induce sequence normal lead tractable equation compare dirichlet prior update step q qp influence vector p account sample role indicator multinomial tp bernoulli kb rs node role interaction account node role influence benefit node specific describe easily influence conversely node co evolve datum p write simplify pair describe resort parameter eq q factorize variational distribution multinomial compute problematic upper variational em iterate calculate expectation variational updating locally maximize tb guess initialize variational minimize take variational normalize eqs couple covariance similarly generally depend q equation simple iteration process converge compute expect compatibility block compatibility compare evolution dependence use
hybrid monte carlo hmc algorithmic detail broad exhaustive relevance application body derive relevance ard iterative optimization nice ard variational sparse cca relevance real method exponential yield product paper relevance ica blind deconvolution develop also basis pursuit engineering pursuit useful uncertainty spike sparsity establish statistic describe gaussian type model hierarchical uncertainty necessarily spike network unsupervise unseen appear handle miss observe miss create test select case predictive nlp create data metric aside benchmark type add image probability common optimisation optimisation need demand execute many combination approach separate make sensible quick overall learn setting variable demonstrate tradeoff run spike iteration well reconstruction human budget spike use figure well reconstruction various poor sparsity show spike compare induce shrinkage relevance problematic certain reconstruction restrict contribute difficulty set regime ep issue hybrid ideal leaving demonstrate consider accurate reconstruction paradigm broad develop generalise provide sampling importantly comparison unsupervise norm demonstrate diverse application model sparse prominent wide modelling advance cifar nsf fellowship unsupervise diverse area signal collaborative work method performance infer unsupervised latent variant induce bayesian factor accounting principled unnecessary shrinkage practice outperform budget need assess care unseen last decade parsimonious significant significance tie scene sense importance efficacy sparsity among provable relate optimality property increasingly diverse application norm behaviour compete intractable norm closely match prior spike spike spike similar impose penalty prominent vast modelling completion amongst toolbox real underlie observe gaussian unsupervise desirable underlying introducing framework develop comparative contribution generalise strong sparsity provide sparse sect mcmc efficient naive sampler sect present comparison optimisation spike bring benchmark sect interestingly strong outperform concerned search factor weight datum often isotropic covariance model explain explain datum increasingly deal heterogeneous consider probability shorthand exponential family distribution natural parameter subspace take distribution belong prior notation conjugate item notation distribution form row generally family latent recover family correspond consider latent variable indicate highly heavy tail gamma laplace mixture encourage notion mm parameter none element zero exactly spike place suit achieve learn average rather search good use continuous mass enforce place penalty thus binary latent dimension contribute bernoulli spike form component specification denote applicable sequentially variable step slice decide contribute integrated pz nk nk exclude slice evaluate computing q integral family approximated integration laplace bias approximation laplace latent behave show require definite information matrix boundary parameter latent alternate integrating describe result fast reversible specification indicator e slice thought proceed sampling slice randomly draw interval relationship gibbs full easily derive encode connection density appeal optimisation broad provable exact recovery rip optimisation linear semi definite sparse laplace norm generalise modelling n negative use framework control latent variable loss unsupervise
plus minus sampling estimator n certainly since expect numerical approximation bias true occur bias move actually fall another suggest idea gain employ estimate dramatically impact respect bias shift design bias overall chemical spatially homogeneous dimensional frequently leave couple ordinary specie specie specie molecular heat capacity pressure production rate define elementary reaction rate side reaction reverse th reaction activation k change standard pressure reaction mass express compactly subscript ratio fraction specie perfect pressure experiment consume expensive conduct design base employ statistical foundation inference indirect mechanism incorporate source construct measure expect monte gain stochastic approximation feasible intensive setting algorithm demonstrate model parameter inference detailed quantification bayesian shannon chemical role physical example knowledge discriminate compete observation laboratory datum expensive even consume perform experiment dramatically model design question experimental question receive community application depend criterion experimental design write functional include optimality minimize maximum variance instance utility shannon optimality may derive optimality counterpart nonlinear exact evaluation optimal design tractable criterion obtain impose additional model approximation lead involve locally require select unknown model maximize fisher though broad significantly normality assumption prefer rigorous theoretic criterion throughout seminal gain information provide justified maximum shannon range criterion introduce gain material kernel equivalent chemical criterion design design limitation expense design space infeasible gain require discriminate design application simple design objective chemical criterion objective strategy suitable broad theoretic formulation propose curve wherein intuition utility surface space markov chain combine integration idea extend simulated annealing expect utility metric direct information enumeration consider design space information via coupling criterion open issue realistic advance state art yield design formalism assumption parameter inference shannon gain criterion naturally incorporate parameter probabilistic relationship experimental need generality computationally tractable capture dependence nonlinearity dimension quadrature exploit parameter dimension link approximation result objective experiment design rigorously figure component embed design cycle upper box focus experimental formulate construction surrogate intensive section come experiment tool chemical experimental reflect relevant consideration specify objective objective reflect infer appropriate would reflect discriminate consideration motivate notion course expect utility develop goal add cost broadly resource interest cost constraint formulate offer inference noisy indirect incorporate constraint heterogeneous information assessment natural decision exploit discussion approach bayesian paradigm treat probability space measure I measure endowed appropriate hence uncertain observe realization datum change give bayes evidence reasonable vary simplification experimental design form utility support function reflect usefulness value precise choice utility root put kl generic support difference carry unit inference note utility involve internal represent simplify equation yield prior decrease amount informative maximize parameter design apply optimality maximize allow multiple simultaneously well repeat gain experiment equal experiment incorporate likelihood give carry condition expect information gain simultaneously objective often experimental would always single carry experimental e help next approach design experiment approach necessarily design programming sequential experimental design least design due gain must numerically rewrite inside special shannon drop maximize sampling retain equation accommodate example likelihood error carlo e evaluate analytical yet carlo sum bias estimator proportional control control sample see monte outer inner monte produce zero small contribute bias effect numerical bias value grid clearly exponentially monte carlo objective available na I optimization expensive effectively monte evaluate thus suit function simultaneous perturbation stochastic nonlinear simplex stochastic receive estimate two perturbation estimate regardless problem dimension recommend value inverse gradient justification gradient average proof randomness difference allow position feasible perturbation nearest omit discussion magnitude gradient minor modification improve noisy function simply project point near base take objective require step finite however level slow constraint false approximately optimization pose sense expect utility stochastic optimization complex embed equation evaluate repeatedly value task expensive enter function discrepancy leave drawing evaluate calculation replace support entire design option reduction polynomial expansion latter exploit output uncertain extensive define space field generate measure random variable measurable infinite multi expansion coefficient orthogonal distribution expansion convergent purpose infinite dimension truncate order truncation influence degree nonlinearity relationship freedom availability pc output depend impractical depend suggestion proceed increase dimension put component simplicity affine transformation uniquely associate vector inside hyper uniform convergent expansion pc coefficient alternative difficulty step strongly character equation prohibitive arbitrary expansion project quantity interest function deterministic black box need flexibility functional may depend smoothly orthogonality pc coefficient simply pc index analytical expression quadrature model term projection non forward essential numerator equation evaluation regularity dependence quadrature sparse quadrature quadrature rule make formulation quadrature quadrature take advantage quadrature rule overall use quadrature cc especially nest weight require little difference formula difference quadrature rule using plot span space maxima appear pattern understand examine noise batch adopt respectively single parameter symmetry contour identical second experiment locally design intuitively slope output instead slope great examine restrict prior experiment figure design surprising winner explain combination design yield note optima original could experimental essential role refinement chemical indirect relevant develop model regime new great demonstrate design framework behind reflect sharp rise pressure suitable delay delay carry process experiment describe spatially pressure chemical ordinary individual chemical detail subset reaction associate reaction mechanism chemical nonlinear variable consist temperature specie mass equivalence production depend factor energy reaction table reaction equation help open chemical software formula parameter reaction branching lead net species reaction reaction relevant help target range nominal positivity easily appropriate temperature expect equation maximize choice state state affect delay due front state either dependence desirable smoothly goal easy release intermediate chemical specie peak heat release characteristic actual implementation several order magnitude function design ode deviation error value constant resolution technology note depend implicitly influential expect approach different ode system practice construction evaluation worth total evaluation optimization expect exceed surrogate tradeoff gain input support expansion polynomial expansion projection pc numerator expansion truncate stop degree expansion examine section accuracy rigorously observable ode pc affine pc expansion expensive minimize evaluate use level isotropic sparse quadrature contain show contour dominate exponential observe smooth ideally contour plot accuracy gain difficult contour use ode estimator pc surrogate contour pc less variability due importantly illustrate since design reflect large ode datum perform contour ode surrogate posterior full pc utility design informative three ode precisely model ode exactly generate noise use two second value surrogate remark characteristic informative peak greatly influence even observable full observable force mode observation selection make argument lead designing experiment infer unchanged efficiency pc dimensional grid entirely introduce objective might objective optimum noise existence non negligible balancing understand impact scheme fix maintain loop set optimization evaluation I noisy evaluation perform plot run cross red pair design choose design result less determined influential group final figure indicate case quickly affect evaluation take evaluation shape utility observation far assessment history utility final really end histogram utility final point negligible employ histogram indicate persistence create support bad good design study term quickly reach factorial experimental list design lie experiment use design much well experiment factorial factorial pick corner good experiment factorial would exponentially become impractical quickly experimental capable produce quadrature high quality gain must quality similar pc position history final fact appear even tight surrogate practical reach require ode speedup order full experimental sufficiently experiment posterior contour involve average broadly explore range sensitive optimal experimental issue polynomial experimental employ thousand million combine computational surrogate construct polynomial perform optimal reach construct pc inference expansion capture thus smaller fewer expensive pc easily paper systematic tool design incorporate intensive rigorous theoretic criterion simplify give show cycle experimental criterion prior posterior couple stochastic optimization would otherwise prohibitive accelerate flexible demonstrate experiment nonlinear delay magnitude illustrate design information investigate utility scheme overall experiment informative
go reason decompose batch observe q nc properly scale onto clean part involve utilize analysis validation crucial handle vanish fraction observation clean entry condition simultaneously equality b construction inequality observe step uv te tw four tw uv te tw uv te f uv te uv te write eq ii first use hold second argument thus utilize sign similar self hoeffding te event k high probability k polynomially different prove inequality use ii use separately term I use assumption fail dependence use though random subset un corrupt independence operator term define k ie type bound lemma term turn sign bind need distinguish case term definition expand product term k first type term q q np use theorem collect five give w apply part twice p apply later sphere property operator xy xy I hoeffde inequality xy nd make exponentially union summing sum together e therefore inequality complete line write sufficient optimality convex impose condition indeed optimum part way devise less frobenius repeat time decrease jump phase go increase probability adversarial success change experiment predict deterministic rank corruption adversarial lie hard recover adversarial spread bernoulli possible go deterministic predict grow linearly bernstein useful sequel later let l n away follow te ie proof one simple lemma since incoherence f f triangular fact index np lemma sum zero bound small moment random z ie te ie ij ij te ie ie te ie te ie ie te ie ie ie f ie apply part variable index index satisfy p ie z p set fix j te p te ie np te ie z te te np next lemma te eq column repeat desire condition te three incoherence entry np choose cn similarly side finally iw assumption finally ce independent sign decompose ce b separately te b te express e e te e bound argument te nc te term lemma inequality te b te te te f np side c complete low contain entry observe location unified minimize rank succeed allow component one hand corollary single way work completion hand rank analysis dimensionality reduction area pca collaborative either correct simultaneous presence fraction corrupt recognize fail fail even corrupt light studied fully set guarantee support wise strong present theorem presence scale corollary rank logarithm provide exist provide guarantee support case vanish grow application fraction error assumption recover work allow vanish observation deterministic work locate besides improve proof high neither location constant allow fraction program notation value intuitively act norm surrogate sparsity specify early appear incoherence optimum I impossible identify add prevent scenario approach incoherence proper subsection guarantee adversarial error recover constant quantity set additionally observe entry generate entry column unify universal solution nn conclusion treat treat miss apply weak result study refined manuscript handle case randomly locate provide strong allow small publication conference also random observation good exist completion adversarial prominent adversarial recovery improve sign sign constant equal nn fast agree intuition easy error location sign arbitrarily provide strong require fraction usually quantity dependent deal error e unobserve correspondingly tradeoff improvement provide recovery satisfy satisfied result guarantee matrix moreover another improvement square tight manuscript close completion paper prove unified line low matrix simultaneous deterministic dual proceeding additional notation clean clean element write entry span share projection complement orthogonal sequel might simplicity case square proof five elaborate next five section suffice assume sign arbitrary sign error obey
introduce rw among neighbor node loop move move essentially transition high degree bias rw recently exploit network contrast graph visit terminate collect traversal seed explore visit seed similar except iteration explore seed ff flip coin probability decide ff already forest grow name accord classic name neighbor visit visit hide population drug social survey comment l node fraction cover depend range regular graph concentrated around graph balance skewed several decade www internet ip level give graph graph arbitrary capture approximate topology well classic give match fig ignore rectangular left random run high respectively relevant particular rw serve reference point analysis walk excellent converge equilibrium degree calculation true connect average degree show transition show consequently rw system analyze chain crucial dependency handle adopt elegant connect bias comment work c real interval next node index match plain uniformly two match degree sequence loop begin traversal technique node compatible model queue keep discover yet initial yet discover nonempty discover remove depend scheduling implement schedule last forest line never never edge discover uniformly grow exponentially construction execute uniformly defer theoretically equation bias number sum depend iteration process determine search choose trick dependence tractable ready degree choose independently vertex sample time expect degree normalize eq unfortunately interpret proportional neither match edge discover goal express calculating expect visit although numerically rewrite distribution fraction node expect degree describe bias equation insight nature graph traversal technique ff etc discover select degree therefore sampling simplify process traversal equivalent rw th select node procedure follow concentrated imply say x ff short biased number configuration process stop cover possible efficiently graph previous expect degree biased node potentially arbitrary try estimate node straightforward node rw proportional unbiased proportion simplify bias leave node densely purely easy inherently bias affected explanation likely wave surprisingly connect walk rw distribution fix graph regardless topological recall analysis approximation network property technique graph evaluate approach range life fully internet topology graph average q real node degree well difference visible topology fully capture rw simple alternative coincide rw systematically degree node description matter email large european facebook facebook wikipedia network com web google c correct rw circle node line correct plain line rw average thick analytical line calculate bottom correct eq precisely degree distribution c k facebook fraction average fig facebook seed seed full degree facebook analytical guess explain life fully facebook implement facebook follow rw es node time collection insight collect describe facebook sampling observe k consist relatively indeed start finally short compare facebook reason drop yield almost identical methodology collect describe entire fig facebook underlie connect nevertheless k approximate degree collect report report agreement subject bias relatively facebook rw bias correction apply particular life internet estimator fortunately arbitrary far parameter interestingly unfortunately variance section incomplete traversal start follow uniquely show appropriately requirement account calculate mean whose value order ball size around sampling technique obtain various topology sample extend trivial sample stage half feasible radius try approach k appropriate calculate half improve arbitrary topology feasibility sample translate difference summarize arbitrary estimator huge life topology concentrate precision trade terminology arbitrary whereas inaccurate correction rmse topology email topology facebook topology web google arbitrary topology average seed recommend rw arbitrary topology assume practically rw contrast rw diameter short case attractive plausible carefully affect example drop grow good restrict community component collect full facebook completely www domain recommend introduce metric particular bias random walk monotonically analysis base procedure bias technique broad ready implementation calculate diameter acknowledgment discussion unbiased facebook proposition corollary widely large internet topology advantage random walk plausible observe incomplete biased observe degree traversal technique forest bias procedure capture life base internet topology evaluate demonstrate unbiased internet topology community focus internet topology ip connectivity web online network restriction entire graph impossible topology facebook likely impractical collect representative sample particularly interested naturally www operation category walk category include classic walk rw walk web graph walk bias rw collect rather topology c sample walk rw traversal technique calculate average rw bias rw increase complete sampling traversal technique lead shape degree calculate category node visit completion graph vary visit include search ff sampling popular widely internet topology www use popularity plausible path coefficient walk course seem lattice small lattice unfortunately fail high node confirm facebook degree topological despite popularity bias relatively formal challenging sample node mathematically characteristic main contribution function node describe second propose bias input collect cover arbitrary common graph technique perform broad facebook implementation finding propose correction arbitrary although characterize scope life topology restrict attention self unweighted social link dynamically interaction graph scope bias correction procedure design analytical diameter future outline follow work traversal study briefly paper evaluate introduce correction practical sampling recommendation section explore et social many et interaction facebook constrain facebook facebook rw bias observe facebook motivated field social sampling
view imply laboratory major source information wikipedia serve role may subtle behavioral amount page cc score account people try potential idea easily change behavior become behavior status stable suggest work well community become nearly subject aggregate create strong biased wikipedia formulate form persistent attention power page user wikipedia ill attention wikipedia systematic quantitative investigation nevertheless suggest serious publication medium camera accuracy middle east report ask keep become anti member goal significant wikipedia page prominent retrieve far camera wikipedia may influential source break wikipedia wikipedia even retrieve addition difficulty belief even subtle outcome enforce wikipedia neutral viewpoint viewpoint leave viewpoint vice versa nothing page worth sensitive small wikipedia page page mathematic generate page deal quantitative evidence behavior paper present examine behavior page work assign page identify article fraction minor page attention focus page score particularly assess behavior page especially page broadly across article party past clustered cc score account similarity page user cc particularly focus user act interest behavioral interaction heavy equally validate analyze simple score also tail public topic large cc become status status cc request find focus become population fail well sense score pre low indicate fail actually significantly wikipedia introduce behavioral whether wikipedia historical determine use investigation behavioral rely beyond wikipedia discover focus behavioral status wikipedia behave change upon suggest wikipedia scientific far identify difficult incorporate deal user focus extremely user unlikely automate detection wikipedia candidate choose stand go request user stand late user finally article begin discuss describe cluster page wikipedia article page article unless proportion article fraction minor page page anonymous quantity mean article score transform produce page one manual user page page expect page remove page measure hypothesis concentrate article go able focus entirely broadly number party rest across interesting far interesting concentration could parameter change correct topic concentration article user cluster page page page page page category similar incoming link user page base divide cardinality intersection equal category consider page entire like concentrated average sigmoid condition prevent final reason measure measuring relate ranking extent concentrate tool raw cluster score page page page wise page say fraction broadly wikipedia cc quantity compute wide email message processing similarity evaluate validity provide measure wikipedia goal gain wikipedia score use amongst camera people wish point wikipedia try look cc contribute behavioral metric quantify use perhaps modification behavior deep second wikipedia evolve along
compose choose cause aspect classical cubic order smooth curve able take region monitoring activity make difficult stationarity trend first smoothed curve covariate degree spatially evaluate variability structure input theoretical fit case functional demonstrate characteristic h paris p des j spatially functional technical www co file pdf r krige spatial publication h random ed rand evaluation di spatio york spatially functional environmental di universit functional em pt environmental spatially correlate analyze spatial find group similar center optimize good representative propose evaluate study simulated feature environmental environmental landscape science daily environmental phenomenon area explanatory essentially analysis perform functional contribution tool datum focus spatially knowledge hierarchical group dissimilarity functional two alternative univariate functional datum characteristic approach kind define alternatively suitable al method spatially dependent achieve krige spatio prototype cluster minimize spatial variability krige curve get curve minima local krige propose krige base definition site representative several attempt discover linear regression two spatially organization relation residual instead dispersion unknown decompose semi large scale spatial variability although handle densely domain rigorous present difficulty belong environmental area spatially prototype optimize group function variability issue procedure environmental application stationary model accurately paper organize spatial illustrate real spatially measurement observation dimensional et distinguished marked point focus functional site spatially dr compact tt site nh nh small empirical involve simplify expand coefficient function functional obtain consider tt tt dt tt gram identity spline basis integration distance due empirical ordinary ol weight validity spherical domain change within sensor sensor potential error describe spatial variability component center lag estimate nh nh average center estimation center express manner functional clustering find characteristic find representative name fitting formally find partition cluster set cluster follow cluster representation measure prototype represent heterogeneity dissimilarity goodness perform allow element concern tt ts spatially locate partition spatial optimize center center dependence curve site lag evaluate membership mention random reach curve ordinary function compute allocate variability consider location pair separate spatial distance functional variability partition allocation rule tendency spatially correlate especially consistency allocation criterion criterion verify base euclidean cluster prototype optimize study first spatio functional random tt ts scope test performance procedure detect spatio structure largely separable purely purely temporal apart schema control temporal scale parameter six make locate field belong three cluster include spatially
cluster exceed proof proposition modify fashion word cluster go exponentially consistency see surely decision incoming signal precision determine value level determination error nan hypothesis accord plain simulate critical region triangular lattice consider level table critical c region boundary region simulate site display h cc low fact ii p p denote configuration associate detect cluster determine nan level conservative implementation picture depth thresholded signal find cluster detect reject statement terminate linear standard distribute give signal correct site given help algorithm threshold follow way perform repeat reject advance make uncertainty finite simulated cluster correspond optimal separate visible noise however signal may infer type sense threshold author like thank discussion appendix explain asymptotic nx let tend infinity definition phone method noisy detect shape presence boundary shape impose weak object interior algorithm linear formalism explore important theory addition behavior finite noisy digital pixel reconstruction quick mathematical look ask diverse automate detection picture run intensive image reconstruction picture majority skip start impose permit detection nonsmooth object especially one highly usually field automate material regular shape advantageous image require reconstruction analysis performance drawback subject wavelet paper nonparametric hypothesis completely necessarily smooth continuous nonparametric term exponentially pixel algorithm stop stop paper noisy focus image serious limitation affect object object detect picture colour boundary differ colour moreover applicable simple rescaling colour organize detail statistical power small subsection critical size present devoted proof auxiliary theory consist connect site site neighbor site bernoulli mild graph qualitative behaviour phase connect finite qualitative size connect site site precise size order infer intuitively regime quite sharp locate critical deal mostly discussion triangular sketch emphasize choice lattice discretize decide six denote think pixel discretize indicate denote indicator subset give variable noise noise signal signal signal paper detection statement hypothesis alternative statement thresholded terminology site thresholded site configuration complement choose critical lattice paper hope cause statement intuitive picture reasonably large equal example triangular subsequent mild condition arrange point approach state make qualitatively formation accord description immediate arise threshold discretized picture say site question super threshold determine detection however choose lattice properly lattice universal satisfying concern start condition entire remainder degenerate critical degeneracy simple degeneracy always probabilitie signal time site white obvious reason pixel site give address favorable black one support inside c free crucial observation let symmetric white first degeneracy consideration assume except degeneracy symmetry round infinitely triangular lattice complex additive root unity eq thus please hold triangular rely assumption logarithmic change quite critical triangular particularly locate site triangular lattice cause packing unit discretization choose ask obtain triangular lattice describe bound type ii mild shape completeness refer attempt triangular lattice lattice monotonicity lattice resolution right statement result please mean make necessarily cluster setup lattice site contain detect subset equation collection inconsistent want suitably depend connect weak contain site ng type contain square type consistent estimator estimator large issue detect correctly noise high fact precise explain size large begin infinite lattice depend contain triangular lattice consequence mean know define introduce ordering say analogously inequality read follow inequality decrease estimate cause lattice precise among configuration mark black depend distribution
flow diagram steady reach extraction pca respective euclidean histogram histogram obtain respect dynamical structure estimate observe orientation eigenvalue provide reference eigenvector adopt subsequent e rich though focus regular much attention drive integrate unfold network model characterize relationship respective dynamic remarkable application methodology dimensional pca integrate unfold average integrate realization uniformly steady axis observe frequency colored degree result along example respective signal interestingly time term pca signal maximum frequency fire instant move cycle along frequency cycle harmonic dynamic weak coupling depict projection also color illustrate significance value simulation reference integrate dynamic histogram log integrate gray fit log distribution significance level confidence specific illustrate methodology strongly version complementary figure histogram histogram figure nan couple simulation strongly couple strongly understand versus dynamic sis map topological separate shape histogram measurement two group signal eigenvalue centrality three measurement tend interior shape network da grateful financial support support grateful author thank cluster work author box reading focus characterize overall average account distinct properly represent identify aspect influence work integrate sis identification highly dynamic sense present network way compose interact effectively map dynamical extract typically quantify topology geometry g angle focus try structural provide real world system grow devoted dynamic global investigation focus aim address comprehensive systematic step principal pca important dynamic unfold probe structural organized box henceforth consecutive order characterize completely uncorrelated optimally align along direction variation principal axis focus attention dynamical relevant aspect signal influence aspect measurement index neighbor markovian assume therefore depend three function topology signal network simulation dynamical feature estimate network node node introduce define node significantly dynamical understand structural dynamical predefine density resort system namely sis box steady enyi er investigate neuron integrate fire node record pca project onto projection color remarkably fig average entropy pca projection supplementary main densely dense complex observe peak low suggest dynamic order sis use investigate node figure f pca sis map color respective plane result simple projection see dynamic yield signal tend supplementary illustrate significance nan condition integrate density small influence f k greatly affect integrate sis remarkably vary result sis dynamic mostly affect strong coupling dynamic result network identify methodology stronger verify coupling obtain entropy explain
generalization ds offer powerful aggregate uncertainty interval structure vice versa discretization propagate propagation build expansion introduce intuition response extract homogeneous represent generalize scheme characterize beta compact polynomial mathematically attractive separate polynomial orthogonal polynomial particularly characterize uncertainty dynamical represent ordinary equation uncertain solve polynomial transform bernstein range bernstein response optimality interval demonstrate interval nonlinear function algebraic challenge investigate ds interval conclusion future discuss primitive evidence represent give set understand evidence evidence member define interval structure world quantify amount amount complement measure argument frame combination aggregation mass due derive belief aggregated formula mix reliability source xx represent cumulative box induce box uniquely structure exist body n cumulative plausibility similarly complementary plausibility thus upper cumulative plot h evidence b assignment find propagation solve develop conservative evidence automatically evidence problem propose bernstein polynomial ref arithmetic inaccurate present index transformation power bernstein th bernstein bi polynomial ref box transform intermediate operation power give bernstein bernstein bernstein expansion range form estimate experimentally ref bernstein small univariate tight choose minimum coefficient sub efficient range computation vertex get structure interval interval element body way response prove concept select algebraic challenge literature use ref concern correspond ht obtained aggregate source use ds basic assignment polynomial expansion total great numerically quadrature bernstein provide genetic ga box bold indicate small large upper genetic provide ds table wide box represent narrow box bound return line interval ht fig interval properly include sum properly interval intersect region box ref give along region reason due large low interval quantification polynomial random support way find basis exploit map transform expansion bernstein form dependency accuracy propagation proposed investigate et propagate obtain basis build bound function polynomial expansion intuition expansion discard information provide randomness range bernstein form property bernstein polynomial propagate nonlinear algebraic
c rectangle pi north south north document description north pi edge pi w direct model dirichlet allocation identical construct generative vector dirichlet value mean potentially evaluate deviation proportion word topic everything part correspond lda part challenging pass connection share everything include estimate topic use analogous cite accumulate gaussian likelihood connection main introduction derive broadly speak inference collapse form belief benefit opt lda factorize entail loose slow integrated since form topic integrate per adaptation brevity construct given assign dirichlet q divide dirichlet dirichlet distribution approximately independent approximate graph gaussian latent product independent py h mean marginal posterior gaussian q writing message inference contain particular without change generalise hyperparameter noise likelihood evidence amount pf f separate evaluation less evidence numerically stable optimisation straightforward algebra identity ht leave softmax basis approximation correspond laplace mode softmax simplex basis link part inference connect belief task amount uncertain form probabilistic belief show softmax jacobian proportional real number parametric arbitrary ensure restrict basis unimodal mode fall aspect good laplace numerical convenience q hessian kronecker stem dirichlet derivation identify mean gain analytically introduce fit capture dirichlet ht wider unsupervised learn topic gaussian process dimensional generative learn mapping document define topic generate usually assume kernel distribution discrete sample dirichlet topic perform conceptually cite model topic consist state union address joint annotate identity combine feature discrete one identity fall power drift topic represent radial function evenly period topic rational hilbert identity time additional author rational square kernel assign nonzero scale exponential finite function expressive section optimisation difficult consequence expressive interesting american interior development red topic fast development american external linear radial author either predict date office figure see optimisation every visible negligible effect intuition hyper optimisation two dataset require numerical invert external external external iteration kernel document cause optimisation directly topic point subsequently dataset show considerable optimisation relatively elegant topic experiment document take wikipedia list document set element short link document link assign arguably present nonparametric kind allocation conditionally efficiently laplace cubic corpus offer sized corpus analytic elegant side effect laplace marginally paper replace estimate bayesian uncertainty infer maximum evidence acknowledgement like thank david david circle minimum text width pt sep fill fill text draw black fill black text white white fill size sep size latent dirichlet allocation discrete belief document mixture weight document hierarchical social document challenge efficient cast around laplace approximation transform type variable topic feature work latent allocation lda comprise collection collection discrete distribution document constitute popular corpus treat bag ignore collection document word specific real exist product communication corpus amount author augment additional popularity vary west poor old lda extension sequential document description function space link softmax assume stay change round link
process demand importantly target basis em analytically multiple anneal idea converge step counterpart chance maxima copy draw monte therefore apply multivariate probit option draw hand may lie ensure identification meet relation probit expand simulate suggest multivariate probit acknowledgement thank associate greatly manuscript thank implement numerical six probit give normal constrain domain inverting integral respect independently normalise simplify domain lead write smc sampler smc random walk metropolis p weighted mh smc loop incremental normalise weight degeneracy ess covariance mh update current particle available multivariate student sign n algorithm cycle draw effectively mean ensure proportion preserve truncation sure particle initialization proceed update metropolis run particular automatically implementation resample adaptive ess smc key smc probit possibly weight eq core repeat noise loop step cycle conditional p implement current smc move likelihood sum probability smc noisy zero keep expand sum towards normal sum fourth error quickly variance correct pt probit model appeal dependence multidimensional binary modelling matrix latent multivariate probit regression carlo avoid intensive multivariate normal gibbs sampler normal proceed stage draw truncate far towards em nature exploit em update iteration computational conditional step iterative em probit embed algorithm equivalent method validate widely high simulate approach correct treat probit monte carlo multivariate probit originally bivariate particularly useful dependence typically high dimension likelihood analysis previously augmentation nature expectation iterative classical optimisation iteration ideally implement analytically version truncate chain monte gibbs particular art different option sequential weighted particle mean weight though originally introduce dynamical kalman smc static art multivariate normal gradually truncation normal tail student particle require truncate normal gibbs main difficulty standard optimisation large simple extension guarantee identifiability constrain normally difficult identifiability issue intrinsic regard expansion approach em fact probit make idea smc version parameter burden validate previous high example monte sampler produce sequence obtain refer particle interest static scenario purpose element control degeneracy resample perform effective fall move next normalised formula incremental weight backward kernel respect suggest convenient backward walk metropolis hasting give iteration moving proposal scale practice mh sampler arise wrong art extensively investigate mcmc literature original adaptively monitor acceptance target metropolis adjust gold realistic situation purpose diversity sensible avoid move degeneracy lead smc metropolis type quantity stepsize logarithm computation maximum likelihood posteriori problem rely let datum us estimate eventually correspond term mm certainly decrease comprise respect e replace sample augment suggest iteration analytically difficulty multi turn ideally wish time probit formulation vector component associate observation probit parameter density random identical setting care identifiability probit variable discretization multivariate gaussian variable think obtain unobserved vector z I function probit component elaborate inference matrix lead positive ensure positivity eigenvalue probit trace ignore correspond completeness derivation provide expression provide trace convenient intractable gaussian density expectation weight j normalised sample provide smc truncate multivariate particle obtain sequential manner split obtain zero condition value approximation current trace write condition satisfie already update step remove intensive approximation generalised lead impose step probit model phenomenon link invariant change decompose perform space large conversely ensure identifiability could high art wu principle local zero decompose practice likelihood though space subtle proxy likelihood unconstraine would conditioning change arise exactly create unconstrained preserve inequality increase lead though necessarily conjecture agreement example seek incorporate expand examine neighbourhood maxima expand converge standard suggesting give significant demanding appeal probit rely smc sampler evolve take iterative discuss identifiability consideration probit almost computational walk metropolis towards tail exponentially method involve normal inefficient low rate acceptance tail distribution respectively replace denominator correspondingly actually acceptance allow degree grow variate unconstraine student distribution truncate intermediate target smc nc probability c ultimately probit reach student freedom enough truncate distinguished perform could truncation since reason student aid region overview truncate gradually oppose increasingly truncate distribution student relative conceptual adaptation framework notice section artificial address functional form section move intermediate target priori determine dynamically local difficulty adaptation evolve parameter quantity solve q efficiency inspire page stochastic dynamical design keep threshold update observe introduce scale maximum correction term target end ideally smc sampler fraction slightly define resample resample threshold number reach reduce small also volatility binary smc algorithm advantage sampling truncate multivariate normal proportion probability reach behave like use target dimension perform one single smc target cutoff interval fig smc form give diagonal element reduce include rescaled element p inside correspondingly log essentially tie apparent though independent inside constrained keeping eq expand impossible likelihood preferred remove ambiguity replace replacement effectively invariance rescale whose element root appear result higher see unconstraine preferable define nature invariance obtain identical identity obtain depend fix diagonal vanish find solve repeat identically allow involve invert multiply relative optimisation grow fast diagonal dimension package parameter identity start method fast time improvement vanish along translate likewise solution turn depend constrain unconstraine associate drop scale matrix element invariance mean transformation shorthand form ex ik p c ik identifiability invariance identifiability give crucial constraint implicitly attempt formulation probit model clarity table probit model section vector coefficient diagonal response model may keep special case share single allow slight row give reduce yield invariance write compact alternatively covariate may share dataset covariate response represent compact row correspond response coefficient fix probit model fix account impose modelling reduce invariant seem six impose unnecessary identifiability vector sub transformation positive constraint avoid sub vector multiply factor unchanged rescale direction fix consideration slight change trace need constrain impose ensure model formulae smc sampling truncate multivariate normal gibbs chain draw truncate variable efficiently reject efficiency near gibbs correct region slow converge correlation extreme smc move truncation sample become preferable truncation truncation occur four variable correspond dimension moment gibbs sampler statistic distance rejection sampler burn sampler plot inter fig rapidly smc sampler moderate correlation smc notably well large smc gibbs pass obtain sampler provide discard roughly correspondingly growth still smc rather value course computational time efficiently sampler sample error scale treat widely longitudinal study air probit bayesian geometrically multivariate oppose probability likelihood readily produce sample far expectation six model status age consider analysis refer child child value indicate present covariate namely age child term probit j cumulative distribution standard normal example space sufficient covariance correlation constrain em invariance stochastic stepsize gradually importance innovation learn weighted reduction long within neighbourhood monotonicity convexity practical cause issue estimate value multiply correct likelihood obtain increase particle tune acceptance root likelihood run comparable slightly correction discuss run close likelihood region evaluate feasible accuracy smc find notice seem close close exact use sample optimisation sampling optimisation involve evaluation require form numerical routine page variable efficiency indeed exploit significant complete conditional single complete iteration benefit smc method particle update necessarily column smc estimate much reduce computational updating particle approximation fast draw latter factor keep number particle strictly six correlation reason dimensional impose invariance fix
expense burden quantity carlo repeatedly entire appear bootstrap encounter massive quite demanding costly modern trend seem ideally suit straightforwardly leverage processor bootstrap parallel massive cost independent processor compute node operate even resource largely devote reduce bootstrap complexity eliminate need comparable subsample book closely introduce bootstrap appear repeat consideration drawback sensitive subsample subsample variability rescale rescale knowledge drive optimal great eliminate gain reduce cost conjunction however series expansion estimator automatically execution bootstrap motivate assess scalable result bootstrappe original bootstrap subsample little subset individual weight form sampling subset replacement computational rescale overall favorable profile quantity much small dataset suit modern generic favorable property correctness show bootstrap formalize statistical discuss subsequently share bootstrap study compare bootstrap subsampling discuss distribute present superior massive hyperparameter finally bootstrap real series sample corresponding indicate compute random underlie estimator computation estimator assessment notation place standard practice base knowledge estimator directly addition operate normalize example compute distribution statistic obtain direct dependence nonetheless involve allow standard deviation development generalize straightforwardly exposition notation drive compute generally amenable straightforward repeatedly subsample subsampling book subsample subsample form approximate correction prior knowledge increase bootstrappe give uniformly disjoint q analytically general compute manner bootstrap repeatedly compute estimate b substantial benefit nominal particular generate suffice vector point count estimator respect storage space hold use estimator result resample tb disjoint subset predefine b n avoid problematic repeat computation contrast contain contain mb dataset tb gb subsample gb substantially total multiple fairly suffice small size amenable distribution allow distribute parallel enable additional simultaneous compute partition large cluster thus quite costly computational computation quite indeed straightforwardly permit multiple process naturally leverage intra parallelism subsample thus bootstrap assess natural single possibility prohibitive full assess point compute returning inferior average discuss empirically applicability favorable bootstrap addition bootstrap subsampling choice subset size statistical correctness identical bootstrap bootstrap consistent degenerate limit study asymptotic procedure center additionally account carlo approximations standard estimate generally take hadamard subspace measurable pf continuous space metric practically hadamard regularity condition satisfy compute also generalization assumption work g beyond consistency characterize high great devoted showing order many case converge driven limiting follow degree correctness choose importantly value bootstrap via asymptotic expansion power expansion compute expansion term expansion expansion fisher expansion full expansion well function mean version define obtain moment p nb enjoy correctness bootstrap natural power estimate represent moment like bootstrap sufficient additionally k p probability decrease applie highlight substantially slowly disjoint datum q enjoy correctness consideration divide standard hold generally empirically characteristic statistical exist experiment datum correctness property bootstrap subsampling setting parameter vector model procedure compute marginal parameter compute boundary confidence wise define averaging assessment task ground generating realization dataset compute collection fidelity realization assessment parallelization record produce bootstrap subsample deviation component confidence estimate confidence width repeat process realization trajectory quality relevant variance interval control quality assessment procedure probability large execute maintain notation subsampling regression I I draw j skewness vary quadratic vector encourage numerical dimension estimate parameter approximately bootstrap middle row ss trajectory linear generate right linear succeed converge middle fail bottom perform bootstrap subsampling relative large suffice implicit axis range large quadratic beyond order estimator quadratic regression via method penalty encourage numerical confidence parameter linear approximately row bootstrap trajectory logistic regression middle logistic show result classification varied appear set comparable bootstrap converge higher still converge fast robust fail even superior qualitatively bootstrap experiment implicit axis require range relative subsampling performance bad legend leave plot logistic generating examine converge relative explore relative vary value procedure report achieve many output increase consider accordance indeed grow achieve relative substantially lower quickly precede intend investigate statistical insight see figure processor require time less bootstrap ability via platform exceed storage capability individual use parallel architecture assessment tie ability effectively resource exposition due bootstrap partition node least potentially feasible bootstrap incur associate overhead repeatedly compute system website disk memory physical size constraint exceed memory incur extreme cost read disk disk magnitude read disk read acceptable slowly compute prohibitive bootstrap require permit simultaneously implementation enable gain small process subsequently intra parallelism compute subsample full read disk store disk potentially store implementation benefit advantage available parallelism parallelism compute multiple cost super subsampling worth subset relatively disjoint access distribute large platform modification accommodate partition allow truth substantially realization dataset numerically single base resource require storage prevent degeneracy large logistic bfgs large implement process simultaneously processor compute node simply computation upon partition utilize resample avoid though little qualitative outcome amazon implement either disk available gb leave show disk full worker store disk disk quite worker gb memory compute core cache repeat access bootstrap improve disk exist bootstrap process result computation work bootstrap largely address computational issue generally process reduce subsample amount dependence influence provide achieve seek value study value confidence leave plot insight influence set datum particular small low select ci min hyperparameter linear vs parallelization adaptive trajectory mark trajectory hyperparameter interval ci component wise case expect hard large avoid hyperparameter validate inner whereby subsample continue note form series well behave many case unknown suffice increase use processing increase value adaptively independently batch parallelism output tb z adaptive hyperparameter representative set early parallelization selection illustrate hyperparameter compute converge low unnecessary computation degradation though priori interpretable yield generation apply conjunction different easier help hyperparameter selection value efficient desirable could bootstrap subject future fairly investigation seem choice real absence correctness assessment output various procedure knowledge dimension yield figure bootstrap uci connect uci study use bootstrap change procedure qualitatively six slice range see selection bootstrap legend move stationary liu bootstrap setting variant bootstrap identical context briefly procedure stationary suitable assess series extend subsample mechanism generate particular suffice select length generate bootstrap subsample length hyperparameter bootstrap select subsample follow series length probability next beginning reach subsample subsample execute describe bootstrap introduce task rescale approximately stationary bootstrap stationary improve performance characteristic ccc method standard stationary series rescale aggregated trial rescale approximately provide powerful alternative automatic quality suit modern compute architecture share property consistency generic applicability bootstrap computational demonstrate computing platform subsampling analytical correction enhance adaptively datum open extension remain though hyperparameter subsampling resample validate beneficial adaptively select may reduce note
data constructive unsupervised regression dimensional manifold reverse formulation space optimally reconstruct fit manifold problem point problem hard unknown review knn regression topology learn linear kernel project hilbert locally principal unsupervised regression start algorithmic parametric e denote unsupervised kernel leave loo cv n argue unsupervised regression memory accelerate introduction neighbor output consist pair knn regression compute mean knn locality neighborhood label label close pattern sample knn g section iterative regression datum matrix space seek optimally minimize define optimally manifold unsupervised manifold contain manifold norm word consist reconstruction knn example extension knn move effect still near neighbor extension penalize avoid limitation knn fix far knn absolute position position perspective neighborhood solve high dimension combinatorial iterative strategy iterative assign propose variant test intermediate look embed distance choose remove repeat element low comparison position test near datum neighbor overall constant practice step computable compare neighbor similar color b embed point part topological sorting although observe local optima variant variant show beginning b embed begin color correspond embed see assignment experimentally test problem sake pixel handwritten h digits digits embed digit assign digits digit latent achieve digits show three setting lowest highlight exception achieve exception role fit dimensionality reduction iterative strategy high latent speedup restrict solution reduction heuristic turn experimental analysis achieve slightly fast constant work analysis optima strategy extend global extend latent topology dimensionality simple stochastic strategy employ randomly p ii department von universit scientific high
appropriate prior estimate model commonly prefer solution time family criteria bic widely datum give measure function constant investigate consideration principled thesis publication finding experiment investigate external internal validation alone sensitivity initialization parameterization variation internal bootstrap validation approach internal validation compare real external investigate independent exploratory throughput thousand generate hypothesis intervention laboratory highlight assess uncertainty currently widely genome constitute thesis overview technology section preprocesse measurement crucial chapter thesis utilize genomic database microarray gene chapter organize overview various throughput microarray study external sequence database model evidence across microarray quantitative probe throughput summarize novel throughput come level biological technical add variance natural biological individual nucleotide add technical source measurement rna experiment platform laboratory significant array come activity give probe differential gene expression highly contaminate add variation number affect probe publication elsewhere portion microarray moreover although design uniquely intend nucleotide affinity content likelihood cross target alternative cause position effect contamination probe source poorly understand importance challenge technical process analytical ultimately publication tool probe relative probe level contamination probe probe level detect probabilistic differential target indicate line show estimate source genomic alignment probe rich associate publication datum help reduce improve preprocesse statistical raw throughput preprocessing array thesis microarray array step microarray quality correction microarray quality array remarkable remove degradation array microarray array vary smoothly array arise technical background detect array probe background correction array help differ significantly array average preprocesse background correction global intensity observe gaussian b correct bs background comparable array quantile force array quantile normalization assume concentration population human systematic array final individual probe level single estimate preprocesse difference characteristic systematic probe probe significantly affect widely preprocesse utilize probe probe probe preprocesse probe level effect account improve quality microarray probe expression probe method probe algorithm probe specific characterize probe bind systematic difference capture probe probe help level summary probe outline I probe level probe bind affinity noise probe probe j model need absolute gene drawback order identifiable probe average yield well show accurate extension probe utilize data microarray probe probe preprocessing array design expression gene expression investigate expression level difference experimental expression level method probe perform probe affinity probe potentially suboptimal publication demonstrate calculate already probe improve need formally publication probe publication summarize probe gene specific outperform widely probe preprocesse publication probe specific effect probe preprocessing utilize external microarray design date sequence available rapidly evolve genomic sequence reveal probe annotation body knowledge grow recent include publication detect uniquely match remarkable microarray match database publication exact vary utilize thesis microarray analysis background database microarray collection probe verification increasingly preprocesse microarray probe genomic database construct interpretation array genomic annotation accuracy cross microarray publication elsewhere publication combine probe novel compare combine microarray huge microarray available come microarray microarray prove source probe sequence probe matching array technical verification sequence database confirm confirm publication publication utilize genomic probe remove genomic external database contamination publication probe base datum collection reveal characterize source probe contamination gene average theoretically justify prior analysis verify match gene remarkable portion sequence microarray pearson correlation array biological show probe match array comparison good probe array advantage probe verification array probe collection compare I comparison technical replicate match set measure pearson array publish publication investigate probe genomic publication physical reveal alternatively sequence collection microarray public valuable study publication extract probe level microarray collection preprocesse assumption probe datum collection independently external level preprocessing expression publication explicit formulation quantify incorporation concern probe reliability estimate probe reliability differential expression expression external incorporating microarray publication summary suffer probe affinity contrast probe expression remove advantageous gene level preprocesse probe investigate probe publication assign probe probe guide probe consider probabilistic introduce publication ultimately expression vary collection ground assess signal assume characterize target normally probe quantify probe reliability probe array probe probe distribute probe reliability probe parameter probe affinity probe level differential expression avoid probe key probe probe preprocesse affinity user array array array probe j probe array write noisy observation true probe probe affinity publication partly explain level improve analysis probabilistic preprocessing method aim quantify effect interpretation probe reliability probe reliability differential expression expression probe specific variance simultaneously prior select control array latent obtain search relate gene connect correlate across potential global interact gene gene sensitive role gene predefine decompose predefine pathway module rely predefine classification gene drive gene directly prior genetic need refine suboptimal enable discovery demonstrate measure differentially express measure activity characterize change publication address de tool increase cost false finding gene discovery function prediction visualization widely cluster approach hierarchical patient novel activation association poorly gene self visualize coarse collection gene problematic whole genome study reveal sample small coherent subspace particular interesting detect gene module database subset coherent define relate tool condition compact summary aid interpretation collection enable discovery expression signature central global signature describe expression establish signature however establish signature typically classification particular signature association underlie process publication enhance signature ignore use feature technique connection gene typically handle hundred insufficient throughput use guide interaction gene coherent cluster model condition include orient additional include orient utilize information discovery module instance publication introduce complementary concern relation gene genomic applicability limit network gene construct gene genomic collection collection enable discovery genetic probabilistic dirichlet computation tool characterize expression set molecular interaction network thousand protein small molecular cell reflect gene response general drive proxy network distinct functional activation information gene global context pattern publication introduce general provide model interaction response subset cell biological focus analysis introduce publication first publication base gene coherent response coherent publication state detect find subgraph network gene identify step suboptimal condition response publication introduce purpose algorithm criterion publication search coherent interact gene interaction guide classification involve amount protein gene context state reflect unique expression signature expression level state gene precise give state state observation r include specific measurement state index association treat gene model leaving justified considerable gain response primary gene cascade expect help distinguish partially cost triple fluctuation bayes interpret soft membership hard deterministic distinct agglomerative use interact merged gene singleton merge neighbor joint gene value gene likelihood comparison give merging size possibility constant dimensionality effect criterion increase optimal g cg pair direct pair iteration continue merge modeling yield genome part support power agglomerative find step lead high merged mixture note globally response often publication activation pathway base database pathway provide package gene expression publication outperform unsupervise use search method figure publication response coherent suggest potential functional confirm detect p highlight exhibit term response alternative highlight connectivity publication response co gene group individual global activation use context particular condition reveal help formulate novel context specialize cell biological induce co gene produce specific datum condition specific proxy identify distinct network potentially functional role publication tool genome scale predefine classification bring miss drive readily applicable pathway protein available therefore scope rely implementation implication require result highlight response provide activation human disease j chapter present thesis genomic measurement organization understand organization genome ultimately various level genome system biology key property integration evidence across discover mechanism interaction statistical integration heterogeneous genomic variety technical challenge approach accord investigate measurement limit genomic suitable become increasingly house public observation highlight novel modifications micro rna bind nature biological uncertainty prior biological subject issue need principle minimal knowledge model overview standard throughput integration connection develop category evidence accurate ii order guide primary source characterize dependency order discover new contribution chapter publication cluster category introduce exploratory tool dependency tool guide bayesian publication cluster dependency source applicable integration apply dependency activity evolutionary human study occur observation genome micro valuable resource investigate functional property layer genomic tool relate develop biology community sequence genome structural organization accumulation research challenge create picture genome volume genomic intervention element historical public challenge intensive associate complex system availability observation solve challenge combine power public exploratory material genomic concern underlying phenomenon robust facilitate provide theoretical framework thesis exploratory integrate evidence genomic mechanism investigate small contribution improve throughput ii activation iii integrate measurement genomic contribute challenge genomic develop understand biological genetic evolutionary variation thesis carry publicly access guarantee tool original thesis extension develop integration functional promise line future readily genome wide disease occur datum concern various genome become micro gene modification study layer organization contribution health fundamental challenge biology wide exploratory rich idea mm science sciences public university school technology university technology computer school technology sciences science p box series http mm thesis http www accordingly free display purpose assume figure separate http thesis school science integration exploratory advance throughput sharing genome encodes program functional biology organization thesis computational strategy cell computational extract genomic observation probabilistic multiple genomic thesis key preprocesse genomic database collection throughput exploratory cell activation pattern functional across information genomic interaction database derive help gene approach occur source mechanism evolution analysis layer genomic functional measurement source implementation facilitate far rt ta te py sis lt n min sis sis ep te ne mi ty si ty ep si sa I k te I ne ne mi ty I ne I en la work carry network centre adaptive centre laboratory science computer school science technology work computer science university year I also part institute information technology school engineering project research school bioinformatic support thesis work give truly field scientific work necessary essential learning like express thesis expert feedback research biology excellent traditionally distinct science biology particularly grateful daily laboratory institute year belong co extend former member mi I department provide valuable sharing publication datum impact thesis express contribution thank scientific life I demonstrate rational go science friend science want thank discussion grateful share life child evident remain understanding nature life I year share aspect grateful parent accept I I create tie thesis overview probe level array probe reliability array transaction biology bioinformatic network bioinformatics detection constraint international machine page machine conference machine cluster dependency transaction bioinformatics bioinformatic summary author contribution author author thesis effort key contribution author thesis summarize publication genome study genome author study external manuscript probe gene combine multiple microarray experiment gene analysis probe improve microarray author derive formulation probe level manuscript open implementation review computational biology introduce activity genome network provide tool collection genome design perform author publication introduce dependency cancer major role task design jointly author author carry open publication introduce principle integration dependency author design manuscript publication extensive consideration work sensitivity analysis comprehensive bioinformatic design comparative experiment technical list thesis symbol denote vector capital symbol symbol vector scalar trace mutual parameter vector parameter vector ac comparative cca complementary dna dp dirichlet algorithm ib bottleneck kullback posteriori ml maximum rna rna lead introduction life process new technique small object molecular dna lead pass sequence human dna publish researcher large volume concern accumulation genomic database accelerate research structural organization poorly understand characterize regard system paradigm biology transform genomic collection new bring challenge throughput mind phenomenon challenge relevant statistically uncertain volume observational make life study massive set thesis principled exploratory central functional rna genome time protein synthesis cell level gene order associate genomic available public combine heterogeneous background public relate statistical uncertainty well form picture genome starting point research poorly observation toward proceed particular question support refer visualize identify facilitate new otherwise poorly characterize adapt extract automate oriented require process may require wide thesis increase high throughput combine statistical database across microarray ii specific interaction normal body information genetic guide iii integrate measurement occur strategy recognize challenge obtain first strategy side genomic microarray collection probe microarray preprocesse genomic sequence use remove extended publication introduce principled framework probe probe combine probe microarray number contaminate probe level well poorly contamination unknown could control probe level principled incorporate prior introduce outperform alternative activity genome interaction publication contribution thesis search genomic interaction database detect finding activity biological third thesis human genomic novel datum detect dependency publication use association knowledge use constrain latent improve introduce tool open contribution thesis wide access tool science proportion embed traditional valuable public thesis organize chapter genomic exploratory paradigm chapter contribution thesis chapter reliability throughput microarray present wide dependency introduced investigate functional activity conclusion thesis summarize law program double feature life purely world year evolves maintain adapt environment external hierarchical organization replicate reproduce life mechanism molecular suggest common evolutionary genome encode program technology science view organization genome molecular biology investigate organization genetic thesis new investigation functional overview concept resource background molecular genome life carry genetic program cell carry copy genetic code genome genome cell constitute dna rna ordering carry property synthesis paradigm molecular biology organization life dna set gene activity consequence code code carry protein synthesis despite identification protein gene genome detailed protein key entity cell protein synthesis refer cell biological protein product synthesis double proximity gene gene dna sequence convert code code segment respectively form rna convert sequence universal genetic triplet form sequence constitute protein stage protein synthesis fold post protein ultimately key synthesis key synthesis translation pre modify produce carry translate universal code nucleotide triplet correspond organization material nucleotide pair double carry material locate http file rarely attribute ultimately activation change cell biological environment activity level synthesis major functional genome protein gene refer chemical structural modification dna instance bind modification pack dna around cell combinatorial modification access source protein synthesis bind element fashion post modification variant variety protein contribute diversity cell several mechanism degradation micro nucleotide sequence specific complementary base degradation modification degradation mechanism cycle protein protein biological interaction complex life functional organization genome rapidly genome overview human level resolution nucleotide attribute dna traditionally point call nucleotide snps protein dna increasingly recognize structural variation genome contribution variation structural variation organization nucleotide large variant genomic modification directly influence health publish genome pair gene code less human genome million year majority half genome highly sequence genome sequence element mobile dna genome recent dimensional dna remarkable human highlight evolutionary specie human protein synthesis gene contain gene law law copy small express thousand copy source microarray study discuss dimensionality size form challenge throughput microarray greatly exceed sample study sample leave considerable uncertainty analysis concern phenomenon different part overfitte multiple form considerable challenge automate thousand hypothesis concern finding characterize prediction dimensionality functional challenge challenge control remarkable technology development procedure thesis combine increase rigorous genomic individual remarkable extent contribution characterization population genomic mechanism disease discovery medical throughput disease mechanism genome disease take sensitivity signature reveal lead diagnostic disease cause change chemical genomic new response genomic genome disease genomic disease understand genomic change continuously micro rna locate genomic relate non dissimilarity p give mutual kl theoretic employ thesis inherently couple different euclidean potentially lead equally naturally invariant variable demand refer scale transform suitable far select stand technique problem inherently equally feature interpretable particularly important genome sample phenomenon advance dependency consider interact model centroid publication selection genomic decompose reveal publication maximally set central dependency biological goal exploratory analysis technique summarize facilitate hypothesis research poorly characterize knowledge analysis toward hypothesis test data procedure exploratory traditional hypothesis light exploratory particularly standard biology discovery differ target mutually exclusive generalize partition unseen approach without cluster discover mutually cluster popular computational biology comprehensive application beyond thesis expression profile cluster suggest formulate concern discover novel cancer visualization another exploratory compact summary projection principal optimize self cluster probabilistic popular set sample uncertainty account rigorous particularly sample infer employ learn mixture process alternative infinite relevant extent infer variable component tractable characterization generative model distributional relationship probabilistic interpretation pca observation model normally distribute latent variable parameter noise define subset typically task instance describe process irrelevant nuisance integration marginalization marginalization latent give marginalization find account yield speed marginalization marginalization goal thesis utilize publication discrete publication latent dependency occur observation specification distributional practical task parametric principled structure gaussian publication proportion example relation describe theoretically principled exploratory flexibility potentially overfitte less moreover interpret probabilistic stochastic process prior structure process prior space dirichlet process dp dirichlet partition ga ga pa chinese restaurant crp provide intuitive description dirichlet crp define cluster crp customer arrive restaurant infinite table choose subsequent customer table state restaurant table customer proportional customer table pi probability proportional table customer analogous cluster intuitive cluster potentially infinite ultimately base method number component datum potentially infinite let replace intuitively call stick break stochastically break process I stick truncate stick break stick assign prior proportion learn help increase observation use increase model certain cluster selection include comparison theoretic seek insufficient approach uncertainty uncertainty lead refer ignore element posterior characterize encode match particularly train considerable parameter informative sample converge provide robust uncertainty
partial specific include model somewhat contextual multi bandit provide round determine pick still action policy choose adversarial select choose adversary choice round regret simplicity focus horizon advance expectation sequence choose multi armed bandit get reward receive construct unbiased action jt jt jt action essentially information action member depend assume advance intuitively think viewpoint remainder split clique meta expert action present build exist deal round clique choose reveal unbiased reward clique exponentially action expert standard exp meta action combine expert provide appear appendix round forecaster regret stick clique clique correspond clique action reward action correspond attain empty reward action standard attain graph regime follows fact clique computational hope well bad specific class compute clique relatively easy disadvantage structure stand exponentially arm exploitation uniform action carefully exploration solve program name bind regret concern action information theorem j kt lk jt jt j jt jt jt well choice undirected graph discuss regret undirecte via compare thm partition attain change case computational involve require np create large course advance solve treat copy run armed construction meta action incur factor deal undirected choose versa set condition choose equal thm rely tight conjecture case note weak superior graph clique determined quantity compute trick graph provide term number link randomize every least proof appendix intuition independent armed bandit play low armed bandit result follow undirected match regret upper logarithmic direct difference rather huge briefly discuss concrete notice performance partition choose choose seem reveal everything current improve feedback essentially undirecte gap third learning probability fourth case devise inter center market split exponentially forecaster action forecaster pick receive observe reward update unbiased estimate reward actual reward completeness forecaster define convenience hold q taking sum series fix rearrange get pick forecaster meta action exp meta action incur meta single lemma first definition combinatorial unable occurrence third lemma examine particular denote independence number size large adjacent node hold prove allow convention contrary exist follow either repeat guarantee eventually arrive show kp original fix vary adjacent everywhere namely node increase discuss state lemma graph define adjacent include exist otherwise choose value adjacent large since value either lemma rather armed bandit e get definition take fact sum rearrange get expectation side slight use simplify get thm difference subsection identical upper invoke long counter opt weak small upper bind independence let set two expect use resort know lower bind armed bandit game action side pick action assign action show learn cumulative adversary armed bandit describe independent choose one armed bandit feed reward back neighboring feed back well node round neighborhood exploration fix choose stochastic matter feed obtain reward identical implement round achieve goal provide bind choose action outside whenever choose receive highest big increase round time possibly pure round exploration round regret exploration round expect expect regret overall q rearrange standard multi armed bandit select obtain plug maximal node exploration rather replace allow oppose corollary microsoft adversarial maker observe action get side encode graph link naturally decision maker armed bandit maker reward practical provable regret theoretic information basic setting framework assume action performance term reward iterate round total good action accumulate framework generating might full reward many world canonical ad display whether ad constraint lead choose set adversarial price provable factor factor bandit expert rather bandit receive theoretical reward study rich various reward formalize bandit set intuitively assume obtain reward reward action way round feedback direct action action sufficiently good reveal action well complete expert set scenario motivate web standard armed bandit ad two ad package ad display contrast ad run ad unlikely clique sort motivate sensor certain sensor cover may cover centralized controller receive model cover area sensor reward choose obtain sampling come select transmission noise channel adjacent frequency band result picture provide fundamentally superior guarantee theoretically
evaluate show situation aspect nontrivial exploration induce exploitation balance location uncertainty accumulate important insight usefulness reinforcement secondly system trajectory necessarily follow full assigning low good trajectory amount reach slice control control inner sep node white l td kalman filter achieve measure cart study arm compare average controller access kalman gaussian controller approximation td radial free assumption frequency influence nontrivial rarely reinforcement accumulate differ discount learner controller control balance necessarily curvature value well potentially compare exhaustive quantity exploration vary bayesian gaussian nontrivial reinforcement equation easy differential field raise reinforcement solution approximation functional note argument dynamic process inherently determine may loss incorrect replace even concern well uncertain probabilistic allow similarity work extent inference widely utility acknowledgment author express crucially improve pt black minimum black exploitation trade central reinforcement bayesian intractable statement space subject infinite dimensional differential equation give result helpful learning thing spend irrelevant incur unnecessary classic consider trade policy classic reinforcement ad hoc method thompson thesis know exploitation require trajectory along situation depth tree propose approximate ad hoc optimal parallel idea reinforcement area structural recent et al exploitation trade kalman reinforcement loss function restricted invariant system core reinforcement dimensional new approach paper result prior section description system yield amount control control exploitation control rely concept reader reinforcement control action transition reward optimal control attempt reinforcement description optimally control uncertain dd qx qx analytic concern dynamic seek simplify uncertain sample function note g uncertain acquire stochastically sample belief process k word draw infinite vector demonstrate inner point describe uncertainty process scale unit change location utility gain task horizon trajectory equation control law discount definition uncertain rather generation somewhat ad hoc simple assigning acquire dynamic continuous dynamic still system cart realistic exploitation representation reinforcement usually construct bellman sec analytically solve optimal phase belief function nontrivial swap integration controller actual reward access true controller system disadvantage average optima actual ignore integrate drop controller follow adopt greedy effect amount step look ahead imagine extension future analytic bellman explicit upon central affine optimal reinforcement order interpret sign comprise immediate drift first effect phase subspace effect augment former latter dominate govern physical cause control physical knowledge statement reinforcement objective exploration even equation remain nontrivial two although product look integral read remain solve least choice integral solve exponential nontrivial equation numerical answer arguably constitute value future trajectory differential equation raise hope analysis claim hard transition belief infinitely condition joint control form assumption form clear way infer without address specific analogous use break section k approximate nonlinear radial restrictive widely reinforcement kernel convenient integral serve
methodology situation anomaly psd integrate drop allow euclidean string describe class contaminated future investigate asymptotic trade impact different begin first third distinct linearly positive definite subtract multiply multiply equation turn g compact continuous continuous space calculate differential case come lemma solve iw first fact decrease convex imply condition inequality strict inequality strictly happen happen distinct strict monotone define surrogate function define argument q next monotonically lie limit come inequality g sequence indice contradiction give fs generalize however thing care deal summation suppose give probability q exchange integral valid h g strongly well therefore come fact continuity increment still apply need find exchange valid dominate convergence lipschitz g h g x combine df f df simplify substitute x x linearly compare n diag eq found therefore imply sg triangle inequality open ball center radius f unique j sg ann mi usa email nonparametric robustness contamination kernel kde classical interpret kde radial associate reproduce sensitive robust iteratively least sufficient give global minimizer density trick kde multivariate relatively little improve kde situation method contamination training international conference machine learn follow contamination eq nominal contamination assumption nominal distribution many nominal density spatially relative although state condition precisely capture intuition behind motivate application anomaly imagine multi collect example volume traffic along instant measurement collect nominal detector unfortunately often anomaly vs intensive estimate nominal distribution desirable propose robustness density estimator kde radial definite psd hilbert rkhs yield robust density weight necessary first weighted kde influence exact formula sensitive contamination outlier third conduct several dataset anomaly motivation kernel know without kde tends low motivated dependent bandwidth adapt dense aim aforementioned outlier estimation psd refined treatment adopt hilbert embedding approach attempt match design contaminate mind estimator square problem rkhs estimation develop song et optimize squared space whereas integrate square design contaminate primarily robust surrogate loss apply feature around spatial median depth contour influence function contain proof matlab implement available estimate ensure kernel condition e borel q example satisfy property kernel laplacian dimension psd exclude psd kernel hilbert completion reproduce thorough psd purpose critical property reproduce state evaluate inner radial kernel kde kde sensitive presence find q unlike huber huber loss quadratic huber loss detail huber many tb htb argue valid density contour f x iteration sequence weight experience initialization view optimization transfer perspective characterize every grow quantify recall influence scalar influence represent assign probability contaminate mass measure bound consider concept scalar express estimate function standard kde empirical influence kde influence fs huber find matrix ii addition strictly extend matrix solution near contamination kde agreement robust view less sensitive mention boundedness influence robustness htb show univariate tail value kde additional kde kde decrease training confirm setup present conduct diabetes f originally www uci set randomly partition f contamination one choose nominal digit contamination mnist nominal anomaly amount nominal compare kde density loss use kernel neighbor well omit c terminate three study outlier third task detail th set rank absolute let rank rank statistic test level study first measure influence exclude comparison x change density impact experiment learn nominal mean performance sign state affect c kde kullback leibler kl eq divergence whenever nominal kde separate estimate infinite
duration population birth rate disease furthermore age differentiable easy eq ode table population furthermore relative know ht hand type ode I p ode analytically life disease rate age equation stationary describe introduction derivation age incidence age usefulness report publish health year whole population member percent take account percent sample gender stay diagnostic code associate gm report age frequent report confidence decide whether group significant due ode general office year order incidence perform derive calculate incidence spline function uniquely bound perform software version table input follow describe mm age person incidence disease ode relate age specific incidence rate ode incidence cubic spline choice one incidence extract sophisticated restrict optimization intensive perform analysis group generate incidence treat six hour pc ghz cover reporting period author try estimate rate member get diagnosis third potentially positive early see case account diagnosis incidence ode ht age person per person compare incidence visit newly incidence rely free hand systematically increase year diagnosis instead value hence death incidence proposition diabetes incidence rate relate age incidence allow study applicability age agreement publish disease cubic spline equation basic characteristic disease population characteristic fundamentally incidence observational assess population classical incidence somewhat group patient examine point whether meanwhile investigation mostly complex expensive study fact get lose question incidence know health service resource patient example may population number number see equation later complicated analytical transformation ode reduced equation take incidence solve incidence paper organize newly discover age distribution incidence health incidence age group
interesting density approximate consider recover level close nn estimation technical requirement recover requirement level provably cluster retain nn index discover neighborhood retain sample single linkage nn differ retain result namely level recover cluster procedure correspond note trivially guarantee return salient mode empirical treat false cluster consider cluster provide pruning heuristic typically consist define strong tree size moreover assumption spurious easily appear sample rely prune instead return value empirical notion unfortunately finite guarantee removal spurious smoothness require upper density distribution start operation unless radius center contain nn graph say every continuous call tree denote notice form infinite mutual either versa simplify set pruning control show sample level prune cc contain tree keep look lower tied tree give tuning connect component subgraph tree I assumption continuous connect two pruning essentially large remain subset remain envelope suppose nn subset belong cc recall parameter separate da term apply nn tree mild illustrate practical pruning wide range consistency compact n da separate path satisfy ga disjoint thus high hold establishes remain part becoming create spurious cluster removal spurious letting discover removal spurious cluster one guarantee set actual underlie spurious cluster remove upper light surrogate consistency maintain expect discover salient mode salient salient separate separate every salient theorem follow disjoint na salient leave n sample maximum sparse increase average max setting mode tree guarantee lemma describe happen subset remain prune interestingly although interest sample statement literature supplement intuition combine find probability nx ix provide l say low region remain mutual nn subset r point level path show salient salient contain set point vc establish condition subset denote mode satisfy nn nn mode empirical salient mode set ball whenever bx r kx salient mode contribute map iteratively start establish therefore leaf root leaf subset connect empirical prune near connect path depict path depict connect must sufficiently consecutive whenever scale apart various near neighbor create way get choose possible near ball weak keep track scale path nn belong cc lemma show possibly x x ip bx nx ball continue adjacent must argue must ball center show establish edge relate first imply two imply g ny must bx dr nx iy iy terminate procedure remove spurious hence spurious pruning let disjoint na na n nx low n thank often chernoff ball nk eq choice fix nb bx nb word chernoff lemma combine bound choice long infer event inspire relate center ft combine prove combine bb c fx r consider eq integrate probability kx finally
curvature noise perturbation tangent leave mostly tangent tangent classic cause interpretation suited noise appear u triangle geometric control seek regime careful employ concentration present appendix matrix unbounde care ensure hold ambient dimension avoid dependence ambient hold analysis concentration hold maintain main mean curvature rescale curvature model natural strictly positive formulate follow number sample point observe linear position sampling result result q random realization follow definition ease presentation finite correction numerator finite correction denominator term simplify choose demonstrate accurately angle true computed decrease noise bind minimized tangent recovery optimal scale may neighborhood ensure condition note denominator analyze bound recover angle subspace unable require conditioning impose spectra curvature perturbation quantify scale curvature eigenvector tangent condition requirement sufficient meet numerically track violate bind tight recovery error demonstrate main quantify need geometric principle impose tangent solve satisfied require real derive full allow geometric uncertainty natural perturbation curvature curvature compare curvature ball require note principle manifold less intuition correction interestingly tangent careful perturbation would compute manifold clean perturb remove compute homology noise topological main track scale require optimal recovery practical main demonstrate intrinsic parameter level local uniformly tangent curvature recovery plane norm utilize practically trial indicate deviation hold empirical relevant chernoff far provide geometric encode principle tight unchanged trend regardless accurate condition tangent plane radius flat vertical see discussion show linear subspace increase scale axis monotonically green track blue nearly red b show free three curvature case due monotonically slight numerical predict reach curvature infinity true indicate orthogonal tangent violate long spectra predict compute contain orthogonal true tangent version panel large curvature scale bound track error principal observe behavior large scale encounter free saddle embed demonstrate principle mixed sign red track true tight case except order understand height dependent plot height red curve result nonzero experiment track behavior show parallel logarithmic multiplicative constant far indicate triangle bound norm tight decomposition tangent dash indicate minima dash dash green location occur indicate yield tangent particular location occur flat panel error compute error angle tangent stable examine coarse scale coarse indicate miss exact optimum dimension important experimentally sensitivity tangent error follow experiment sensitivity track one parameter vary hold radius neighborhood radius manifold embed detail hold scale intrinsic shape direction principal panel panel varied panel neighborhood right main show blue track angle hold curvature incorrect value range right way curvature hold thereby green main result compute inaccurate radius entire indicate properly intrinsic sensitivity estimate mild shape principal blue track recovery indicate optimal radius hold dimension range green vary within bind indicating noise relatively stable small curvature set mild curvature manifold normal insight avoid stable b blue track subspace radius hold large incorrect curve right remain stable bind variation curvature high principal principal expect experimental indicate perturbation provide algorithmic tool directly tool translation distance tangent plane estimate clean origin equip two bind space free practice presentation assume recovery origin decompose recognize compute side realization sample size observable ambient approach determine proceed calculate ball give ball radius effective insight gray scalar curvature manifold volume small ball plane volume ball vb vb distance estimate eq approximately capture effect relationship previous derivation volume noisy concentrate surface radius convolution manifold gaussian flat ball center account along beyond follow volume conclude radius plane radius large precise computation remove confirm noise less lead prove along author confirm thorough next rough accurately track figure first embed normal geometry consist noisy manifold embed principal great geometry tangent plane blue ambient show reference geometry geometry indistinguishable relevant geometry compute tangent true tangent reliably therefore observable tangent plane blue equation indistinguishable scale ambient repeat subspace recovery ambient tangent radius ambient radius tangent plane bind track error present ambient ambient manifold c like like vertical line indicate logarithmic axis specification give exclude linear geometry implement choose give principal set gaussian experiment ambient use tangent plane error true perturbation utilize represent good hope mean error mean bar deviation origin show behavior counterpart compute decay axis track true blue nearly indistinguishable red manifold three normal direction ignore instability scale match panel observe due curvature matching geometry principal track dash vertical location curve blue dash fall quite flat figure scale collect ball ambient radius tangent plane show exhibit direction impact geometry green curve normal situation ball ambient necessarily point large small radius plane amount curvature geometry see ambient radius grow tangent grow ball ambient tangent lack geometry indicate orthogonality scale indicate provide sort discover ambient green order project tangent plane red order identical geometry curve principal equal geometry exhibit great curvature note possible future nonetheless present indicate main accord track tangent ambient user result explained previously propose system analysis seem perturbation source perspective alternate use reader apply noisy compute simple require decomposition worth expect universal multiscale present intuition role close clean plane since exist coordinate direction coordinate align goal move normal framework rotation axis merely convenience discuss close right close define blue normal red neither perturbation require trajectory radius grow coordinate recover scale ball n ib r mn b square scale explicitly precisely tail scale reach coordinate uncertainty densely sample manifold apply rotation conventional coordinate unitary coordinate slight modification observe coordinate intercept origin order leave coordinate coordinate proceed without generality show small coordinate derivative least square enforce model estimate anchor examine detail confirm scale smoothed stable must observable radius radius presence informally measure point let denote offset calculation similar show yield scale hold introduce presence yield hold high take indicate necessary geometric trajectory slightly consider scale point let note rescaled curvature encode scale eq probability trajectory decay replace rigorous analysis mean g understand scale produce result replace discard discard behavior scale trajectory quadratic explicitly interval compare choose error could carefully could compute one expect satisfy decrease procedure scale interval small trajectory center dense algorithm demonstrate parameter implement sample dimensional embed manner origin support reference specify seven experiment origin reference error trial report interval use c c e curvature saddle saddle offset table accurately origin setting curvature initial offset large occur high setting produce curvature relatively noise quality beyond careful driven dimension noise coordinate origin error decrease error pca manifold receive assumption growth work focus recover therefore confirm closely pca spectrum estimate free develop multiscale detect sample analysis level much result set angle regime although curvature treat drop main result form recover curvature way noise perturbation datum density neighborhood analysis analysis yield density lift proceed limit requirement associate separate regime correction recover require expectation imply agreement size choice ensure small denominator decay numerator limit infinite absence density study multiscale pca spectrum detect manifold generally cloud random empirical localize noisy close population covariance high author estimate population noisy noise effort center multiscale noisy appropriate present population control center localize lead moving allow curvature affect point experimentally main experimentally cause theoretically center rescale radius localize close localize origin recover center true origin neighborhood offer algorithmic practical curvature analysis recover suggest intrinsic dimension introduce approach recent pointwise fashion perform svd determine examine growth value interesting remain coarse exploration approach dynamical perspective exist present set experimentally several neighborhood multiscale value suggest denoise method estimate curvature g vision explore tracking center rate pca spectrum tractable plane choice conduct experiment plane radius manifold tangent plane rigorous need tangent tolerance may chart ensure requirement meet similarly able neighborhood may tangent neighborhood local global name optimally cover national foundation dms department de acknowledgement grateful anonymous comment suggestion manuscript inf mit qualitative chen little advance multiscale wavelet analysis ed pp j adapt wavelet j eigenvalue number shrinkage curvature measure transaction american determining flow find intrinsic dimensionality manifold r deconvolution hausdorff mathematical introduction several variable geodesic riemannian journal computation eigenvalue principal ann dimension sample ann component analysis manifold process tangent plane th international mathematics tangent plane noisy avoid curvature plane statistical pp mathematic surface pp functional ann lin riemannian manifold pattern multiscale ph thesis university multiscale multiscale curvature mit unify conference surface noisy cloud perturbation ann composite locally science orient filter journal intelligence global locally neural inf pp mit vector isotropic conference pp wu vector diffusion map connection tail matrix smooth embedding matrix compress ed pp wang j finding dimensionality fast active contour alignment scale patch dimensionality linear modeling pp zhang manifold appendix calculation particular probability norm norm tight bound ensure independent ambient dimension proceed bound result start proof comment notice sometimes simple interpret merely upper interpret notation often eigenvalue square standard event seek nonzero nonzero equivalently ignore utilize find self adjoint expectation eq yield set eigenvalue notation great hold large great soon necessary eigenvalue remainder spectral use eigenvalue eq frobenius delay section tight however hold proceed denote small singular respectively result give tight control let whose standard normal variable divide coordinate soon reasonable sampling similarly soon direction curvature quantify note bound matrix nc center expectation distribute original involve frobenius merely euclidean unique modifying slightly arrive complete compute tc tu set expectation take matrix u nc q finally eq great random seek form realization dimensional wishart since block indeed let partition equal frobenius differ rotation explain next first entry extract norm control norm matrix get norm frobenius loose equality probability random proceed conditioning coordinate gaussian matrix prefer row euclidean conditioning know diagonal copy eq bind derive replace full small finally happen moderate mild condition realization last consider depend tell likely e imply depend finally bind l tu fm previous proceed precise envelope te tp correlation coordinate term clearly upper top eigenvalue variance measure along expect rigorous value singular orthonormal column size define project vector technical difficulty check rotation proceeding realization eq concentration te v fashion overlap block row orthonormal orthogonal matrix zero circular shift right construct stack construction nonzero every column allow action everything together conclude orthogonal previous norm proof remove condition p conclude n indeed great joint selection random realization q nu te tu give brief outline get q nu tc tu component great realization department mathematics usa set lying consist parameterization tangent plane return optimal basis sample nonlinear manifold small scale stability pca space adaptively reveal plane stable purpose provide theoretical real tangent principal model curvature noise lie manifold parameterization datum represent parameterization may inherent discover geometry sample remain research study parameterization give svd datum near pca linear case subspace higher proceed nonlinear embed consider fashion linear latter subject linear parameterization manifold tangent tangent plane corrupt curvature local pca tangent characterize local tangent perturbation theory estimate randomly within neighborhood adaptively neighborhood give analysis proper recovery intrinsic curvature however pca locality partition g adaptively size perturbation vary locality hand approximately avoid large effect simple criterion tangent plane set color angle form plane define adaptively accord neighborhood neighborhood exhibit tangent recovery curvature vary adaptively orientation due curvature stable quantify high angle recover subspace
rsc glasso rsc rate rsc regardless new rsc joint rank row proportional reduce set nonzero row generic optimally square coincide construction knowledge bound model obtain immediate restriction single estimator satisfy q inequality constant select countable rank pattern essential matrix satisfie n computational responsible selection existence force address propose norm two estimator define minimization restriction glasso synthesis strategy refer lasso mild say satisfie index iff row design literature depth diagonal pt eigen tuning minimizer large enough nonzero pt stay adaptive predictor prior contrast regular glasso differ glasso need refine three suggest consistently rank rsc singular exceed span glasso two mild restriction strength detection problematic prove rank technical rate let satisfie restriction small pay efficiency initial analyze validation consistent square estimator true discuss present computationally two construct b conclusion provide satisfie q indicate comparable perhaps stage select glasso rsc adaptive select consistently prove follow consider row rank estimator spirit involve becomes couple c j minimum j matrix equal directly nonconvex constrain may way lasso orthogonal abuse notation optimization r n v refer iterate statement monotonically stationary suppose fs k grid rsc estimate single tuning criterion select em block convergence stationary require crucial need optimization denote kronecker I subproblem instead perform low thresholding v soft operator jt k resource need flexibility uniquely strong arbitrary accumulation monotonically simulation I normal matrix entry row feature resemble thus rank constrain variable report similar consideration even correlation rsc glasso minimize influence observation tune similar lar restrict result correct path suitable pt set table summarize mse histogram turn asymmetric computed goodness comparison l r experiment pt automatically useful surprisingly yield new predictor associate fold cv find complement analysis score rsc variable order associated eigenvalue instance important accounting roughly simply quantify cognitive set infection cognitive cognitive performance typically measure via cognitive five domain work processing explanatory contain clinical imaging measure several matter initial rsc massive especially derive run new score select aside mean rsc newly propose method demonstrate existence cognitive establish suggest perhaps fractional derive appendix generic coincide column furthermore space favor hold penalty writing c imply b inequality k b inequality conclude entry consequently write q find reasoning side row notation imply second e e b f p e inequality complementary case inequality hence eq ep conclude proof fix globally respectively tucker j j take combine ep find decompose remain projection column chi degree use schwarz right invoke complete theorem yield term penalty b r take proof global lemma map contain outside convergent subsequence solution pt refer characterize let k fs composite set v analyze minimum globally start pt fs glasso attain von inequality svd globally uniformly pt fs fs f describe point prove set r fy x sx j sx fy fy sx lf fy f fy sx similarly close map theorem far converge begin context view unconstraine manifold explicitly fs page continuous since uniquely attain directly lemma denote limit converge fs j fs point suppose fs contradict strict apply algorithm system recall minimum stationary without scale kt fs fs pt know describe similar ms j jt j ss argument pt accumulation fs lemma apply verify equivalent minimizer give global remark remark support nsf grant dms support nsf propose reduction regression exceed size sometimes complementary predictor matrix article gain prediction obtained consider motivate new rank least square penalty impose restriction prove extensive analysis response q measurement response subject unobserved set need denote index nonzero counting observe free furthermore span column either parsimonious model propose popular rank tailor approximation value second reflect belief method class effective parameter higher especially study model suggest analyze strength reduction square tailor article impose sparsity restriction coefficient aic univariate penalty rank result estimator coincide rate cf
fitting potentially region unique region error paradigm turn effective behind region output combine overall partition input combine expert general well derive variable derive necessary gradient reformulate address expectation maximization consistent responsible paragraph standard learn activation activation classification output layer node hide unity competition different competition nan probability combine superposition usually together mlp observe try take strength account discuss implementation show network stress sake simplicity require partition space color follow regime observe dr vs express source compact lie vast fact break optical arise bi particular source color fact two inside window change sub domain paragraph regime region heavily mostly determination employ fuzzy fuzzy hereafter sharp oppose fuzzy counterpart give metric find centroid minimize cluster maximize reach belong different sharp work sharp counterpart find centroid belong distant particular identify centroid membership determine coefficient equal partition membership large threshold assign soft introduce redundancy translate pattern allow belong optimal although address mlp introduce negligible identical train prediction choice randomly couple normalize unity plot network variation precede optical galaxy optical optical use determine number network determination case threshold reach reach optical optical optical digital survey dr confirm optical retrieve survey galaxy source classify galaxy clean estimate survey band compose retrieve dr table kb optical source subset sample database source kb confirm optical source counterpart optical band kb subset kb query database method encode magnitude correlate color represent measure thus discuss change color encode partially distribution dr plane error color evaluate individual correlation color color case window correspond error color corner error vary finally spread plot high color symbol source exploit contain color uncertainty uncertainty experiment produce less color color yield experiment involve determination optical galaxy magnitude color error magnitude galaxy obtain matching source fit two image extend source band profile well fitting magnitude magnitude correct accord provide third magnitude use optical color uncertainty far magnitude attribute source kb distinct class perform vary parameter one yield term statistical discuss output well use galaxy variable hereafter give word hereafter evaluation accuracy experiment table physical motivation characterization mean paragraph choice total respectively experiment involve hereafter galaxy diagnostic equation diagnostic select criterion separately respectively class optical optical clustering mark optimal high optical galaxy optical optical uv expert expert expert expert neuron gate epochs gate learn gate gate expert treat explore expert experiment evaluation galaxy retrieve color single magnitude training describe single distinct c mean cluster perform kb uncertainty color uncertainty magnitude determine fuzzy dimensional space add color uncertainty fuzzy member account membership show member kb galaxy histogram optical confirm four color associate uncertainty determination cluster kb source color expert generate optical color uncertainty fuzzy fix determination carry feature distribution kb histogram optical uncertainty color optical filter filter magnitude therefore whole available color determination final reconstruction optical experiment show figure galaxy database obtain galaxy way kb galaxy clean band observational retrieve identification sake completeness format give uncertainty service name description unique object right degree color double double color double double error belong determination galaxy extract show figure appendix galaxy kb certain galaxy kb galaxy galaxy significantly galaxy hereafter call band employ deep galaxy case evaluate contamination high galaxy census galaxy galaxy release include span survey overlap galaxy cross galaxy apparent magnitude filter galaxy fraction assign method consistently uncertainty quantity evaluate statistic bar magnitude cause low statistic dr galaxies apparent band symbol optical candidate extract compose source type classify extend clean filter band source query retrieve galaxy change candidate extraction additional quantity derive method candidate common available retrieve matching original optical candidate dr service long unique double degree degree long double r I color color double kde estimate relative double kde estimate p double kde kde f opt uv optical extraction include candidate rely characterization confirm employ cluster achieve separation combination algorithm includes follow perform salient confirm extract kb source extraction source close candidate associate dominate candidate candidate confirm probability associate distribution star kde refine reduce completeness extract yield source first third subsample optical available optical optical detail contain source query retrieve reliable counterpart appendix describe cone c unique error error long double double color r color color double double kde double kde p p double opt uv characterize observational use thorough description bias comprehensive z average number overall variation reconstruction measure spread source z z galaxy hereafter use estimate alternative accuracy paper wide ground survey optical survey optical optical optical galaxy table galaxy optical knn achieve method bias method reconstruction optical knn much aim determination empirical color optical optical consistently method exception normalize uv optical column describe determination optical optical galaxy optical optical statistical discuss diagnostic exp exp belong kb kb possible large coverage reconstruction value show sigma function source show axis percentage source kb source set optical galaxy optical set randomly draw subsample minimize effect source behavior experiment increase set create kb exp exp exp exp kb always evaluation accuracy advantage ability statistical useful scientific error represent difference evaluate color evaluate absolute train carry approach slightly different use expert except employ table reconstruct source belong distinct plot variable low panel panel color code evaluate source vertical dash represent emission line characterize spectra shift plot show color code dash emission characterize spectra line filter resolve l c min opt cluster max epoch expert neuron gate gate learn gate gate error involve error estimate lie inside degeneracy vs plot upper occur characterize shift system shape globally error compatible c optical right average profile black plot emission similarly vs plot optical optical sample contain region optical plus error estimate real uncertainty extent reconstruction heavily yield completely phase generate plot absence belong hereafter provide object optical unlike error source sample evaluation inside kb spaced bin error spaced presence lie prominent source reliable involve plot associate optical optical effectiveness reconstruction depend value explore determination efficiency completeness source value completeness third maximize efficiency e product efficiency completeness couple large efficiency second experiment optical n respectively optical reliable estimate interval determination retain reliable inspection kb optical color symbol quality histogram subset difference reliable vs experiment concern estimation optical c exp exp diagnostic reconstruction source determination contamination exp show plot right source expert determination working galaxy able distribution kb template besides give description work also determination optical galaxy optical accuracy optical galaxy express dispersion variable optical reach thorough apply provide perform result experiment optical galaxy produce optical optical associated detail method relatively optical achieve priori convergence newly acquire process acquire new kb requirement become mining cope stream optical survey produce amount total collect part drive framework new old particular traditional provide hypothesis accurate galaxy survey put nature call group group appear well fact drop employ technique drop non template hypothesis drive fitting link employ complex consistently use together expert base gate exploit hybrid architecture learn traditional physical neural interesting architecture different integrate strength belong domain dm consistently predictor dm technique training gate predictor hybrid approach empirical classical method method template review template acknowledgment take author acknowledge mostly library foundation http www project retrieval publication tool protocol international publication publish cone search service service center ia version anonymous comment improve retrieve whose use describe server system g galaxy example query retrieve dataset query dr server p bad retrieve counterpart p join join p distance center department sciences university associate california institute technology usa availability survey become crucial expert datum technique technique exploitation base compose optical galaxy optical square accuracy galaxy extract optical uv paradigm virtual field observation galaxy mining survey ever digital survey carry extended human approach mine advanced technology etc stands recognize scientific theory shall task galaxy etc observation still demand still abundance observation branch stem long rich technique example colour colour classical consider number member uncertainty possibility extensive effort year candidate instance determination visible scale source trade derive quantity e arise reliable select advantage obvious latter select digital release dr extract effective source magnitude range find theoretically lie limit spectrum galaxy determination galaxy type structure investigate reality physical characteristic galaxy worth less template differ among aspect knowledge kb kb accuracy hereafter interpolation modern wide mixed survey combine band thus number significant subsample digital survey remarkable mixed survey benchmark optical select characteristic effectiveness evident encounter critical tune galaxy template fit part degeneracy bias final mining degeneracy minimize derive belong kb shall relevant exist color see emission mechanism approximation example present dimensional training
q message miss factor message marginal generalization factor cycle bc bc bc indicator box later consider version channel function one go fp generating kp ideally gibbs produce sample remark introduction base address turn capacity channel constrain method turn resort describe layer importance q verify importance useful feasible feasible obvious choice factorization helpful base gibbs acceptance require find may address several auxiliary part inside outside importance also importance j particular efficiently gibbs follow index fix factor cycle cycle partition fig configuration x backward draw chain irreducible gibbs fast naive analogously tree simply drop e estimate graph fact gibbs sample importantly direct cf easy noiseless capacity horizontal graph fig likewise follow easy pass factor graph noiseless bit vs plot path fig capacity vs several symbol fig issue mix improve vertical variable fig product still equivalent time mix result reduction computation fig respectively bit figure single loop loop carry task draw unconstrained I however draw p loop compute q follow fig principle estimate handle slow channel additive noise horizontal dot noiseless channel noisy constrain channel additive white signal snr define specify db symbol vs db different path information rate symbol vs sample noisy constraint plot different grid compute db capacity noiseless bit symbol fig db different path snr snr decrease isolated mode good importance value require show db number limited plot limited constraint db function point temperature partition snr low snr layer temperature note carlo computing rate model method channel open case input constraint without belief information guarantee gibbs de product method knowledge channel
actually least probability ce corollary several constant line phrase depend context lot sub row contain matrix index row contain row index contain column actually column adjoint denote element support define fix set cardinality introduce deep absolute provide give scheme introduce david later constant value first two replace provide hold sufficiently small assume throughout prove appropriate requirement inexact optimization explicitly inexact duality exist vector satisfy x f plug moreover construct construct f block divide c mention deterministic let absolute replace high provide sufficiently construct imply independent moreover calculate know follow etc orthogonal satisfy recall incoherence ie ie j suppose pp slight proof inequality hold os subgradient nuclear q h construct also prove notice l high enough construction q large enough uv ie ie iid random high n z uv z l z z provide provide therefore z j sufficiently notice similar scheme method factor least square part dual actually apply acknowledgement grateful ph help manuscript pt corollary example mathematics stanford university stanford exist compressed sample introduce recover signal tractable fraction corrupt provide fraction corrupt nonzero entry tractable fraction keyword robust isometry cs approximately acquire numerous field cs represent establish solution guarantee original iid occur provide dft discuss cs compressed sense totally absolutely entry want recover signal accurately mathematical nonzero coefficient recover discuss frequently acquisition device typically nonlinear occur component portion sensor measure outcome typically measurement report totally measurement collect error investigate recovery cs noiseless cs rank suppose square year nuclear sum original low meaning iid guarantee probability positive numerical dependent concern cs broad applicability wu et typical true model defer section random gaussian variable choose numerical everything else nonzero need imply adversarial moreover lasso pursuit model iid obey model introduce matrix support cardinality restriction concern sign symmetric solution provide constant support recent randomization first fix sign de randomization row dft numerical proportion sign know would hold general emphasize little strong free important remove sensitivity introduce good however little decide noiseless rank write originally support fix either specifically k model numerical sufficiently available approximate follow recovery recover actually slight modification argument large suboptimal completion tight noise noiseless take validation compare exist common choose notice motivation satisfy bit li set sub sensing x recovery obey restrict exact recovery sparsity requirement standard however go piece later appear form matrix incoherence independent uniformly randomization sign nearly optimal word extra condition optimal model discuss tight continuous paper plan include common assume similar require condition least impose bad differ prefer may proof scheme exploit another matrix always assume interested strong condition al
construction central limit variance observe evolve odd consequently evolve geometric parameter assess criterion order reversible kernel consider apply irreducible acceptance set result reversible stationary metropolis yield maximal inferior variance assume distribution transition allow euler discretization advance diffusion set hasting bernoulli execute however hasting alternative hold claim metropolis view acceptance order example sampling algorithm normal example transition geometrically reversible trajectory setting cn normal normal normal sample q sample normal sample cn normal markov report variance corollary stationary substantial normal become inefficient cn order cn big operator operator asymptotic walk bound increment metropolis metropolis metropolis thank helpful eps stroke department university uk department transition integer transition chain derive spectral formula limit assumption e depend theorem fundamental chain reversible ergodic chain associate refer spectral markov probability nonnegative integer sample context set irreducible proposal move accept metropolis ratio big roughly speak motivate recent advance equal reversible stationary nonnegative integer kernel geometrically analytic spectral kx explicitly corollary reversible ergodic transition kernel markov dominate stochastically dominate pe moreover conclusion another direction study markov chain bound pt chain bound bounding hold geometrically meet bound
paper whereby hypercube connect close error bound provide rigorous particular robustness notice determine rotation translation inter matrix position shift throughout one position original position center note sense invariant transformation generality yx yy xy yx f cn hold high distribute connectivity necessary localization problem connect h dimensional vice regime far surely nonzero semidefinite localization follow sdp minimize compute ne th th abuse notation sdp step gram node key obeys constraint solution eigenvalue minimization coincide remove degeneracy eigen center reconstruct point origin high p converge hypercube connectivity sdp coordinate conversely constant general stress see next use stress matrix random graph see prove background localization attract significant past name propose localization guarantee prove noise group distance reconstruct matrix distance short pair localization scheme sdp crucial existence sdp efficiently check whether uniquely find realization maximum unfolding sdp problem local metric interpretation lie manifold ambient low representation attempt apart ambient maximize total distance give large method broad paper model relatively broad class remainder organize brief notion property implication application theorem contain proof use prove reader convenience symbol table give distance finite node coordinate transformation brief overview definition theory interested thorough discussion undirecte correspond stress mention establish geometric preserve satisfy call equation anti derivative span account freedom transformation correspond degree correspond dim say dim framework scalar stress stress set edge length configuration recall coefficient connection follow result prove stress globally stress stress geometric graph lower singular value stress prove basic number hypercube volume node defer bind ball symmetric vertex eigenvalue stress computer present geometric let laplacian degree node eigenvalue constant restriction notation subspace span orthogonal denote throughout constant frobenius singular set edge convention adopt represent vector convention notation center position match note connectivity threshold logarithmic numerical idea show namely immediate bad measurement perform dx c r reconstruct position question motion set hand vanish hypercube sdp recover graph namely poisson globally w already establish phenomenon geometric globally globally phase geometric exist globally high area determine position sensor hardware rule system sensor acquire interest global environment semidefinite network localization develop advance inaccurate shall place ambient dimension either connectivity various interference nearby accuracy measure technique measure wireless device signal arrival measure receive ratio proportional receiver use dominant receive measure receive denote received measure magnitude per nr remarkably accuracy average compatible connectivity time signal difference distance receiver velocity measure lead proportional theorem obtain dx nr suggest design system stress bound prove adversarial similar ambient lie implication sdp equivalent assume little generality unit hypercube estimate pair try find position reproduce geodesic nearby euclidean reduce mathematically localization whereby distance measurement estimate depend curvature manifold vary unit radius minimum equal show lemma ij support claim estimation paper focus sdp noise direction take zero variable instance another due consider improve way instance manifold maximize pair add constraint short reconstruct geometry cloud incomplete inaccurate local angle arrival gram base lemma numerical w q constant p proof lemma frobenius assumption triangle take eqs eq q prove claim term psd stress construct psd psd stress combine eqs next construct psd node clique clique define establish property clique true ready stress zero next immediate psd almost last fact h corollary low bound p finally remark node thesis turn author show smoothly vector remark claim claim markov proof next claim clique defer appendix ij w q near I I n nm x establish graph appendix theorem decompose I x u j obtain matter explicitly constant clique degree claim defer probability di prove suffice claim obtain complete da xy yx ty tu u tu yx ty tu di xy gx compare key ingredient subsection proof appendix prove define e gs step whose tuple entrie tuple use set whose tuple set adjacent graph subgraph see illustration chain consider chain choose choose number line illustration vertex construction partition bin h bin vertex choose obtain perturbation width setup lemma chain appendix let exist defer position theorem width force flow number chain edge term accord describe contain bound p therefore equivalently xy chain node side desire number set vertex number x lk lk thesis notice value convenient generalize system variable write anti eq force impose force state net u interpretation mind find constant simplicity proceed defer appendix assume direction easy solution show force suffice observe force value write lk dd one form lipschitz function satisfy node node contain hence force respectively low map hypercube curvature radius instance slightly hypercube map let distance measurement measurement also ij crucial point step dimension input return gram eigenvector eigenvalue algorithm project estimate position separately taylor provide let unitary refer small defer appendix p tm p small eigenvalue addition small entry eigenvector small applying remark small z iid surely eqs consider worst uniformly introduce map claim algorithm run sdp bar configuration define accord let summarize fig respect stanford award nsf dms grant thank comment width dx plot width plot delta dx plot reference otherwise application right side become bin non overlap length total bin hypercube mn n lb apply union get line bin imply thesis orthonormal k w k l ti ki therefore vector j configuration generic imply adjacent graph dropping prove nr dc markov sequence consecutive possible choose follow bin side discuss node path node bin bin bin uniformly
observe th moment fourth observe var jt proof continuously differentiable real e first q triangular first pt first triangular inequality substitute item substituting give I differentiable work change amount I sum difference order expansion martingale q drop convenience ease ten p tt proceeding e n hence dominant note ta second term however order remove ny dy dy dy eq moment dy dy satisfy lemma prove divide respectively give n cn hence give time continuously x nf diag var entry nf use eq approximated li give uniformly q proposition lemma give proof variance mean variable assumption imply first equality second start give remain bound equality proposition obtain u tu j r jk n identity term twice v establish five cauchy schwarz establish bound upon partition define eq property fundamental pair say break pair fundamental index pair contribution symmetry representative assumption case contribution discussion concern fundamental impossible fundamental fundamental fundamental eq q turn break pair simultaneously break requirement break change contribution break assume pair break must contain remain assumption pair supplementary law iterate scale corollary thm robust es anonymous associate first la la sciences research author school economic finance material cm universit e du mail tp weak selection box statistic consistent critical lee tailor autocorrelation coefficient provide many coefficient simulation experiment classification primary secondary keyword white inference automatic test optimality white context autocorrelation regression confidence interval correlation residual residual improper autocorrelation diagnostic tool finance test noise hypothesis fan wu derive white extended goodness fit test kolmogorov er von test hypothesis refine residual setup sum lee white wu contribute propose nan nan test implement lee extend lee include origin kernel improve result appeal feature er von detect directional converge nan parametric detection type slow local spectral conclusion er von powerful box type universal exist er von point magnitude detect still describe type alternative multipli correspond scenario term policy effect term alternative expectation past give process close difference alternative finance alternative test intuitively explain wu normal critical suggest suggest box provide nr box provide enough order square box correlation smaller show er von important limitation critical ad hoc detecting instance unlikely give test power detecting need properly development drive various test detect alternative converge fast class unknown chen nonparametric hypothesis relevance le equivalence spectral continuous ready white exception fan maximum standardize box et drive choose order selection procedure west simulation selection justify west although optimal testing differ therein choice base aic adaptive white drive white noise alternative mention suitable optimality compare er drive wu report propose calibration automatic test drive west order conclude material observe covariate stationary mean nj residual density test kernel division evidence grow normal shall subscript n multiplicative propose select maximize ep k n penalization penalization differ aic penalty term term follow base maximize equal differ side critical drive covariance base formal statement variance e pn n bias trade value statistic lee run lee residual twice continuously motion rejection use sample variance tu j tn rejection region turn mean variable z alternative na n n dependence state nan follow wu stationary satisfying moment contraction wu measurable univariate vector copy change assume fast mixing use et al satisfy nonlinear volatility bilinear wu reference main away differentiable critical twice continuously differentiable support regularity c c nu maximal process brownian full ii admit u n n assumption interval impose coefficient condition come high order finite base cauchy schwarz implement propose test principle alternative role proof restrictive suggest focus since various process comment ns tu consequence wu verify ols version b lee show condition restriction value exclude next issue restrictive describe requirement white alternative important stay test statistic hand level nominal substantially address nan asymptotically sequence asymptotically asymptotically therefore impact impose compare element since impose either hold technical contribution without power allow bic drive view potential negative see side behave standardize exact b supplementary achieve alternative go alternative similarly sequence alternative exist consistent lag construction statistic latter sequence admit penalty sequence condition impact impact use fact alternative situation detect correlation well np favorable sparse coefficient allow coefficient converge cover independent variance setup possible detection achieve wu n wu test lee region preliminary replication quantile respectively penalty behavior white noise standard report level replication simulation draw small ensure observe size close slightly less test use benchmark drive lee west lee pilot bandwidth remain differ valid use standard er von distribution three chi chi lack chi square reveal sensitivity skewness size slightly strong chi except chi experiment consider white martingale examine I normal coefficient expect behave uncorrelated report bilinear label examine process uncorrelated dependent finally residual test adapt thank test even small suggesting distortion critical lee behavior either pass behave follow adjust desire normality alternative expect alternative especially simulation impact decrease decrease lag similar characteristic power test drive surprisingly outperform test outperform illustrate nine example undesirable correction lee consideration compute preliminary label figure nine except process suggest test nonlinear white noise power noise well bilinear second randomized simulation moving shape coefficient p infinity covariance scenario report experiment test show drive test lower really affect propose automatic noise observe residual new test statistic lee estimation alternative coefficient autocorrelation moderate lag enough alternative average autocorrelation test wu scenario impact may cause finance rule deviation martingale alternative alternatives simulation new cope white empirical finance also confirm regard alternative nonlinear process goodness test goodness criterion autoregressive integrated move average series university empirical test check goodness test process detect martingale automatic serial test significance diagnostic uncorrelated white drive specification regression minimax specification consistent serial unknown conditional form parametric without estimator bandwidth autocorrelation presence covariance matrix spectral origin bootstrap noise goodness fit adaptive testing wavelet asymptotic strong principle dependent red red dot lee coefficient range sample replication cm robust pt pt supplementary material proof lead line integer part copy condition ensure supplementary material proposition impact deal residual thank proposition proposition k cp l kl kl suppose j q let p nr ensure enough give ensure retain value level critical p np chebyshev inequality proposition eq stay away first nan show imply lemma eq observe residual give alternative enough q eq set spectral density center coefficient define old constant spectral r ik b g j j ij md eq enough kk f give correlate alternative p pp lemma log brownian process depend family sequence behavior lemma satisfy level test since hold old equivalent brownian motion bayes base choice schwarz op nj n n therefore
sense aspect bound recently area extensive contamination svm synthetic briefly heuristic amount contamination mis collect contamination understand co situation context mapping yield many contamination datum contamination model bind statistical rule throughout function decision decision replace rule denote substitution fix among call bayes risk distribution stand surely simplify quantity distribution bayes learn minimization classifier differ sample totally question classification train testing generate wish access contamination vanish classifier state different state theorem sharp achievable contamination eq contamination uniformly lemma rf x rf contaminate xx gx continuous e completely membership hold imply lemma conclude contamination induce apply generality decision center write let take universal consistency imply risk arrive sharp contamination indicate contamination make small suffer contamination explain svm fraction label rely fortunately currently popular early stop statistical contamination research statistic back survey extensive estimation primarily problem relevant sensing however al investigate image mis work purely underlie mis form forest form patch type additionally al impact mis xu error location richer broadly divide stage stage roughly deal contamination learn bag work include related deal broad contamination explicitly difference source long small whereas contamination numerous publish domain adaptation beyond give reference particular david follow error indicate source nature replace vc rademacher theorem nature learn vc often loose still loose asymptotically vc bind small underlie vanish way component contaminate easy contamination establish contamination david et particular david bind quickly approach whereas apply consistent classifier dimension polynomial neighbor classifier without consequently svm kernel boost bag nested decision dataset test average generate aa set test respectively classification contamination figure ignore second adjustment eps scale eps class originally boundary contamination repeatedly multi bind contamination simulation eps eps total dataset take repository detail htp c set training include split test size training rest aside train rest contamination result contamination average image dataset use example reference cite description linearly slow optimization include contamination svm plot htp eps eps eps eps eps image image interest interval day year optical relate acquisition sensing image scene offset cause bilinear index mis mis ii mis roughly contamination sample mis sample take fold validation fold c I mis relative area correspond pixel homogeneity homogeneity number mis since field effect mis half may cause far svm adaboost applicable collect table adopt result htp convert original contamination contaminate contamination leave future mis heuristic pixel mis thus pixel serve underlie boundary roughly mis say proportion pixel fall boundary inspection result train contaminate boundary follow pixel patch center two patch pixel patch mis capture formulate contamination nice feature bind distribution free datum extensive contamination tight adaptation literature classifier infinite already contamination mis beyond use contamination useful empirically show boost training carry success co ht eps small amount example originally high build noise f w clean denote clean co clear case rate label typically small potential benefit training co train feasible example training help limitation contamination image I mis image way desirable account work derive contamination desire impact datum classification leave interested reader acknowledgment thank square department environmental science management california google view problem mis determine result pattern contamination phenomenon mis model applicable many measurement etc contamination classification study statistical contamination apply classifier extensive simulation replacement random derive motivating example mis occur mis phenomenon map wrong usually cause acquisition device underlie mapping map datum collect scale take angle mis image eps eps primary sense monitoring acquire characterize liu classification xu etc never perfectly make mis acceptable al achievable thus important assess mis round weather additionally type part occur amount contamination broadly contamination cause datum
branch scientific mining analysis decade author theoretical practical base amount interest numerous investigate art technique hierarchical cut method certainly methodology regularization theory author use canonical define differential apply potential task two joint spectrum treat layer respective believe concept spectrum share base processing way classical certainly focus future partly author thank figure grateful bfgs de mail de mail observational multimodal nature represent address layer propose combine eigenvector result joint multiple cluster social dataset superior art common baseline summation layer extensively year usually object unweighted edge equal weighted value object subset one wide numerous reader survey mobile social bring challenge modalitie interaction modality represent whose set fig mobile phone mit reality mining mobile phone different proximity physical movement iii phone communication contribute meaningful combination possibly layer mobile leave cell assign edge phone seek propose novel spectrum spectrum layer view eventually generalize decomposition apply laplacian framework eigenvector share graph vertex cluster spectra enforce regularization characteristic cluster graph alone theoretic generalize evaluate world compare art technique baseline base summation graph result show term cluster outperform baseline competitive art contribution limited cluster spectrum multi layer lead rest motivate iii review building block section consider layer weight undirected associated aspect combination index explicitly point embed reveal intrinsic mean easily layer build mit reality mining vertex graph participant mit mining represent relationship mobile phone user term different proximity proximity phone call relationship three layer indicator blue nature insufficient achieve cluster mobile fact membership isolate vertex two entry rich information layer well unify joint combine provide addition effective group spectral novel method joint inspire popular spectral cluster main novel reader familiar skip section spectral become promise describe undirecte represent laplacian degree vertex along unnormalized combinatorial version laplacian define keep version detailed discussion choice adopt cluster algorithm spectrum dimensional form transformation eventually cluster illustrated toy show weighted graph number compute walk compute correspond small eigenvalue contain th row algorithm assignment first eigenvector new representation cluster trivial theoretical process theory multi find joint spectrum perform embed eigen eigenvalue eigenvector eigen contain diagonal eigenvalue graph layer eigenvector provide decomposition laplacian layer multi minimize write represent play role capture characteristic identity dimension function fidelity error add third constraint enforce inverse additional finally regularization balance trade convex alternate find optimize fix consequence give initialization suggest informative eigenvector initialize inverse loop solve variable quasi newton memory bfgs namely multiple layer spectral eigenvector eventually layer work eigen base average information tends treat equally layer propose target degree l w get eigenvector column th cluster assignment section novel method cluster treat base consequence help preserve layer examine eigenvector laplacian matrix weight eigenvector constant mapping mapping line vertex stay mapping satisfie scalar minimize orthogonality introduce view view condition rewrite equivalent usually illustrative mapping construct cloud keep connected vertex importantly view value objective graph eigenvalue illustrate b see stay quite mapping form embed property imply special smooth function graph process combine multiple equally highlight propose smoothness laplacian vertex enforce layer jointly smooth share layer spectral process graph follow eigenvector seek scalar function smoothness smoothness regularization fidelity term term objective eigenvector jointly algorithm worth role eigen regularization layer generalize propose clearly discrete membership information graph end jointly layer weighted target compute walk laplacian w kl n ki spectral replace cluster mean prove propagation one whose vertex propagate neighboring vertex make inference exactly regularization clearly solve iterative initial represent value parameter update convex value neighboring notice initial value relaxed graph problem membership interpret cluster way take make base task disagreement source source example algorithm disagreement multiple enforce layer disagreement reflect optimization fidelity term explicitly disagreement solution come regularization term structure indeed small two end function therefore total disagreement multiple formation disagreement model different individual respective performance three social compare mobile phone one dataset explain graph mit reality mining mobile phone three location phone specifically physical location service aggregate month weight addition phone call assign weight take ground cluster self medium school student group intend mobile phone currently research include mobile work source mit difference pair mobile layer graph third object mobile user still reflect human activity experiment come natural mining manually label one category truth cluster represent title abstract take corresponding citation reflect citation paper paper create mit much truth cluster observational intend cluster dataset reflect physical proximity phone mobile user make difficult moreover imagine dataset even mit email nevertheless choose ground well rich mobile phone challenging explain briefly propose eigen iv two solution experiment enforce inverse choose select part mutual layer mit dataset cell mutual combine act cell third phone incorporate final relative importance information share second mit dataset representative spectral apply adjacency weight layer use summation normalize summation vertex represent eigenvector graph laplacian regularization regularization propose art multiple graph
linear acoustic array communication relate objective use predictive location regression observation classification briefly theoretic relate seek expect location test monotonic entropy point include yield experimentally expensive posterior inductive theoretic require integration probabilistic close active active svms volume vs proxy improper volume vs fact posterior become intractable observe vs propose approximate approximate laplace ep vote vote disagreement posterior disagreement vote confidence exhibit mc ep ccc width title xlabel dim ylabel dim axis axis leave mark option coordinate black mark square option title xlabel dim ylabel dim line color mark mark option coordinate height title xlabel dim ylabel dim color mark coordinate color mark consistently amongst match good performance condition fig approach test line range expect knowledge noise dataset g reduce rapidly bad noise free indicate strong algorithm often small often poorly noisy dataset fig criterion maintain notion inherent uncertainty exhibit poorly greatly yield choose cluster influence point point reveal perform access objective empirical weight picking assign weight fail poorly hyperparameter fix fair select location maintain hyperparameter done mcmc far pick point randomly demonstrate theoretic gp date apply learn directly naturally learn hyperparameter active notable include ep online thereby trade theoretic computational appendix supplementary taylor expansion even term preference compute laboratory theoretic active widely probabilistic policy tractable however task classification hard achieve gain entropy apply make information experimental performance compare active low computational well decision theoretic secondly develop preference extend preference inference design internet expansion vast quantity costly call former minimize decision collect minimize risk know test advance want extreme exploratory hard motivate agnostic hand inductive seek reduce possible either heuristic margin study quantity entropy although decade criterion complicated infinite entropy perform present yield kernel interesting theoretic theoretic approach apply extended approach compare care contrast approach machine address datum directly goal active key learner choose query observe response within dependence output parameter infer active reduce maximally datum problem np made optimally seek data decrease expectation unseen g parameter often compute entropy eqn become poorly avoid exponentially dimensionality entropy introduce bias approximation calculate computational difficulty potentially update eqn insight eqn unknown insight entropy space eqn entropy usually entropy require intuition seek uncertain confident outcome learn disagreement apply build entropy around posterior approximation minimal additional knowledge algorithm represent fast discriminative gps tool challenge information quantity entropy infinite function consider probit give cdf inference intractable assume sparse though care indicate exploit query eqn expectation posterior mean variance compute quantity eqn handle probit second f pf eq reflect integral intractable perform material approximated curve convolution finally close eqn depict incur tend zero yield author previous context approximation monte consist apply predictive mean point select query practically function gradient maximally square parameter set nuisance I setting want maximally integrate nuisance gp spatial maintain posterior hyperparameter parameter certain determination hyperparameter variable primary interest preference label prefer denote task predict relation special building pair latent preference predict preference denote become define inference entirely probit perform operation gp supplementary k
suitable kernel statistically likelihood perturb define q analogous abc mle smooth nz ask analogous hold smoothed smooth noisy abc mle careful read analogous continuously reference measure condition smoothing note comment remark smoothed noisy mle smc conditional law drop law hide particle smc respect likelihood smc approximation use mle approximate perturb hmms smc I random extend perturb hmm since conditional give state likelihood standard smc evaluating likelihood computation likelihood k l lx k n density lebesgue abc likelihood manner particle see reference detailed analysis smc resample sequence price hmm economic factor directly price become distribution return asset price likelihood stable difficulty financial noisy toy transition conditional stable growth bad ie log asset stable distribute drift asymptotic bias abc mle true intuitively due perturb hmms hmms position lastly small size seem ie order theorem investigate abc mle suggest fisher abc mle decay large value indicate fisher abc mle decay fourth power large accurate use achieve use particle abc small value even bad volume around quickly abc truly practical neighborhood rapidly overview investigate abc hmms mathematically collection arbitrarily noisy mle noisy mle increase finally mild abc result help extend exist investigation firstly paper weak assumption find mathematical relax secondly suggest mle technique sample current support lemma sequence continuously cauchy uniformly second concerned identifiability additive give zero lebesgue measure zero denote characteristic follow concerned state measurable suppose probability connect see section support probability add observation general complete common dominating support equality let variable observe variable appropriate dominate identity density variable apply jensen inequality immediately jensen measurable hold v measurable measurable p since curve contain sequence open ball less b k measurable correspond dominate continuously p proof observe dominate measure computation follow dominate convergence c b establish property component hmm perturb hmm k let hmm hmm identity assumption follow b certain well property hmm extend sequence law consequence hmm extend dominate exposition give conditional obtain py l r thus bind express hand corresponding hmm leave hand conditional conclusion corollary replace property infinite concern collection hmms extend hmms l p n surely define corollary eq far first part boundedness density bound collection finite positive mass still since positive finite measure equip doubly kn equation follow letting apply function clearly continuously derivative derivative uniformly corollary sufficient inequality appear analogous manner side bound individually use fact straightforward history likelihood converge infinite likelihood continuity continuity r mapping rest fix dominate mapping g thus dominate immediately map dominate theorem also term shall prove proof completely identical difference value likelihood identity eq imply converge definition limit bound lebesgue theorem almost follow lebesgue dominate continuously differentiable eq uniformly uniformly bound eq satisfy condition conclusion identity continuity follow l p sufficient perturb respectively process stationarity observe perturb eq stationarity last dominate eq function p likelihood converge hence apply obtain finally law one prove sequence integer assertion follow order assertion since connect law probability finally densitie analogous definite sequence show every ti stationarity follow straightforward consequence apply law integral w gradient g since dominate assumption remark physical sciences college abc popular approximate likelihood often likelihood analytically intractable abc many investigation result estimator abc base estimation markov normality discuss provide implement markov bioinformatic e also recent often hmms observation particular maximize q unless simple analytically variety sequential wide conditional despite sample process parameter g lead approximation carlo another technique problem indirect hmms filter density inaccurate extend kalman filter likely expensive deal computation abc reference methodology summary statistic assume metric fold reflect estimate intuitive volume radius around factor ignore likelihood order result behave purpose issue investigate ease exposition paper result continue statistic see end section conditional state likelihood sequence radius markovian simple monte smc implementation abc experience compete see reference deep approximate parameter firstly could commonly take frequentist maximize approximate henceforth bayesian computation mle abc become estimation either frequentist particular mle distribution concentrate true abc seem mle converge place mathematical purpose bridge theoretical develop justification mle standard normality likelihood perturb hmms abc behaviour observation asymptotically bias arbitrarily small mle rigorous justification complete picture mle noisy always raise abc result suffer efficiency perturb additive independent finally study herein help provide field notation approximate mle abc overview smc implement give qualitative estimator support shall letter letter observation variable various infinite brevity shorthand notation integer variable sequence give integer denote denote shall use sequence notation combine doubly infinite random j j l essence property operate hmms ensure consistency space hmms compact furthermore denote kernel dominate state space conditional density dominate chain recurrent write law expectation law dominating mutually absolutely respect long term kullback present arise try understand behaviour extend unfortunately require perturb likelihood w dominate measure essence despite associate perturb sufficiently need order take limit constitute conditional perturb process give past expectation stationary hmm far let asymptotically let accumulation perturb alg conditional law law bound conditional log part argument mle bias show arbitrarily sufficiently provide justification mle standard mle small theorem whose mapping q continuous exist sequence follow result additional decrease positive constant theorem whose give invertible continuously differentiable hence case markovian abc summary ns markovian moreover suppose mapping identifiability system hold hmms preserve reasonable choice summary statistic show perform abc choose maximizer likelihood likelihood inherent sequence noisy nz law correspond perturb estimator standard hmms result mle asymptotically light remark observation noisy noisy method investigate abc section mle consistency normality mle provide mle
implicit improve multivariate assumption form prior restrictive appear function objective reader adopt improvement candidate acquisition rate convergence prove thompson option portfolio strategy combine appear optimize acquisition function version project find acquisition objective acquisition evaluate observation parameter draw policy policy gp call boltzmann sample candidate transition one sample generate optimizer find maxima unnormalize boltzmann exploration slight sophisticated greedy draw optimizer function follow contextual bandit dimension popular block gibbs system trivially vice interaction positive tend opposite sign landscape interaction consider behavior gibbs wang regular model boundary dimension periodic boundary square interaction bias phenomenon arise experimental protocol trial compete sampler store energy visit energy comparison analyze energy trial rapidly markov parameter discard sampler connection unit visible vice versa visible illustrated learn one variable figure well wang perform slowly carry compare popular like would connection tree address present considerably gibbs bit lattice computer depict wang minute ise model minute minute second minute minute second computational spend flip length sparse possible speedup store l updating flip entail replace result variable flip contrast densely connect om om simple despite rbm densely densely easily proposal parallel unnormalized flip unnormalize flip flip divide evenly among processor speed updating unnormalized near take sampler sampler simply rough could happen find small move sampler affect degree connect compete wang demonstrate extent indicate sampler range model already nonparametric optimization parametric carry point ise develop acknowledgment thank modify slow research support joint wave proposition wang specialized value avoid walk state bit mcmc self dynamic integer remarkably broad task boltzmann machine arise computer ise also boltzmann machine deep apply rao integrate example effective discrete great statistical efficient monte inference ise vast domain challenge sampler make notable hamiltonian monte effort deal space domain wang well densely graphical latter acceptance trial simulation possible rejection accept move favorable get present specialized equilibrium monte move self avoid walk mcmc unconstrained system tune trade exploration bias proposal distinction seminal author concern inherently namely system another priori idea impose self generate rapidly state class know yet propose sequential coupling importance boltzmann machine sampler mcmc proposal enhance apply consider meta wang system consist state boltzmann system statistical learning particular energy mcmc gibbs heat sampler suffer mix minima dramatically rely mcmc idea present relate mcmc local previously focus advantage particular state ham clearly agree I I extension act sequence uniformly via acceptance low single flip move tend move incorporation requirement asymptotic procedure force exploration equilibrium take type biased length sequence visit procedure bit select equivalently f bit flip call show element sample follow likely sample principle choice behind energy bias propose configuration final state imagine uniformly high extremely unlikely accept begin bit sample yield reach point self avoid process imagine space lie flip avoid walk state induce acyclic subgraph lattice move twice stage process obviously avoid note construction impose transition occur neither seem ask less molecular try yield traversal landscape avoiding visit substantial portion reach I multiply straightforwardly move delta result term ideally mh would evaluate propose f accept mh unfortunately length marginalization massive k special case obtains follow accept state care detailed balance still mh reverse straightforwardly map example see somewhat involved support coincide turn balance balance side enforce balance k summation ready chain balance propose accept computational evaluating accept ratio order proposal numerator completely involve proposal look transition kernel evaluate path right side accept call accept strong f reader detailed balance long nonzero example visit fortunately set visit trivial collection flip set consecutive integer separate occur two length enforce correctness section discuss choice discuss matter ise type interaction propose move idea continue familiar note term mind present equilibrium note restriction unnecessary readily consider principle concatenation straightforward iteration move flip proposal intermediate avoid evaluate flip generate distant favorable unfortunately priori guarantee intermediate state visit often sequence especially potentially pass automatically proposal segment iterate procedure operate value encourage new two bias unfortunately likely rejection rate numerator ratio enforce particular low temperature deal three type p frequency choose sample encourage local towards desirable minima last seem somewhat since reject purpose acceptance move carefully reject due help accept ideal candidate explore effectiveness adaptive tune free parameter length respectively previous group free symbol define probability tune parameter fortunately challenge candidate increasingly stochastic obvious function minimize auto specific lag previously
furthermore specificity ratio low alarm respective alarm observe network attribute limitation structure induce effect evident specificity decrease grow confidence addition appear grow sample specificity sensitivity show fig also alarm hoc usually exclude dynamical level size ad hoc certainly conservative choice impact sec incorrectly significant ad hoc fig plot systematically ad hoc low threshold comparable specificity negative across hoc attribute separate entity concern identify biological set abstraction underlie mechanism pathway context pathway full commonly ad hoc identify significant hoc correctly aspect strong dependence hoc effectiveness propose expression datum al package determine significance significant sample empirical cdf early across identify fig respect edge false positive spurious application et contrast et study identify influence cell arrive posterior learn present flow vertical dash investigate functional record molecular intervention perturb perturb study sec fig estimate threshold non significant present sample size big algorithm exclude use et conjunction direction edge direction direction edge datum graphical considerable biological medical community especially interaction entity identify significant ad hoc edge learn noisy negative propose statistically identify graphical cdf cdf effectiveness synthetic expression define setting learn uk technology sciences national library lm also thank useful suggestion reference association molecular pathway mechanism graphical especially ad hoc often conjunction significant propose statistically association identify significant association graphical edge counterpart effectiveness demonstrate datum publicly molecular protein sensitivity specificity result also demonstrate across vary specificity linearly logarithm systematically maintain level specificity reconstruct paper study use structure ad hoc threshold significant association ad hoc significance association motivate hoc spurious conclusion association model average molecular statistical relationship compose refer express relationship refer graph express relationship semantic principle graphical imply variable find acyclic nature focus univariate former parameter interest usually estimation structure determine encode present coincide structure thank optimisation difference terminology group class goodness score al et al assess arise algorithm measure datum unknown systematic assessment particular identify bootstrap resample average original datum edge present eq mu mu degree another structure accept unknown serious limitation lead use ad hoc define assessment study setting first minor accounting edge distributional require addition make latter either hybrid computing use offset propose statistically cdf observe confidence cdf subsequently al al q intuitively clear element ideal either non significant arise happen learn element equal significant edge provide separate importantly datum provide motivated one amount cdf way depend common norm divergence leibler grey change straightforward programming common q norm compare robust variety configuration identification thought follow follow even individually example model undirected q grey edge great satisfy test synthetic proportion edge correctly proportion miss correctly proportion edge set vary commonly use benchmark alarm network provide alarm message intensive care possible severe structure edge network design evaluate car risk compose incremental constraint child network grow algorithm independence test mutual information unlike mutual consider separately hill hc score equivalent arise uniform approach consider min hill combine max parent hc test illustrate performance measure network alarm possible bootstrap similar edge ignore edge network specificity compare threshold approach
separation bss bss exist prominent worker achieve sense hyperspectral image ica bss term matrix describe boltzmann shannon temperature internal principled free minimization free energy much suitably generalize scope extended compression communication accurately model commonly biological paper formulate variational source maximum dual entropy normal average via unsupervise formulate aid independent superiority separation g single information duality implication duality permit say infer logarithm derive previously ref possess property exponential form important concern expectation expectation define linear employ satisfactory molecular approximation successfully utilize iv describe employ separation observe pixel scale prominent utilize characterize exponential exp entropy introduce note equivalent g entropy govern calculus apart provide analogy expression calculus statistic derivative logarithm wise clearly lagrangian relation sum condition q j eq canonical n ij entropy substitute result potential relate substitute aid aid cast henceforth information update matrix ascent result taylor yield characterized exponential one face perturbation un ref additionally derivative replace expansion turn procedure double pseudo force dual break logistic vi follow pseudo value provide constitute experimentally fact former initial merely input far simple unity signature number iteration separation model b readily statistic exhibit correlate separation employ paradigm
interior surely weight previous close convergence make close surely slightly sequence allow satisfy make root rate eigenvalue unclear eigenvalue helpful case sure available consequence taylor almost surely compact become fast considerably fast albeit support weight message vanish convergence previous support know gap pr handle consistency example illustration simulation elsewhere finite accurate generic produce estimate mix whose simplex u leibler role support argue appropriate minimize adjustment start finite huge element optimization minimizer model cluster galaxy velocity galaxy datum finite location mixture mixture galaxy methodology component consideration component interval grid candidate pr method identify six galaxy cluster closely count substantial pr present show aic scad hellinger distance count five include besides find zero include grid let decide pr fit c estimate aic three take ccccc exp exp exp author suggestion department mathematical university portion complete recursively design pr simplex boundedness trivially assumption theorem exist lyapunov ode differentiable eigenvalue part corollary chen recursion pr distribution know pr mix mixture consistent condition misspecification finite theory converge well kullback nearly pr know modify pr estimate compare rate kullback function approximation challenge algorithm fundamentally different way hill learn sequentially like pr dominating unlike surely pr depend dominate pr density goal little shall approximation author knowledge fully therefore shall analysis pr possibly support case pr surely parametric close leibl also one choose algorithm naturally suit finite mixture unknown show pr yield estimate unknown support establish unknown example illustrate detail reader refer nonparametric borel present estimation sequence compute return pr connection distribution take connection pr unknown pr marginal section support pr recently become let set mixture density build show respective topology show close leibl divergence identifiable almost topology bind pr suitable rate leave suggest upper bind correspond boundary nearly root conjecture hold finite fy py xu two assumption mixture model identifiable one define leibl henceforth shall infimum closure lemma ensure particularly rather pr asymptotically pr generic mapping pr design root nothing conditional expectation density equal consequence martingale investigate ode ode limit purpose definition plug follow vanish equilibrium ode converge regardless useful lyapunov ode differentiable neighborhood f lyapunov lyapunov equilibrium show slight kullback lyapunov context map lyapunov ode calculus reveal fu
simulation analog digital cs synthetic digital algorithm approximate achieve algorithm depend approach solve interior ls investigation produce accurate digital time possible significant issue implementation scope cs parameterize signal depend variance priori empirically observe task variance improve multiplicative desire ensure display inspire index simulation depict relative calculate evaluate solution essentially digital algorithm term optimization digital compare calculate rmse meaning variability comparable digital solution difference row objective bottom plot solver normal digital solver well hard recovery synthetic value mse simulate parameter previous medium produce good converging second whereas converge order individual average case basic though circuit many include power consumption constant analog solver converge approximately simulated speed order magnitude fast solver real recent implementation especially accounting interface analog circuit right convergence hard fidelity difficulty recover fidelity converge support investigate medium cs problem display plot show simulated reliable interestingly digital g multiplication significantly analog like multiply circuit size increase time implementation demonstrate synthetic achieve digital solver improvement several order magnitude digital analog circuit potential image cs recovery section simulate speed compressive short scan improve patient throughput scan furthermore compressive mr medical perform high without cs acquisition dynamic subsample image transform wavelet transform fourier solution transform transform coherent subsampling poor result synthetic two section signal large image wavelet digital image reconstruction mse relative quality simulation take second translate concentrated explore many fitting form process basic variety analytically determine program approximate norm modify norm well group zero coefficient coefficient explore function note program theoretical wide impose condition analytically translate cost function require activation result case behavior imply huber scad nonlinear activation regularize program consider square form widely function include e count regularization benefit norm cs activation special analytically activation large huber piecewise cost calculate activation separately region interval activation put piece thresholde show function converge early barrier norm scale mean poor amplitude cost one draw intuition various norm jeffrey determine q show activation activation cost early increase signal community coefficient also block purpose encourage sparsity block penalty cost pointwise input output effect calculate cost yield directly simplify back relationship simplify eq range yield activation thought type group input group recent sparsity solve tractable program minimization update q coefficient drive small increase weight update q establish interpretation drawback require immediately iteration save significant time digital solver advantage gain modify analog involve weight modify steady represent analog scope system reach digital method plot relative recovery measurement sparsity variance signal iteration modify system achieve plot dynamic weight dynamically clearly iteration traditional digital algorithm modify optimization iterative weighting scheme digital comparison weight modify discrete solution evolution simulate term dynamically converge approximately play processing toward computational tool toolbox difficult application power analog analog circuit demonstrate real solver solution scale beyond implement wide result analog dynamical system certainly investigation cs expensive computational bottleneck cs wide variety increase incorporate sparsity structure improve would dynamical already establish analog circuit traditionally benefit illustrate achieve actual development analog issue analog circuit platform preliminary simulate issue future inherent load implementation work include solution explore design exhibit system hybrid analog digital achieve benefit note prototype achieve less rewrite equation formulation essentially depend write constrain program barrier convert approximately program strategy algorithm barrier solve increase norm increase activation derive ideal soft thresholding operator relaxation fit used solve find invertible straightforward soft relaxation function soft figure plot extend give occur less activation function back l macro indexing vector coefficient coefficient dimension tradeoff sparsity symbol derivative derivative vector coefficient index subset index inverse inverse energy log edu pg berkeley mail com present grateful valuable child improve significant effort algorithm specifically problem advantage system sparse analog recover synthetic acquire compressive sensing system scale support fast digital furthermore analytically problem basic norm classic approximation analog rely apply incoming circuit significantly improve processing strategy noise common formulate thought search signal e improve signal image rely linear linear eq represent signal transform fourier represent corruption fidelity solve find signal frequently assume result separate cost p decade focus number zero draw significant interest appropriately actually problem simply count coefficient recent heuristic approximate performance guarantee relax date guarantee involve function optimization name include pursuit surprisingly many recover tractable compressive cs performance inverse highly zero show certain generally take random recovered signal use despite long optimization field application utilize highlight optimization solver real mention real power imaging channel wireless communication application problem art include computer vision well research focus dramatically reduce solve challenge smooth function recent fast specialized solver unable solve moderately fast storage time requirement time steady design efficiently induce circuit thresholding etc analog significantly power example quickly recover thereby optimization process cs main highlight benefit wide applicability analog efficiently analog system digital context recovery demonstrate analog architecture support order magnitude digital arise community norm norm classic mention provably optimization program smooth system locally competitive algorithm comprise drive approximated steady way generality
segment delay nonlinearity sequence reservoir construct record reservoir state often helpful mask symmetry mask present also mask nonlinearity exploit intrinsic chose reservoir delay loop allow explain detail mask periodic piecewise e value mask reservoir dynamic approximate regime reservoir correspond dynamical similar become couple rich instantaneous nonlinearity transformation reservoir linearity provide feedback delay turn representation architecture depict nonlinearity implement place light generate adjusting intensity change bring system instability dynamic variable obtain rescale lie interval version eqs derive material stage process reservoir experiment round usually reconstruct reservoir recognition concatenation wave wave top reservoir line bottom essentially well system obtain discretize check discretized version choose architecture experimental code take component agreement measure dynamic efficiently explore validate supplementary reservoir task use reservoir art digital implementation train move drive noise precisely reservoir produce output detail method measure normalise similar obtain digital size report reservoir consider receiver reservoir digital reservoir obtain apply reservoir isolate digits reservoir community performance computer reservoir present report yield reservoir detail material reservoir computer art digital implementation relevance speech channel flexibility computer new operating reservoir change possibly reservoir experimental reservoir computer successively dynamical system speech nonlinear introduce relate report input respect reservoir use internal introduce low pass filter reservoir enough processing convert analog process multiplication mask output multiplication digital constitute towards build optical reservoir computer computer implementation help understand analog processing acknowledgment van suggestion discussion begin project researcher work reservoir project acknowledge de task nonlinear auto move average simulate recurrence uniform aim predict know task reservoir community channel go channel yield yield range task misclassifie ref metric isolated digit recognition task ti ten record time digit reservoir take mask take mask ten associate target otherwise winner digit word train reservoir four average word fraction misclassifie correspond digits red green part optical optical setup operate nm determine operate feedback input signal arbitrary generator computer generate input signal task record optimize read feedback adjust average inside loop optical output adjust operating point operation fully automate diagram depict plot reservoir label bottom go mask reservoir get get new multiply add desire ph panel normalize ph blue take account red square three agree within statistical bar bar point might material practically obtain reservoir progress accomplished rise science digital analog apart remarkable find science incomplete cell process elementary organize system issue conceptual huge machine progress approximately order consumption computer presentation analog biological system rise artificial neural abstract develop machine attack reservoir propose paper date exceed reservoir analog particular biological rise computation present exist review rather point build reservoir computer reservoir provide detailed road building system architecture understand computing reservoir road experimental computer reservoir consist internal evolve evolution variable external evolution reservoir independently instance gaussian distribution variance call reservoir fix reservoir large proper reservoir past traditional mean aim reservoir computation internal state reservoir denote given header digital another would predict input electrical load reservoir computer state reservoir time reservoir perform input winner take optimize logistic discriminant analysis function support vector end optimisation prefer accomplish e ridge reservoir ready input reservoir input correspond datum check reservoir new reservoir target draw normal feedback gain gain necessity understand coefficient state behavior adjust appropriate contribution term input reservoir eq find choice matrix happen try coefficient optima linear highly advantageous list biological neuron interact sparse resource add reservoir modification behave dynamical time could come evolution system trick outperform predict time noise simply reservoir remain beneficial add ridge use simulation well linearity digital simulation give realization may experience extent sub continuous simulation far evolve system natural reservoir operate continuous write progress line reservoir analog system new insight operate reservoir computer base present look digital high consumption programming method first nm standard c band optical time pass light minimum light adjust rewrite light optical enable adjustment loop variation value mode optical detect integrated rf produce intensity experiment hence round act rf act filter unit transform cutoff intensity intensity lie becomes experimentally change experimentally tune range typical optical optical optical digital increase intensity diagram observe diagram evolution diagram reach affect diagram simulation way setup measurement setup verify branch diagram addition term input evolve discrete mask continuous reservoir mask may changes rf place rf intensity combination intensity experimentally vary amplitude generator take place rf affect filter time negligible sn depend exact intensity integer physical upon effect discretize equivalently dynamic coupling line appear reservoir experimental discretize component nonlinearity preserve operating system moreover discretized error traditional allow validate nonlinearity topology computer continuous matlab experiment signal discretize generator transfer operate gain response exact measure configuration dark add agreement simulate diagram agreement simulate performance continuous easily sensitivity shape nonlinearity always reach post light intensity loop convert record discrete duration associate affect efficient synchronization system estimator computer send reservoir reservoir reservoir task detail performance system random sequence discretize mask interval reservoir size wave ideal experimentally obtain perfect agreement simulation practically study error misclassifie nonlinear channel reconstruct noisy nonlinear wireless output channel gaussian yield signal ratio range reservoir computing could mask uniformly reservoir task mse obtain estimator discretize close estimate symbol fraction calculate form set set study noise figure text bar note number bar close suffer measurement db study give reservoir snr system give reproduce nonlinear input distribution target task mask reservoir train step system input reservoir average pair sequence performance discretize regime digit recognition isolate record time five digit reservoir time input mask probability mask reservoir ten train one digit average time winner take digit divide train reservoir repeat system digit give recognize digits reservoir reservoir reservoir size winner take report experimental reservoir measure diagram vertical optical measurement de optical multiplied gray optical intensity feedback feedback gain unity light intensity possible feedback gain unity intensity gain nature transfer lead behavior determine inside branch diagram experimental reservoir reservoir reservoir depict measure normalize lie whole length red discretized reservoir output show reservoir reservoir leave panel panel input feedback reservoir instantaneous input influence previous recurrent neural technical report technology term memory technical national center nonlinearity predict system wireless york processing chapter make dynamical mit stable perturbation neural international conference cat page david reservoir journal international reservoir advance recurrent european page www competition com david reservoir speech international joint conference page advance process optimization network neuron network international advance computer science edge advance neural systems mit david van toward optical reservoir van dynamical complex complexity network transaction apply delay large dimensional delay letter delay optical system physical review letter k scale physical recurrent neural network automate page bilinear david isolated word state european artificial page source recurrent transaction adaptive processing page mit develop word isolate corpus ti speech international conference speech signal electrical k cat bag sc new york usa david capacity dynamical david versus linearity international conference page david intrinsic temporal neuron visual advance memory neural international gray rgb service universit information universit cp author work author ac reservoir computing processing datum basic reservoir computing recurrent couple architecture non node delay importance nonlinear remarkable capability attractive decade exploit computation concern optical optical digital suggest flexible approach call digital reservoir inspire intelligence reservoir computer architecture element nonlinearity integrate intensity store internal reservoir report implementation task practical channel speech reservoir highly provide notably financial speech task reservoir computer nonlinear drive reservoir computing reservoir depend experience case computation randomly choose
pearson paradigm nonempty note enough check state small integer solution eq go hold right hand zero go message motivate standard learn independent couple eq let I subsection previously symmetry lead q condition satisfy moreover implement pearson binary classification connection constrain reference constrain optimization problem vector robust latter problem essentially simplicity take value function chapter convex surely chance derive sufficient call copy consist big author address unlikely term control attempt follow instrumental convex problem conservative replace f nonnegative take choice loss name idea employ cast simplex however directly scenario eq theorem entail fix moreover cosine complete proceed property eq q two display complete build upon r proof h proposition together imply since increase convex hold previous probability decompose treat proposition apply indeed yield combine bound find hold statement theorem definition complete n q eq complete lemma binary two binomial parameter hold bernoulli hoeffde yield binomial easily particular resort derive introduce scope fix binomial variable take derivative eq interval eq inequalitie variation hard admit hence inequality increase follow imply q fact scale theorem definition edu problem anomaly detection implement error combine specify ii minimum possible solve technique consequence chance program binary paradigm anomaly constraint empirical chance optimization pearson classical latter focus minimize slight abuse language type ii essential kind expense illustration medical diagnosis severe consequence spam machine true error enforce hope dependent goal design bound pre high show good performance excess type review main classification emphasis framework main proposition theorem proof secondary technical illustrate couple dy therefore define indicator expectation joint clearly reference rewrite replace decrease commonly convex surrogate hinge logit loss hereafter relaxation treatment problem overfitte mapping one small empirical risk resort way first classifier specific sometimes candidate collection setup result remain asymptotically meaningful naive tree satisfactory classifying individually decade boost exploit suitable consequently restrict classifier combination flat em rule sign rule majority restrict search natural measure bound formally inequality hold scope candidate classifier class result know bound vc dimension however argument bounding rely proposition convexity classical em type error paradigm constrain significance user np closely concept work theory provide thorough treatment statistical two statistical hypothesis determine randomness come randomization bias coin arise occur occur pearson amount significance minimize ii solution sufficient respectively lx kp lx merely proposition nevertheless paradigm recall classification goal solve directly unknown I x n x n mutually assume deterministic subsequent investigate subsection extensively proposition paradigm remain well theoretical np paradigm find vc solution pearson high bounded consider family accordingly setup sensible np beyond kind detection np work matter paper ensure pr follow principle pearson constraint concern difficulty seminal justification program seem unlikely estimation proposition confirm opinion define unknown base regardless sample excess pseudo fact come technical view go resort surrogate particular empirical bound involve summarize type construct type high excess result process optimum optimization consequence np chance programming explain section solve classification subsection simply counterpart classifier empirical type
plot sparse able recover nonzero prove small nonzero procee turn compute schwarz inequality nc distribution subspace constant small almost sparse solution nonnegative nc section euclidean c appear recovery n subsection connection robust correction compressive sense compressive sparse sense multiply q compressive sparsity perfectly noiseless compare almost euclidean noisy geometry tool stability transition compressive sense mainly focus error perturbation measure remarkable sensitivity phase provide compressive average gaussian paper derive arbitrary limited compare restricted isometry bind correction happen good isometry small proven sparsity illustrate almost show sparsity outcome system th measurement nonlinear nonlinear true nonlinear iterative nonlinear subsection performance recover correctly recover q part h let differ pick cardinality certainly optimization h lead exist certainly computationally costly nonlinear may non set h variable ideally minimization function convex apply h optimization jacobian state estimation update reach specify additive additive recover true condition true secondly correction additive condition jacobian nonlinear jacobian need assume iterative start generally additive precisely algorithm guarantee v state local jacobian aa k iteration nonlinearity capture term estimate follow plug denote gap generate jacobian suppose norm function reasoning k respectively h local know algorithm true additive jacobian satisfie local point aa verify correction nonlinear local jacobian simplify focus true state though first jacobian subspace condition hold fact difficult convert check explicitly condition error correction verify condition local jacobian trajectory input check gets guarantee converge neighborhood system neighborhood every kk bad requirement solve subset objective take solve large k upper objective enable relax semidefinite second nonlinear recovery jacobian state row local picking value fix jacobian property meanwhile h h inequality norm h nh long get summary parameter neighborhood true h decode algorithm true get size specific fit linear jacobian measurement noise also apply recover nonlinear bad power gaussian measurement independently denote choose estimate I dimension dimension randomly denote averaged run choose reduce recover even percent contain percentage fig power monitoring operation base measurement state magnitude angle operation power device measure limited current real flow contain error modern technology exist attack incorrect system result consequence massive scale measurement power angle magnitude system one reference angle reference magnitude rest function minimization test fig vector magnitude minimum nine angle take flow characterize level randomly contain independently noise apply iterative minimization recover nine take final error relative contain example system prove condition exact next indicate indeed case first know weight iterative test detect measurement pass test potential repeat pass statistical run verification method jacobian state row figure cardinality always bad datum around true iterative conservative broad predict recovery nsf theorem claim xu university wang recovery bad detection mix locally measurement property error bad noise sharp linear subspace mesh solution though linearize convex nonlinear electrical use programming inspire power additive nonlinear static power vector magnitude angle indirect send central state power common error sensor communication adversarial introduce call bad datum usual estimation need detect eliminate measurement power us problem suppose make function additive measurement dimensional zero element assume nonzero entry real error reflect nature bad generally party may sensor moment adversary party able error rare know ls effect estimation try minimize l bad magnitude bad denoise decode sharp almost mesh estimation error propose programming perform measurement guarantee iterative result network convex programming linear characterize decode optimizer restrict
group bound early cite logarithmic assumption dictionary design oracle aggregation weight achieve prior discrete development start general oracle sometimes overview oracle derive opposed turn mirror finally quite sparsity suggestion langevin carlo inequalities noise deterministic design use drive truncation characteristic one add group compute term resort exploit fact mh g describe mh note example hypercube define instrumental neighbor mh instrumental simplicity speed another potentially accelerate mh generate generate chain define pattern estimator mh solution estimation prediction integer block value elsewhere readily recall define triangle fuse version fused report measure figure screening poorly illustrate yield miss fact minimum zero sake risk minimizer risk yield fuse fuse fuse prediction performance turn eq theorem entail nonempty realization weak isometry constant nonzero let brevity enough risk random variable infimum test take random back selector denote acknowledgment author part dms corollary fuse gaussian potentially exponential take derive sparsity popular framework include ordinary sparsity fuse aspect principle successfully review several weighting emphasis construct scenario exponential weighting deal estimation cf presentation framework consider pair gaussian norm notation dictionary function preliminary estimator detail possibly give measure average square wish satisfy property remainder term characterize aggregate characterize aggregate often consequence measured coefficient inequality indicate dictionary reflects also lead systematic since wish small possible important obtain lead oracle find literature however infinity additional assumption oracle call truly salient reasoning goal appropriate whether viewpoint parameter indeed inequality offer coincide dictionary sense risk way attain minimax sharp oracle mixture exponential weight select minimize major theoretical organize weight penalize risk combine combine function estimator oracle adapt sparsity derive inequality popular sparsity implementation pattern procedure empirical minimizer broken exhibit intrinsic minimization minimization inequality selector selector exist follow function sense exhibit term turn well cf choice convex namely oracle inequality minimization receive lot choice consider obtain oracle front oracle sharp overcome limitation combination dictionary combination belong potentially good penalize look since proxy inequality replace combination carefully choose ideally remainder term difficult simple eq q follow minimizer kullback penalty lead flexibility result associated let probability measure divergence sequel adopt convention weight explicitly kkt multiplier kullback solution selector achieve desire average select dictionary aggregate aggregate balanced side contain gaussian necessarily exponentially aggregate infimum kullback leibler divergence take discrete consequence restrict vertex precisely denoting lead remainder term prior uniform section suitable methodology worth terminology average set aggregate preliminary hold initial randomization analysis aggregation enough work aggregation deterministic limitation randomization carry many estimator aggregation aggregate observation aggregation estimator affine direct independent identically distribute first randomization still lead expectation randomization big remainder inequality square kernel ridge estimator filter long list equal row rather deterministic exponential modify risk deterministic risk weight long unbiased linear aggregate naturally nf consistent family aggregate bind immediately mainly estimator particular general spirit apply projection mixture suitably however element square terminology come fact interpret indicator feature pattern give square moreover variable gaussian least guarantee square pattern small least square aggregate clearly function projection na risk selection exponential weight thus select resort probability distribution performance last projection pattern fact remainder term balanced inequality choice depend information candidate exist piecewise prior knowledge fit frequentist setup prior candidate nonparametric candidate difficulty meaningful basic favor prior main difference exponentially polynomially exponential least plugging valid penalize another sparsity yield screen aggregate sparsity screen oracle often classical one method fuse idea mix sparsity appear reformulate difference fuse invertible possible difference combination
diag extend logistic regression belong penalize likelihood logistic mixed ig genetic effect statistic matrix make approximate variance statistic function degree linear kernel model treat differently corresponding statistics university california division public sciences research logistic machine derivation response affect include intercept snp snp goal score statistic
refine intuitive trust agent sophisticated machine learn capable similar trust trust base differently focus transaction agent overall viewpoint group share useful feature perform machine efficacy generic system machine transaction set transaction work common discriminant analysis many mechanism rely specific agent historical predict use make approach quite party local sharing e traditional feedback aggregation system mechanism possibility avoid privacy save communication overhead surprisingly positively discrimination feature interaction interaction trust especially party reliable quite rely local approach detect future work investigate tool discriminant etc improve accuracy direction concrete site etc grateful dr technology interact carry ability assess risk provide safe transaction involve traditional trust trust transaction knowledge transaction capable distinguish transaction algorithm transaction previous transaction experiment accuracy especially specific agent rare inaccurate system trust ingredient reliable participant diverse system multi agent dynamic interact past trust mechanism traditional approach rely knowledge instance past trust agent agent insufficient trust trust absence agent learn aim complex decision exist may transaction argue investigate distinguish transaction one apply sophisticated transaction successful transaction transaction use give beyond validation explain trust behavior want say restaurant guess possible restaurant past friend give restaurant recommendation restaurant friend among friend friend restaurant drawback friend feedback restaurant mechanism experience make restaurant price location know city country bad want essentially proposal rely success assessment hypothesis experience learn new transaction experimental scenario proposal predict proposal trust machine integrate work effective machine discriminant lda presentation proposal belonging find example measurement physical characteristic prediction provide tree popularity visually human language lda generic framework discriminant may trade paper proposal agent local past describe feature generality successful interaction set feature quantitative historical interaction group successful perform lda two group allow likely classify root move produce investigate recommendation please design decentralized agent might involved framework local repository multiple trust assessment knowledge propose construct information generic trust design suit generally rely e g dt argue transaction efficiently good moreover confidence recommendation reliable information compare base trust local thus inaccurate party much need knowledge difficult suitably also information simulation evaluation trust party quite demonstrate efficacy proposal detect particularly evaluate transaction conduct trust know trust management organize trust framework proposal dt study subsection discuss sharing proposal real regard future unlike mechanism behavior decide interact base local avoid risk inaccurate third party agent request service agent customer service transaction transaction happen quality service transaction binary transaction customer generality unique transaction transaction happen new virtual characteristic transaction record agent trust transaction transaction take online camera transaction collect country physical distance transaction phone profile average comment number friend etc item category comment different place bid age already place bid regard collection note aforementioned maintain exchange platform ii historical successful trust extract transaction suitably historical trust directly historical trust feature traditional request entity much effort difficult transaction proposal attack trust fig depict proposal component storage sc trust engine knowledge responsible transaction party transaction record transaction relate engine item already dataset past transaction item trust application correspondingly machine trust calculation engine responsible predict knowledge conduct trust calculation otherwise dt suitable restriction suitable across confidence measurement section suitable reliably transaction however insufficient learning address propose traditional mechanism share exchange result sec advantage easy privacy identification difficult bad specific avoid local knowledge detail trust assessment transaction agent transaction transaction repository extraction engine machine discriminant recommendation obtain transaction transaction offer transaction represent historical transaction reliability potential transaction historical transaction disjoint group accord transaction successful transaction transaction please perform linear transaction transaction group service transaction mathematical one group average across transaction centroid averaging transaction lda class external separability extent successful distinguished transaction internal centroid scatter matrix internal fraction transaction regard external actually two represent mean scatter variance sum lda aim projection inter minimize maximized eigenvector associate group similarly transaction transform classified measure transaction group centroid transaction collect transaction filter distinguish transaction one final classify sort root leaf classification object object put classification classify child node node process continue transaction past agent tr rule potential transaction decision tree np heuristic generality hereafter select classify tree gain binary categorization successful past transaction denote transaction transaction q characterize collection transaction belong class successful past transaction evenly measure entropy feature enough corresponding gain high appropriately choose transaction transaction otherwise classify different datum e past gain respect transaction classification correspond decision transaction transaction inaccurate accuracy quite recommendation less impact also trust learning algorithm bootstrappe join interact propose dedicated exchange share help agent transaction transaction request knowledge share p fig depict give share weighted agent trust provide knowledge trust local trust fall trust evaluate usefulness update trust please since trust although discuss trust reliable sharing transaction transaction trust service vice versa provide act transaction transaction agent transaction historical information available rely aggregation web etc scope paper collect request agent trust provide useful maintain list agent provide knowledge initially past request explore request familiar e friend agent request super hybrid key special agent e agent detail use lda issue share among agent etc detect potentially transaction two knowledge knowledge help likely successful projection direction centroid successful group eq weight knowledge determine knowledge projection combine centroid potential transaction euclidean transform centroid behavior e provide perspective share update trust score adapt trust knowledge transaction inefficient unnecessary produce prediction transaction adapt trade computation overhead transaction evaluate local distance transaction centroid transaction predict successful mean trust score trust rating trust score statistically beta function posteriori trust priori trust denote trust score eq prediction correct incorrect transaction party avoid unnecessary transaction successful party good adjust third party site transaction million transaction successful feedback fraction performance dt study internet transaction conduct unknown I historical available transaction perform transaction environment transaction label explicitly environment trust synthetic consist service behave service target transaction target transaction successful transaction outcome feature construct node maintain list record generate transaction describe let transaction factor tune transaction transaction completely mean successful separate transaction base feature approach proposal three approach reject network aggregate node trust false form experience derive trust rating group belong please refer related section exist model lack metric performance trust transaction predict transaction would rate experiment bar add deviation transaction past one transaction start evaluate transaction successful vary investigate effect volume fig demonstrate dt evolve expect increase transaction grow prediction quantitative threshold individual past note logarithmic quickly minimal transaction transaction much two transaction experiment empirically transaction transaction relatively reliably overall rate base dt simulation thus capable quickly transaction online knowledge insufficient initialize accord item item third party knowledge helps predict transaction introduce inaccurate information instance transaction transaction inaccurate demonstrate lda number increment trend false false increase transaction collect inaccurate hence accuracy able achieve corresponding trust local three transaction rely integrate low well service keep negative heavily environment thing false large reason amount increase transaction false negative accept reject benchmark evolution trust management outperform aggregation especially predict request thus relatively accurate lda feature quantitative low outperform
e stay bound see impose measurement variance also verify arise mix derivation error expense complicate asymptotic estimator band obtain asymptotic survey issue quantity community theorem check rigorously far os enyi prove thompson asymptotically size extend chen quantity hold equip supremum g van say converge bound functional existence limit follow state normality zero provide variety condition size soon regular flexibility proposition controlling bias establish normality local linear smoother control uniformly easily handle property estimator prove entail number inequality van finite convergence smoothed process establish argument converge consistency distinct fulfil expectation remark fulfil section band process taylor chapter set difficult stationarity fulfil band coverage compute mean showing stem weak practical importance via notation conditionally one simulate one obtain conditional gaussian process performance width band curve consumption simulate realization basis orthonormal main consider random without replacement population number consider quantile consumption unit half determine allocation suggest get thompson trajectory small discretized trajectory control situation independent random variable variance ar illustrative discretized curve corrupted smooth trajectory corrupt noise deviation noise l trajectory case study strategy smoothing crucial performance drive compare noisy consider denote build easily check weighted least weight new justification kt minimum validation estimator far criterion value bandwidth furthermore suggest criterion define denote bandwidth keep partitioned size take value curve error bandwidth sampling l lin lin different median choice select mean median lin lin third improve interpolation smooth oracle concerned cross adapt select interpolation hand validation effective bandwidth dominate moderate choice lin lin focus due summary summary impact individual trajectory roughly load really interpolation criterion sample account choice median lin lin level bandwidth median lin pt band bandwidth unit l median median lin pt lin band area high turn band quick implement satisfactory bandwidth band application raise straightforward error build band functional quantile automatic however take make lead substantial improvement reduce principal shaped sampling relate score al normality approach confidence band appendix letter vary place decompose stochastic measurement bias e h old turn see pointwise square uniformity write local weight kt kt fact weight smooth lemma deduce prove process pseudo metric hold sub van cover ball radius packing number inequality smoothed stem normality random boundedness increment cover number satisfy get detail eq guarantee convergence nd logarithm look establish finite dimensional start limit function sample function derive fact dimensional light show check boundedness trajectory preserve well asymptotic normality inequality corollary van notation kt increment constant observe maximal weak factor integral employ cover number satisfie integral x c adjust deduce terminology van covariance pointwise pointwise uniform e lie compact equip sup norm close maximal establish mean inequality strictly index vice versa mostly thank slightly simpler reduce triple sum l ki kt iv calculation already main ks kt k moment finally like ks call inequalities van first norm increment justify hereafter mesh index van ns nx arbitrarily first inequality k theorem namely grow dimension affect consequence obtain term start ns ks ks reduce van pseudo ks kt diameter classical n cauchy schwarz moment mc deduce diameter covering c tend probability term result simultaneous dimensional trivial similarly theorem metric arbitrary moment constant vary across dominate thank sum form denote root vector identity vector study remain variate wishart freedom hold one apply away deterministic tend calculation one nx apply van theorem increment usual integral covering tend hold real sequence anonymous review lead improvement r collection survey quantity discrete noisy smooth trajectory mild population observation time regularity trajectory limit establish consistency former band band attain nominal simulation covariance account highlight sample development automate sensor collection exhaustive transmission storage analysis assess appeal trade accuracy cost particular competitive compression explanation although survey long establish motivation recently regard examine theoretical principal component sampling relate population read minute week assume interpolation discretized signal consider thompson greatly utilize see however property rely contribution present interpolation smoothing interpolation improve accuracy prove simultaneous inference estimation vast literature van therein ball exploit goodness fit functional regularize estimator build conservative confidence derive equip supremum confidence band ball approach adopt bootstrap band vary obtain bootstrap band establish rough supremum unfortunately generate correlation cause empirical coverage differ innovation easy attain procedure bootstrap lie contribution provide nominal coverage covariance mean desirable bandwidth survey necessity account usual cross survey total aim set base devise cross validation weight proportional simple replacement study paper organize
take either length scale length general principle dimension couple typical technique go high dimension infeasible high elliptical slice overall scale generality prediction generalise wishart process make since popular arguably predict volatility return mean dimensional white noise condition model specify parameter part length parameter hard positive discusse definite point effort lead general notably conditional variance covariance correlation entry full implement toolbox choose general variant use order low triangular multivariate see predict follow rigorous except make ahead forecast historical forecast prediction account historical covariance different financial distribution parameter parametrize place length scale posterior align though procedure wishart classic like generalised wishart process assess mse always never true proxy component intuitive thorough univariate analogue proxy use assess log likelihood forecast although proxy use historical also sort simulate common financial daily repeat critical easily reconstruct historical datum forecast table identify accurately poor change mse historical forecast forecast financial proxy square forecast datum point forecast thin marginal variance unfortunately assess natural mse proxy historical assess consideration vary daily five period multivariate figure return behave like index return normally distribute compare make advantage another critical benefit true forecast historical square forecast suit generalised wishart capture variance compare advantage procedure complex volatility whereas stochastic process generalise wishart alternative easily depend research apply extremely dimensional hope effort volatility general interpret high dimension acknowledgement thank cm cm ccc engineering university university ac uk wishart dynamic matrix capture diverse dependent readily parameter interpret introduce outperform multivariate process accounting correlation imagine price composite index dramatically rise discover etc rise anomalous index make prediction concern model price security volatility risk management financial key risk come leave fluctuation capital indeed economic economic time volatility think change zero generalization arguably return multivariate volatility understand correlation index univariate portfolio allocation say return asset may wish return volatility receive understand transmission financial another science otherwise useful know correlation importance volatility generality estimation difficult impossible constraint multivariate volatility volatility unlike generally effort lead simple make correlation leave vary machine introduce wishart marginal semi recently multivariate wishart limit dependent scalar restrict velocity particle motion autoregressive learning procedure generalise address arbitrary covariate like one miss easily smooth make aspect require reason scale yet provide general multivariate specification section wishart review wishart present procedure prediction experiment include composite set subsequent present dimensional also gp account correlation predict correlation review process construct detail joint arbitrary kernel mean smoothness kernel square popular function trend value time also generalise wishart conjugacy could inference learn transform formulation outline volatility kernel control variable covariate interest introduce first procedure prediction wishart new base regression generalise wishart process dynamic posterior datum posterior distribution relevant low cholesky decomposition scale model show dependence generalise wishart cholesky scale sample posterior monte cycle successively discussion assume step dimension finally also change describe posterior value fix freedom increment dimension gram function gaussian covariance form kernel depend dimension degree short posterior elliptical update especially posterior prior place axis align slice dimensional metropolis hasting reversible simply wish take divide cholesky decomposition parameter distribution must joint covariance pair otherwise row change freedom dimension dimension represent conditioning construct avoid explicit parametrize
information audio metric similarity metric music song query song traditionally set information ir query query song receive back relevance isolate example may focus new aim collection intuitive advantage importantly mind advantage specific song ir operating quantify detect item coherent isolated answer query contain group advantage guarantee alone usually increase song system advantage rational item correctly detect retrieve community identify song group music employ reader detection cover song community inside set matrix dissimilarity couple assign cover song g originally rank content involve network never apply retrieval task previous work considerable improvement overview outline article build analyze song apply similarity house characteristic expect reasoning cover song detect cover art contribution assessment computation improve obtain group stage sec increase particularly confirm intuitive exploit answer internal organization song perform author cover study first show tendency community conclusion sec house music refer consist non label stage per expect cover song piece content therefore compute couple present measure allow track piece promise far far processing brief outline descriptor employ thesis feature amount short window component ambient independent specific independent piece volume fluctuation characteristic song strategy song delay representation recurrence crp recurrence finally extract sensitive song characteristic previously adapt allow track potentially trace crp restrict application song fig compare song around perform interested comprehensive overview cover measure pair quantify previously black trace maximum symmetric similarity fill proceed dissimilarity duration song song threshold pair cluster cover inside evolve threshold correspond density component node display tb look evolution threshold link network cluster top right represent formation cover song begin form maintain bottom network note isolate remain expect threshold middle group nearly cover community evaluate unsupervised approach implementation asymmetric dissimilarity dissimilarity explain classical computation solely operate distance advantage noisy drawback algorithm implementation package incorporate use try provide implementation agglomerative hierarchical cluster method test single linkage sl linkage cl group linkage default check inconsistent maximal clustering experimentally next algorithm collaborative network outperform maintain pm link unweighted dissimilarity certain low node experimentally cover I show dissimilarity computational approach improve triangular checking component pm try triplet coherence triangular suppose g result triangular one cover either create triangle measure objective triangle incomplete triangle vertex connect penalization incomplete triangle experimentally tb implementation sequentially situation create maximize keep adjacency update necessary process set pm substantially whose seem pm dissimilarity extremely song identification detect accordingly edge weight closeness margin experimental aspect evaluate cover identification collection include cover add cover wrong accuracy estimation analyze cover set cover cardinality setup fix cardinality classical weighting go bad average measure song typical measure entropy representative song positive belong community positive e cover negative actual detect optimize grid search try definition cluster community parameter algorithm turn obtain accuracy broad range near km pm report accuracie high majority nearly reach effectively detect possibility coherence answer enhance retrieval sec accuracies pm comparable pm tb km sl cl pm pm detail denote world music collection qualitatively evaluate spend setup see completely process collection spend hierarchical raise usefulness algorithm huge music cluster well km take matrix music million handle aforementioned exception pm pm pm furthermore subset link notice sparse e link million link investigate community dissimilarity refine ensure community refine rank song initial accuracy common measure average depend precision song retrieve answer sort list relevance song otherwise increase assess optimize instead threshold different one imply method high particular cluster community community achieve increment role positive optimal might illustrate reasoning regard positive rank answer query real song group belong cluster suppose algorithm recall precision answer refined answer evaluate relative increment eqs high table overall reach pm poorly km sl cl pm detail denote within audio cover song identification evaluation international task assessment song identification participant black accuracy collection publish participant tune algorithm collection version obtain accuracy achieve respectively algorithm comprise plus pm dissimilarity achieve former increment task might detection improve particular final accuracy high music basic device represent human exist evidence brain category around category prototype piece would prototype feature prototype form membership cover song gradient category song firstly cover prototype lead song song good manually setup discard song song mark state actually original avoid song popularity regard cover follow cover set employ direct graph support evidence song community fig strong see define usually community song cover make blue dashed line tend dissimilarity tb ability automatically version community extent consider
mdp play crucial minimization empirical loss proportion satisfy previous compose transfer error transfer third dimensionality decrease linear account target task bias towards wrong approximation target average target still effective bellman two operator g dy solve task bound logarithmic use much big estimation suffer additional transfer whenever advantage great bias sample come introduce estimation small select source report thorough arbitrary cp gram dx let whose correspond greedy bind display difference single bind account decrease well bellman mdps enough carefully design feature eigenvalue whenever well tf bellman reward transition might introduce tradeoff approximate close error display similarity introduce run iteration play crucial role proportion bellman operator define define minimize transfer case learn quantity training transfer fig simplex iteration pair return return bound achieve transfer large auxiliary fact task whenever task relate task particular small call generative fair dimensionality dependency optimize simplex may transfer depend space well necessarily transfer well previous show notice depend action slow state bellman operator error approximate monte auxiliary possible advance access source verify target source begin auxiliary learning include auxiliary compute mdps introduce source sample share notion try identify target transfer consider obtain notion try always image bellman operator notice minimization convex quadratic simplex task run iteration prove well approximate bellman operator nonetheless implicit solve tradeoff reduce proportion source properly solution tradeoff task remove favor pool task tradeoff maximum sample vector n optimize tradeoff error first account transfer induce error performance bind main difficulty previous consider task assumption def constrain proportion sample accord multinomial whether experimental effectiveness tradeoff parameter reward parameter ccccc task reward preliminary transfer introduce previous finding challenge variant walk action move make affect intend define interval region elsewhere transfer algorithm eight function report function radial spread episode start uniformly bar single iteration transfer problem use source leave plot compare performance transfer task sample refer auxiliary error thus task add run notice target auxiliary good policy existence linear produce approximation reward four task combination reward able transition task different drop among significantly task happen leave plot transfer beneficial bounded quickly target sample auxiliary training together source select proportion significant behavior possible preferable actually match much performance highlight tradeoff task provide address sample reduce large transfer enough interesting notice improve keep unchanged source avoid comparison sample task formalize first performance rl show access source task consider target set principled algorithm confirm finding work future problem increase make transfer effective adaptive transfer direction future balanced target action see definition generate task access source design specific acknowledgment project education de european receive european fp agreement besides introduce introduce norm pair empirical x orthogonal unique n tf already detail explicit since iteration explicit straightforward regression quadratic call fix design set n ny nr x qx single fit satisfie fig noise observation project function variation inequality put together analyze performance generalization action nx nr value measure space bind rewrite minimizer big variation together q finally proof sample l q qx p v f final due thus affected term refer inherent bellman error bellman relevant error recall bellman previous interval bellman image projection finally possible relate transfer statement let q return probability chernoff true bellman inequality previous series inequality statement available available transfer represent iteration consider source scenario target obviously task percentage sample task constant task decrease reach iteration behavior attempt early still suitable force drop lot source small introduce task optimally poor q reduce proportion weight try lack task approximation proportion change favor finally use task decrease start iteration effect tradeoff realize available tradeoff realize tuned analyze performance value
collect eqn precede display cd dd cd g cd exponent g g divergence infimum measurable conjugate dual convex variational characterization convex mix inequality via desire claim w imply almost identifiability condition contradiction immediate consequence sufficiently small boundedness boundedness thereby extend claim admissible way outline step suitable modification inclusion hellinger condition measure proof existence turn contraction wasserstein apply le discriminate give infimum hellinger p rw g complete turn ii maximal packing fact c r p g p p first fact p packing equation test due monotonicity proceed least w condition satisfied equation x term precede display bound thank display large probability vanish third vanish np gx g np gx bind denominator theorem combine display thereby conclude proceed suppose form cover denote dim simplex shall couple display construct combine number probability time volume k e tp kp argument dim g wasserstein metric combine cover simplex single kp q kp mi I kk conclude px p sr acknowledgment author thank associate anonymous valuable comment suggestion point version lemma grant behavior infinite wasserstein metric relationship wasserstein distance space hellinger kullback leibler distances mixture provide natural rate distribution notable mixing help relatively rich year extend mix construct model fit datum several book distribution include mixture measure atom vector probability combine dominate kp distinct behavior heterogeneous datum probability proportion behavior compare assess mix direction cumulative measure chen result set mixture unbounded multidimensional even distribution see past decade model bayesian set b primarily convergence hand concerned latent quite arise deconvolution mainly zhang fan note recent consistent parameter certain primary wasserstein mix model establish rate number well wasserstein although divergence leibl variation hellinger distances wasserstein utilize g atom equipped wasserstein q infimum wasserstein atomic assessing model worth note probability atom must converge permutation cluster illustrate mixing interpret convergence relevance distance chen special wasserstein mix know divergence functional wasserstein mix strong identifiability density entail mix wasserstein theorem divergence mix measure atomic generalize unbounded convolution latent mixing nonparametric endowed generate accord study namely truth contraction theorem prove notable rely likelihood typically formulate term density wasserstein typically weak hellinger rate possibly number include mixture multivariate distribution dirichlet mixture convergence logarithmic minimax dirichlet ordinary smooth achieve ease place place distance include dx hellinger kullback divergence divergence ball cover entire mutually distance equip distance inequality measure equip borel sigma algebra abuse write atom likewise probability atom measure denote discrete measure infinite support ij q denote discrete measure g take wasserstein employ may metric role mix dominate ease use combine function relationship wasserstein divergence divergence play role hellinger leibler instance broad divergence know divergence let f divergence distance hellinger composite distance kullback finite highlight aforementioned mix measure evident inequality enable test establish bound leibler mixture density probability metric latter former identity j similarly kullback leibler kf kp kf kp g g lemma choice yield topology induce space may imply convergence distance property specify identifiable strong version discrete support finitely identifiable almost imply rate notion herein adapt identifiable q finite identifiability satisfy chen identify broad gaussian p p suppose compact wasserstein family finitely identifiable suppose likelihood identifiable thm ordinary thm pt cd ingredient prove measurable indicator function I subset wasserstein ball write highlight hellinger metric fix totally bound bounded exist condition need loss lack capture packing ball pack bound exploit hellinger information wasserstein test interesting control pack wasserstein ball quantity arise addition pack wasserstein ball whose hellinger lemma latter packing appear analysis lemma provide core establish general contraction theorem concern quantify estimate entropy metric several kullback likelihood leibler hellinger characterization finitely identifiable constant w use function useful handle nonconvex follow family identifiable tend sequence g g probability statement eq theorem augment deduce obtain move class cover bound n kp km bound finite mixture process let euclidean metric kp need kp g bound distance away hold family gaussian density contraction wasserstein kp kp kp assumption atom must g g I let condition trivially provide w g w w assumption bound n identifiability nc bound j large assumption specify bound arbitrarily constant ensuring metric logarithmic univariate g kp kk probability base pack metric dirichlet g rd dp denote ball pack take gamma center dirichlet gm ig support contain measure couple I inequality desire place density constant kf kp p specify g specifically ordinary likelihood likelihood take main first prove condition eqn kp handle ds packing hold cn trivially turn lem lem e hellinger previous verify note respect
initialize eq run budget update reward number totally decrease correct simulate always amount theoretically budget outcome transform constrain price call equation price also x total budget independent outcome eq appendix determine price observe cost everything one monotonically x kf condition price function satisfie inside function least budget unique budget define constant logistic budget could find computing employ price potentially double f nf c converge experimentally usually step number converge use equilibrium happen instance I budget simplify market use method non guarantee exist degree flexibility choosing function way choose logistic regressor market constant market constant equilibrium equilibrium aggregation eq artificial aggregation forest aggregate aggregation however adaboost forest aggregation aggregate forest prediction market constant participant also analytic form budget classifier online aggregation variant model market cx c b x x resemble logistic factor sure constraint observe multiply market thus participant example u x x contract price classifier price give verify reasoning carry market train rbf kernel online boundary rbf kernels participant market likelihood obtain online budget prove artificial market perform maximum version likelihood total batch present market sequentially constant market ascent incremental update maximize ascent ignore dependence gradient market approximate find constrain differ step divergence carlo case much desire maximize weight exactly approximately eq overfitte important market market optimal market participant conclusion artificial general market market representative prediction market capable available market participant train usually reason aggregate could perform instance performance subset classifier train behavior prediction market aggregate classifier participant contribute specialized opinion specialize triangular triangle negative market participant three participant negative outside three lie outside triangle high budget fall triangle budget classifier agreement evaluate positive negative market obtain many construct specialized depend language specialized classifier propose specialized leave decision leaf domain rule obtain linear verify aggregation participant currently different aggregation classifier idea economic bring classifier economic economic market aim market estimator class entire budget participant win outcome market information fusion artificial mechanism odd take strong updating odd instead odd allow price allow relate market market analytical price formulation allow maximum finally evaluate predict class evidence online try adapt observation instead solve regard market solve implicit criterion asymptotically observe market misclassification implicit type reject instead reject aggregated decide roc reject could market define overall reject outside detect rejection instance overall desire classifier market participant leaf simplify take decide relatively would classifier market participant overlap define generic instance type leave however specialized bid discuss artificial implicit online adaboost two mechanism participant market sequentially call maker predefine scoring machine deal participant price utility form price mean participant belief experimental market adaboost implicit online artificial market section market leave first market participant forest specialized leave initialize market market binary market price equal validation bootstrap train market aggregation capability market market simplify market address edu investigate market term repository dataset market incremental update divide thus behavior incremental unless otherwise ht average run see incremental similarly incremental preferred less handle market test incremental market bad experiment probability frequency distribution direction desire bayes error training analytically bayes mm market compare practice approximate market setup error relative forest market obtain forest estimator significantly value misclassification four predict correct difference figure forest see market well error behave conduct uci repository number meta depend take validation market market cb lb ab forest c cb lb ab breast cancer letter recognition voting record balance car connect band hill vs hill tree split name rf rf random use specialized participant market cb lb column respectively market dataset respectively significant pair market rf implementation bad market significantly rf ab significantly outperform outperform implicit minimize update bregman divergence convex function minimize learn rate bregman unconstrained minimizer bregman use market price truth euclidean distance bregman valid update implicit cb cb rf offline breast recognition diabetes voting car connect hill vs vs gene table permutation fold cb epoch cb offline offline epochs cross validate different pair bad implementation implicit cb cb implicit perform offline cb cb offline aggregation capability detection position adaboost construct classifier split participant observation fall participant feature initialize namely budget market initialize adaboost constant bin index much small negative sum weight weight negative adaboost node dataset ct region node inside manual solid segmentation segmentation solid show training test false positive acceptable false training epoch epoch gradually roc adaboost market epoch detection positive per volume improve constant market difference pair present theory artificial market purpose supervise learn conditional novel learn easily certain artificial market inspire life specialized subset comparison real market usually forest implicit online learn artificial market simple update binary classifier train tree meaningful instance manifold market involve face classifier etc involve orientation hypothesis evaluate participant decide combine approach observation currently extend artificial market extension market include future specialized participant adaboost tree cluster could instance specialized acknowledgment thank innovation market grant total hold divide budget exist proof strictly mean assumption thus kn kn k kn kn kn kn price become
section main inference sample depend partition ensure act preferred direction configuration field enough configuration align along direction pattern field model implicitly note positive analyze configuration distribution define scope pattern pattern encode priori infer actually leave coupling example attractive pattern invariance entail infer pattern statistically model pattern statistically distinct matrix define distinct pattern invariance generator one set break convenient throughout paper dot attractive vanish use inference impose amplitude three interaction spin correlation call normalize unity introduce reverse small guarantee respectively small unity index comprise integer range maximize hard introduce scheme derivation expression attractive pattern pattern calculate field aspect coincide presence secondly dependent unity effect easy pattern vanish must low pattern closely inverse mf interaction interaction contribution value depend generalize overfitte avoid consider locally likely expression matrix substitution fluctuation field bar calculate pattern tell instance bar statistically compatible zero formula expect bar rescale eigenvector orthogonal likely arc radius drop geometric criterion reader calculation attractive rescale positive orthogonal rescale virtue likely value indeed zero surely vanish hold variable vanish square represent difference expect norm non zero straightforwardly formula angle fig pattern pattern number uniform break constrain amplitude introduce gaussian presence effective strength term particularly severe transform unity regularization eigenvalue criterion attractive q correction identical first correction find note correction pattern interaction mode interact overlap priori vanish correction projection extract finite amplitude ratio exceed critical refer discover learning compare order order low formula place pattern configuration order pattern vanish order component low formulae first correction require minimal formulae field synthetic various ise attractive pattern specify later component field ensure weak intensity average simple ise vanish connectivity spin correlation estimate equilibrium monte carlo consequence perfect depend spin configuration block growing allow systematic formulae depend amplitude pattern dense hard attractive inference ambiguity infer energy continuous pattern show low eqn sample align along uncorrelated size direction conversely likely align pattern small suffice systematically systematic study inference bottom configuration deviation true infer bar calculate unity pattern pattern deviation amplitude configuration generate pattern due invariance fig size amplitude decrease block ten value error error twice large confirm limit correction j low formulae priori ten block circle square respectively eigenvalue low coupling find contribute coupling dash configuration eigenvector spin penalize line small model compare infer configuration standard low pattern pattern generalize bar true pattern infer bar infer pattern compatible zero discrepancy quantity calculate error bar fig generate extension sampling close correctly confirm interaction infer calculated pattern moderately fig middle pattern add excellent sampling allow inference fig surprisingly systematically correct account correction enough vs dash unity right vs small unity attractive pattern retained differ mf correction test perfect error infer fig section expansion next correction test field zero instance orthogonal pattern simply accord error average four large formulae formulae pattern circle correction test corrupted noise compare component low formulae strong pattern strong relative infer formula order decrease decrease infer correction sample improvement low weak carry expect configuration correction effective full low circle formulae black formula bar small perfect fig correction c vanish second correction correction qualitatively low infer close correction slightly apply biological boltzmann learning procedure top number come f consist wave precede pattern analyze activity binary calculate correlation attractive hereafter one expansion base use select agreement close rather mf clear interaction correction sufficient change sensible next analyze commonly encounter domain bind protein extract interaction sampling site frequency multi briefly speak pca site poorly representation material amount alignment otherwise keep track contain alignment infer pattern recover agreement good attractive number also calculate discard precisely distribution keep red site result interaction denote compare attractive interesting whether affected remain site differ account chain interaction go site nevertheless site see fig strongly retain weight middle panel correspond optimal intend derivation maximize respect minimize appear energy configuration configuration reasonable mechanic systematic expansion entropy power calculation explain formulae reason clear make hereafter pseudo value comprise hereafter infer energy obviously identity fulfil energy square spin dominant maximizing following expand cosine power expansion cosine exponential nx expand cosine low partition low approximation cross accord cosine entropy order entropy magnitude system typical infer order correction straightforwardly generalize pattern pattern model dynamic adequate interaction fix system configuration align along field vector determine along field pattern root saddle solution one saddle point vanish configuration mainly determine behaviour locally stable eigenvalue matrix correspond zero correspond calculate field one affected phase system happen equation equal equal contribution illustration field opposite latter give contribution infer minimization correction coincide value identity condition fulfil limit pattern diagonal vanish cross subspace large square vector large attractive pattern obtain notice pattern unity keep scale accord couple pattern assume add quantity surprisingly add fit determine center attractive denote leading coupling give much small probability inverse mode particularly correlation covariance fluctuation report cross vanish find assume determined come bayesian bic decrease pattern add cost decrease value eventually balance depend correlation need represent bic mathematically justify compare always datum hereafter consideration fluctuation non vanish quantity marginal sum run respectively integral integral dominate contribution root repeat define infer onto eigenvector directly maximize fluctuation eigenvector square maximize vanish mean coefficient fluctuation angle reliable picture draw expression square naturally look correction low order expression first cross low cross entropy inverse hessian calculation correction eqn shift vanish dominant determine draw independently equilibrium pattern limit expect self averaging depend want fast decay infer specific great analytical hard treat field vanish pattern infer measure unique physics zero phase e pattern number reliable opposite reconstruct pattern make arise pattern restrict spin map pattern play role dual spin configuration spin configuration dual dual decay exponentially theoretical entropy infer pattern spin pattern back find overlap configuration overlap random state opposite limit keep interact applie vanish remarkably pattern infer pattern scale denote self e replica result detail solution entropy analytical simulation small evaluate function enumeration entropy one compatible analytical existence effects b reach critical range concentrate pattern existence meaningful phenomenon discover unsupervise replica break negative fig nevertheless may entropy decay ij suggest identical dual semidefinite calculation replica break extended pattern regime remain qualitatively unchanged critical high result sufficient context connection section admit solution spin decrease sufficient zero simply coincide know consist single pattern component readily calculate q encountered send break avoid state suffice come ideal perfect result inverting pattern affect correlation contribution boost large eigenvalue absence pattern eigenvector component ml perfectly recover presence sampling finite corrupt contribution physics mathematic phase check coincide transition regime large eigenvector uncorrelated identical correlation whose eigenvalue mp continuous component mp relate noise large eigenvalue finite converge square analytical formula analytical transform square rescaled identity fluctuation rescale pattern onto illustrate pattern fig formula eigenvalue correlation eigenvalue spectrum ratio spectrum eigenvalue mp agreement formulae pattern overlap vanish repeat find entropy strong pattern accord agree discussion pattern state fluctuation discrepancy correlation entry dominate contribution informally correlation accord pseudo infer field value component equal pattern tangent clearly discrepancy nice illustration correction recall large presence small ratio unity correction quality infer study value generalize interaction encode set technique systematic calculate estimator field variety regime low component large correction validate expression synthetic pattern ise criterion compare study depend sample configuration elementary infer pattern pick sampling scale several consequence exceed secondly large confirm present intrinsic calculation cross inference local suffer study particularly e inference know modify first correction linear number possible
field similarly index operator index correspond axis fields result recursively give formula causal would could utilize equation ultimately algorithm determine much field section give additional field model exact four technique mixture leibl distributional behavior extend field expand fairly specialized adapt vector treatment use length regressor grid lx thing compactly matrix various density field dft field call frequency also dft center frequency proportional dft satisfie unbiased estimate emphasize require assess field treatment differ fourier frequency shrink asymptotic expand domain define term explicitly regression parameter result toeplitz section careful toeplitz approximation assess kullback discrepancy series kl depth treatment field define convenient effect parameter order utilize spectrum assess proximity guarantee minima true cf write kl call minimize assume nontrivial spatial time essentially corner separate corner volume ratio zero corner increasingly dominate lag effect f prefer utilize dft positive frequency although type kind flat discussion bias spectral lag window modify bias fast utilize dft approximate obtain integral kl mesh dft minimized practice accord convenience convenient fourier handle formula prove supplementary propose minimize depend formula although exact log maximize square standard argument q calculate replacement field seem utilize objective effort inversion together estimate proportional recall denote use iterative computed update either exact update iterate approximate biased consistency provide rigorous treatment formal auxiliary datum recall equal limit theorem result expand unique pseudo exact asymptotically mean v h assume fourth expression involve fourth spectral estimate theoretical refine hypothesis correctly exposition misspecification asymptotic likelihood specify parameter fisher particularly elegant case equal twice identity mirror symmetry except equal except lack efficiency building procedure low order jointly coefficient might zero negligible large likelihood ratio utilize ultimately asymptotic parameter residual covariance behave examine residual equivalent correlation hypothesis absence autocorrelation comprehensive regard limitation li supplementary evaluate goodness fit statistic spatial model rigorous process fair theorem gaussian assumption list asymptotic lattice asymptotic paper like together utilize broad process formulate treat marginal moreover provide condition iii highlight frequentist contribution concentration assume likelihood work toeplitz spectral block integrate first entry z w stationary produce structure admissible field consider sum h h f f z spectra minor derivation spatial write stationarity upon difference index sample spectrum spatial field require absolute fourth spectrum via dot bivariate impose product condition regressor specify write correctly adjust mean assume regressor require follow spectral twice continuously differentiable process pseudo exist interior must field model skew index argument extension argument see argument decay boundedness partial see move condition problematic move distinct independently distinct entry interior convexity kl bayesian identifiable imply ensure parameter subset euclidean lie interior accomplish transformation easily accomplish coefficient important extend spatial toeplitz limit average extension quantity g hold continuous ii estimator bias give correction assertion require extend series generalization dimension seem mechanic considerably theorem estimate exact hold regressor n normal l v f denote transpose independent zero process simple application series yield normality condition focus skewed gaussian specify identifiable worth automatic make automatic furthermore together model automatically entail asymptotic normality frequentist result concentration seek estimate exist away infinity restrict range effectiveness estimation outline accord calibrate size constitute grid example denote mean asymptotic therefore asymptotic place c c ii column experiment response column equally space follow effect analysis supplement simulate multivariate simulation circular though account log simulated square model high agree provide difference standard parameter increase go mean finally average great simulation parameter increase extend direction recursive formula setting extremely notable establish consistency normality rigorous platform independent expand simulation supplement readily able acknowledgment associate anonymous provide comment substantially author I statistic interested encourage statistical census supplementary material section nsf nsf grant nsf census central role spatially correlate significant broad paper easy computation comprehensive likelihood field parameter theoretical generally facilitate building greatly inference refine model result yield spatial environmental science area spatial broadly mainly field process spatial among comprehensive stein therein grow spatial stationary impose hold inverse covariance hold generate upon impose structure demand generate define ensure way specify field variance appeal contrast move average identifiability present information hypothesis briefly estimate comprehensive field formula field move field likelihood guarantee move field identifiable without impose quasi field parameter expand particular frequentist propose inversion primary express axis furthermore field spectrum nonnegative value
least determinant class generate st list usual computational switching take start column switch dual remark dual dual matrix walk eight class size find size list mark least exclude class total assume additional sample estimate imply view two cause inaccurate estimate interesting know small st reader bipartite acknowledgement author program hadamard visualization mathematical sciences square case hadamard maximal maximal case maximal gram hadamard class solution give switch optimal hadamard maximal optimal concerned determinant change hadamard hadamard hadamard order hadamard say sign say self say hadamard equivalence class self equivalently class know conjecture interest hadamard determinant order certain candidate generate generally hadamard switch induce switch new order odd order eq due show sharp sharp bind complicated sharp order sharp odd odd may gram via determinant gram gram gram want row row gram involve find solution satisfy omit merely reduce possible search use gram gram matrix determinant generality assume search regard searching level correspond search typically search child leave deterministic searching solution exist decompose may better small search per experimentally unlikely find node child recursively child gram due give determinant determinant maximal fail hour explore reach depth size way strategy matrix determinant sample lack uniformity introduce uniformity advantage uniformly operation determinant hadamard equivalence hadamard equivalence one idea introduce close possibility switch switch close row sign interpretation bipartite preserve product preserve preserve preserve switch four class equivalence switch case equivalence denote equivalence exhaustive know two sum preserve switching normalise ht st consist split yes yes new st dual former case say st split st self st know label paper compose associate st order order occur list least determinant lie least switch website give st generators implement find class generator generator determinant upper plausible maximal improve despite optimisation technique prove technique order search determinant gram matrix st st class gram follow ht st website st class class switch generator generator table class split two c c split ht h small class class order order c candidate gram product thus plausible conjecture difficult reason h determinant even prove incorrect technique class determinant start matrix find duality number start iterate row find program explore switch walk walks switch row column switch connect walk intersect size expect walk intersect probably geometry store check vertex
short set system since quasi order every jt operation prove countable ordinal rr q smooth u tf tt b barrier contradict immediate useful develop language alphabet concatenation sequence sequence concatenation infinitely iterate concatenation closure language closure type require linearization quasi order linearization linearization linearly characterization say say provide infinite conjunction preserve operation converse assume infinite contradict n among set nonempty px I infinite immediate inclusion subset finite principal inclusion quasi order assertion order satisfy condition hence fact assertion since assertion partially aid scientific education technology thank anonymous thank go definition learn teacher abstract independent order tree system theory nash theory quasi introduce system correspond monotone inductive unbounded pattern language system member viewpoint characterize member system inclusion continuous similar positive linearization unbounde target theory quasi order neither employ algebra sometimes employ learnable study lemma consider nonempty member study learnable employ unbounded restrict motivated systematic learnable set teacher datum learner abstract type family learnable positive short consist construction inverse quasi order maximal define order ordinal nonempty discrete represent say continuous identify function many binary unique topological class series title paper enjoy useful continuous language operator closure application induce speak transform correspondence quasi set order image useful investigate language supremum conjecture proposition investigate order type type study e decision net logic characterization definition order upper ordering viewpoint system number analogue review order short closure topological relation infinite closure please hereafter ordinal identify integer denote resp identify exist sequence type eq barrier ordinal quasi order say barrier countable ordinal barrier define countable explain barrier exist small element less ordinal barrier ordinal countable ordinal order barrier countable ordinal barrier barrier quasi study net verification countable
speedup matrix aim successful particular infer internal black box know measurement uncertainty exploit underlie statistical acquire accuracy cost especially roughly since relative map appear thank anonymous comment manuscript imply sufficiently eq average lin tt diagonal routine necessity especially image use improve cost generalize wiener field significantly estimate precise functional exploit autocorrelation develop show situation imagine might necessary engineering give computer act operation probe might infer due internal reconstruction like multiplication rotation solver etc output mathematically linear perform space paper interested input identify g correspondence output vector box operator box scheme probe sequentially nonzero certain pixel repeat read expensive reconstruction deal sequentially always prohibitive accurate feed box output input component find efficiency obtain hadamard improve calculation likelihood extensive corrupted investigate scheme come price result uncertainty completely measurement achieve long computationally expensive spatial pixel etc beyond average generalization wiener determine suitable amount cpu respectively accuracy examine signal reconstruction enable computation marginally prohibitive remainder review sec sec propose derive verify example real sec covariance equal hold zero field define space transpose understood elsewhere refer grid voxel entry square uncertainty effective reconstruction tool filter review review far emphasize also signal eq linear maps ref thing forward signal scenario measurement wiener assume stand respectively gaussians solely simplify mean often forward datum r pn r actual estimating covariance encode result filter straightforward read field signal covariance need covariance spectra project band spherical degree spectrum might whereas often isotropic wiener ref assume alternatively derive logarithm degree freedom spectral band characterize option former derive classical contrast exceed line mean equally probable component sophisticated ref number trace estimation ii regardless choice estimate eq want period residual latter law applicable variety want bayesian exploit knowledge interested confusion quantity appear marked form signal matrix element originally point ref treat separately requirement estimator matrix square underlie generic formula base sec state diagonal adequate suitable since entry considerably already estimator trace distribute diagonal chosen eqs covariance require sec show covariance present ref classical filter equation iteratively compute accord ignore guess spectrum step extreme limit give nontrivial whereas generally exhibit nevertheless present contribute marginally accuracy therefore calculation numerical pose map maximal computation perform open system package estimate estimate dotted starting dash nine case explicitly ensure matrix e sphere norm respectively efficiently expensive computational complex sensible number argue however converge couple pure time amount fig evident estimator reach progress datum finitely diagonal realistic characterize two represent smoothly h smoothing spectrum qualitatively overall gain decrease diagonal rough estimate calculation ten estimator random vector
value function evaluation reproduce throughout semi product duality mapping banach bound operator operator denote great linear operator language value belong exist together semi inner respect functional operator requirement uniqueness satisfy coincide usual kernel reproducing explore reproduce kernel investigation banach bilinear prof cauchy inner duality mapping eq recall mapping bound adjoint use inequality second immediately q property kx bf reproduce enjoy reproduce rkh difference semi firstly additive reproduce pairwise distinct q although still reproduce sampling exceed give vector map converge converge suppose converge get pointwise bound introduce notion adjoint banach banach identify indeed position characterization reproducing value reproduce banach define reproduce reproducing fulfil shall compose eq impose also banach function moreover eq evaluation remain reproduce compatible call banach theorem pair map value banach space endow compatible value give always scalar input value linear complex inner product reproduce via choice q satisfie value show reproduce satisfy sake finite banach compatible inner completeness reproduce first space respectively adjoint corollary form equation span end vector linear get reduce search matrix reproduce value sampling exceed appropriately purpose conclusion value section sensing consists consider find banach banach simple example always imply wise th convex norm uniformly reproduce compatible duality equation translation namely scalar reproducing value banach determine subsection class invariant banach equip space endowed bilinear banach nf l r old notice continuous eq adjoint hence compatible inner value translation reproduce dual product bilinear form equation remark endow gaussian rkhs confirm validity application vector unknown point observation could shall follow regularization methodology form topic form regularization output one choice positive tolerance remain converge regularizer admissible continuous regularizer least minimizer value case topology equivalent continuity topology equivalent norm dimensional admissible deal loss form respect admissible set weakly inner uniformly minimizer uniqueness minimizer subsection follow proposition value reproduce case subject cite therein establish closely relate minimal interpolation start examine fix sampling function interpolation notation theorem subset functional vanish u nonempty good origin known banach characterization approximation banach complete main without effort regularizer satisfy increase let follow satisfy lemma thus strictly consider solve regularize try model parameter substitute convert original many part usually generally additive characterization together constitute system stand function continuously differentiable convention next dimension continuously minimizer product imply suppose point linearly minimizer equivalent linearly independent similarly characterization equation minimization due respect finite basis reformulate leave minimization nonlinear cm example accomplish author zhang banach space propose notion reproduce kernel banach basic property construction concrete especially minimizer scheme value reproduce banach feature map regularize equation establish kernel banach demonstrate rkh machine target vector reference learn task mathematical foundation find framework function output hilbert constitute special banach space space reach banach variety structure norm intrinsic make embed hilbert base hilbert space finally banach banach banach mainly lie banach functional representation discover extend type argument banach substitute product banach illustrative theory basis banach inner product lee study margin hyperplane banach value reproduce banach space scheme theory attempt build mathematical multi banach shall map representation concrete respectively investigate concerned shall space deal banach inner product banach linearity positivity conjugate homogeneity variable inner say compatible banach inner cauchy linear dual functional value semi might role banach space instance q endow
unit slope simulation versus likelihood generation draws consider fast draw increase significance via number draw take model estimate parameter goodness statistic via use square converge exact estimate goodness fit statistic via likelihood generation root converge far fast draw via versus limit goodness via use generation draw converge ratio possible accelerate via asymptotic acceleration four discrete parameterized bin meaning draw suppose underlying experiment significance level repeat calculate realization parameter level less limit repetition experiment reason approximation remark underlie repeating calculate hypothesis distribution somewhat scheme many favorable property goodness form pearson statistic division often lead serious even present illustrate via numerous fortunately availability computer avoid easy problematic division take goodness statistic interpretable black program rapidly calculate identically may consist either parameterize take value finite countable accordance cell common whether I draw statistic statistic bins root difference distribution draw arise root confident quantify confident give square come significance level define significance concern term significance alternative term distribution pearson replace weighted average bin classic however weighted division divide power error divide nearly arise every bin main thesis article use classic statistic long computer root mean square conjunction ratio goodness generally preferable ratio division somewhat fact kolmogorov variant example powerful root certain circumstance discrete kolmogorov complementary square largely compute confidence root number draw trivial via maximum square please report present exact level recommend complementary multiply carry sure subtle bin much problem yet black result goodness fit careful make many advantage substantial draw least every bin subsection please treat terminology old concept long computer technology table fit significance compute really reject threshold significance instead significance probably correct whether large fluctuation outside science typically exactly test validity purpose due contrary goodness even suppose exactly remarkably original article statistical arise set testing association independence contingency cross two review goodness hellinger good member read divergence family bin associate respective bin denote root statistic root pearson standard likelihood common refer hellinger distance define draw substantially analyse monte simulation rely draw discuss testing parameterize family draw actual alternative draw nuisance unfortunately devise distribution none standard likelihood hellinger significance number especially useful draw realization experiment measure repetition repetition course testing seem feasible probability bin typically fit test concern define please mind section nuisance always repetition amount statistic significance level significance assume seem experimental yield combination possibility parameter contain permutation maximum entail sort frequency need freedom monte draw consideration run draw obtain estimate statistic mean square step take simulation level fraction calculate conduct detail procedure work draw measure vector root distance statistic entropy divergence draw respective bin draw necessarily equal occur regard leave probability random observe confidence error monte conduct produce estimate produce statistic significance level produce statistic great corresponding binomial standard great standard fraction monte estimate calculate investigate goodness root perform discuss briefly significance remark accuracy fit toy example q q draw bin unlikely arise like exactly goodness fit discrepancy whether computed monte evaluate take root square extremely find evidence level statistic realization classical classical arise large reject classical statistic display unlikely even suggest sure arbitrarily significance irrelevant bin report statistic contrast square reliably classic behave agree bin bin unlikely arise like various statistic discrepancy plot whether arise monte draw model square classical little evidence model agree draw bin draw bin law analyze datum reference page present subsection goodness english repetition word choose corpus bin plot frequency sort statistic significance hypothesis integer sort first bin contain great bin choose great draw bin contain great remain bin sort proper proper bin plot significance must appear word display significance goodness fit goodness bin word clear priori dictionary appropriate significance root mean several digit classic contrast classical depend know proper produce result report english word repetition word obtain bin frequency sort root unlike statistic discrepancy nan hypothesis seem fortunately essentially irrelevant indeed mean sensitive bin potentially inaccurate inaccurate root whose refer logarithm unlike mention facilitate good relative world monte root square report model match american report various rate physical generating score health significant simulation symmetry square note reveal significant issue symmetry could significance level simulation testing root square calculate table powerful table sensitive model sensitivity bin recommend ccccc good poor excellent good fair poor cc ccccc fair poor excellent fair cc ccccc excellent good fair excellent specie exclude readily identify specie collect via number specie sort appropriate must bin sort subsection sort less likelihood sort please carefully truncate common level monte carlo simulation root square discrepancy datum substantial draw solely log ratio unable detect discrepancy determine discrepancy full include incorporate analyze report detailed report unlikely sample belong order figure bin sort permutation furthermore test alone tail parameter bin contribute goodness fit aside summarize parameter sort real specifying geometric model seem sort frequency great number divide fit remain maximum sense permutation us sort order allow ignore great fit bin fit plot number sort well fit distribution associate great infinitely bin provide aggregating draw calculation goodness fit without draw computing simulation parameter infinitely carlo simulations root figure empirical model draw fluctuation detect section draw detect parameterize quantify success detect draw actual draw distribution compute accord significance level simulation draw say mean confidence subsection principle maximum fit generating plot would level calculated three remark much approximate level parameter subsection remark exactly equivalent confidence time approximation please remark statistic statistic associate square distribution associate equivalent variation eq plot actual arise meaning yield confidence level I root successful fail simulation actual distribution mean number statistic draw require increasingly bin draw figure plot draw actual specifie require increasingly exhibit opposite let specify q draw distinguish distribution define specify root mean uniformly us distribution draw distribution specifie mean distinguish classical square specify figure plot draw required remark distinguish begin present specify truncate poisson draw plot distribution define draw require distinguish distribution specify next subsection time estimation parameter see specify distribution truncate consider several maximum likelihood specifie distinguish specify maximum consider actual parameter specify distinguish specify truncate poisson parameter consider several clearly distinguish likelihood remark distinguish consider q q require distinguish maximum remark
coordinate cut dag construct schedule could obtain graph cut normalization b theorem edu posteriori submodular structured function find cut propagation mp energy mp suboptimal fix scheduling mp cut apparent showing scheduling point mp schedule proper graphical vision biology algorithm solve empirical choose however clearly outperform compete algorithm cut max belief propagation mp precise mp graph cut mp map analogous scheduling mp cut pass message bottleneck capacity let connect decode strong statement mp submodular energy mp show suboptimal submodular wrong solution depth implicit previous scheduling converge bad point give characterization always exist fix energy point mp fix alternatively consequence product tight binary submodular energy point proof believe fix novel due come run ordinary algorithmic degree scheduling construction suboptimal bad avoid inference distinct improve scheduling insight cut maximize assignment energy sake exposition choose present energy energy submodular graph variable potential whose value ij represent potential form energy express thorough discussion notation energy polynomial flow cut computer appropriately construct know combinatorial cut machine computer reference therein convert weighted direct source add edge map direction initial create terminal terminal terminal terminal energy function every terminal graph cut optimization back repeatedly flow equivalently hereafter maintain capacity amount edge opposite edge residual capacity residual minimum path phase path e connect determine source respectively tb unary representation capacity effectively flow energy unary potential pairwise potential unary potential optimum set potential flow identity ensure flow make optimal coefficient potential path phase component path directly positive residual determine directly terminal capacity possibly connect label either within label practice terminal typically strict belief strict algorithm map structure employ algorithm min sum update simple structured energy mp update beliefs b ix ib x variable quantity strict mp asynchronous use mp algorithm formally family belief mp message passing arbitrarily possibly order possibly manner long point strict mp broad exclude fundamentally program reweighte max scheduling include string work scheduling lead improve detail mp cluster definition mp variant scheduling dynamically node flow set potential entry potential useful sequel terminal potential edge potential two factor graph edge potential graph map jk jk connection graph begin energy function balance energy path path energy decompose structure important balanced energy subsequent section convergence suitably energy close linear time rely equivalence cut section energy one path function along message fig balanced compose energy section scheduling cut phase return pass hold message message pass standard apply leave unary message propagate across particularly way leave pairwise factor message terminate strict mp convention term unary chain increment desirable accounting accomplish first unary path constrain increment backward direction increment forward receive increment pass receive similarly direction nx must n else impossible backward message calculate message unary constraint choose capacity unary factor bottleneck ensure message increment propagate receive increment propagate increment path ft ft want unary factor increment message capacity unary potential unary message yield never denominator non iteration approach leave unchanged see message increment propagate chain analogous dynamic message increment backward later schedule combination edge contain residual path minimize potential specifically end iteration residual terminal ij ji cut formulation terminal construct potential unary iteration schedule return path bottleneck implementation residual cut execution since cut finds run finding routine message potential find bottleneck reconstruct note direct cut execution perform residual formulation ij f ij f ji ij ij fact affect contribution c ji ij clear relationship value residual quantity bottleneck capacity find capacity pass pass message strict mp continue reach mp essential reasonable scheduling message phase assume variable receive incoming message neighbor ix variable message increment normalize next condition factor propagate preserve factor message incoming factor ij plug message material jx jx lemma allow message pass execution belief structure unary belief normalization unary unary without normalization propagate factor propagate message exist message value preserve maintain message change product version message use compute parameterization message parameterization cut tree equivalently view mp cluster energy change potential current potential potential product message energy component far remainder message everything else unary potential assign potential component use unary message leave ix remainder potential remainder unary potential ix ix message energy analyze belief parameterization unary pairwise belief consider message message consider potential potential adjacent belief change change could depend variable belief forward potential parametrization structure pairwise begin iteration iteration apply supplementary material detail unary group message leave add end phase unary leaving mean unary change x x unary potential involve total change parameterization pass chain correspond equal flow unary ix b put change exactly phase phase tb potential message pairwise strong draw message give max cut construct product decompose term u message graph modify discuss modify potential change else execution change tt modify potential subroutine schedule path bottleneck capacity pass modify potential maintain see material amount entry vice mp forward backward terminate modify potential every less quantity maintaining cut unary pairwise begin schedule increment increment analogue message reduce negative change unary potential maintain equivalent potential potential vice u ij ij modify potential ft rt ft interpretation begin potential reduce pass backward next message equivalently return send equal case previous run mp holding begin unary potential message pass along pass run mp equivalent easy analyze cut tb arbitrary message normalize belief view edge equality potential potential ij ij involve leave variable message equal incoming message explain fig message edge mp message hold calibrate calibrate calibrate potential pass along receive factor plus receive left factor increment change propagate variable sum incoming message neighboring potential message propagate leave illustrate fig inductive message actual send exact propagate message unary clearly message plus increment increment reduce backward along chain message right neighboring factor essentially lemma message send send backward calibrate reduce section pass forward calibrate potential edge potential base message produce lemma unary belief computing belief increment match message increment belief end mp path edge calibrate follow edge unchanged mp normalization message potential edge iteration additional interpretation working potential specify schedule incorporate potential careful ensure pass lead graph new message free one potential message perform equivalent product tree representation view tree potential max product even message completeness supplementary citation conjunction q message initially edge clear message compute min could initially identity alternative parametrization add one message discard back define edge however mp message thus express parametrization potential recall path increment parametrization group message treat potential justify division suppose message mp converge tu full message step initialize message message initialization pass reach iteration u purpose know guarantee parametrization relative potential leave difference end proceed perform message divide potential potential change potential neighboring could possibly affect since belief standard normalization change message potential case parametrization message restrict also final belief change parametrization potential parametrization pass message backward parametrization message iteration unary b ij know unary belief potential message lemma potential minus reduce computation variable update parametrization initial ij complete proof change parametrization parametrization pass forward parametrization cut difference pass cut simply change cut analysis throughout apply equivalent end phase equivalently work potential presentation simple path unary potential unary capacity cut first use run strict optimal fix existence mp binary constructive rely define term connect belief message amongst vice inside unary illustration homogeneous consider cross max minimum cut path cut supplementary material lemma mp independently homogeneous zero boundary reach internal convergence iterate homogeneous lead belief unary entirely unary potential connectivity eventually variable mp loop message cycle loop unary add pass message stop acyclic structure strict mp obviously monotonically potential I strength unary loop converge pairwise potential prove point submodular homogeneous belief initial seed cut decode arbitrarily previously suboptimal give assignment mp fix submodular function return submodular energy potential define suboptimal reach scheduling fig however message fig decode belief map pass version cut thus update map algorithm view block section normalize potential constant zero show cut see either cut ascent chain structure assume potential standard form chain canonical unary entry know problem primal optimum bottleneck cut path unary value dual block cut follow execute normalize potential dual absolute unary cut local unary potential behavior
depend mdp nature states reflect much expect action optimal approximated parameterized dot action optimal compose sequence sentence n number sentence make content mdp problem triplet document currently read correspond sentence already currently previously assign read iff read category assign read read transition act process reward stop bring case assign classical accuracy measure otherwise must purpose compare text tf global local read already assign category global feature concatenation read sentence trick project high dimensional inside dependent easy classify carlo amongst states size average performance approach able properly training tune generate initial rl svm robust hyper figure comparable outperform dataset visible metric set small reading behaviour explore big corpus hard properly right histogram group notice document read read clearly capable read less different baseline read deal multi remain sentence particular due large multi learn classify sequentially read document collect enough show tc learn document whole document able behaviour fast system performance baseline set new imagine mdp classify read national research sequential reading sentence sequentially learn stop soon reinforcement learning label corpus propose set read tc act label document label unlabele document tc extensively tc generative discriminant mainly lose compute score look document svms classification binary entire decide category inside little word correlate suit category concentrate additionally case associate entire document inform drawback attempt process et base network use linear svms string development text less approach sequentially read sequential goal relevant sentence label reading document last learn correct category read paper fold sequentially read sentence assign topic additionally reinforcement focus stop read classified useful expensive document popular corpora tc baseline portion ability read sentence classification easy training set task small organized follow formalize detail label tc corpora denote document document training document compose iy assume tc category parameterization reduce provide overview formally present manner propose classification sentence sentence decide read document belong possible category choose stop read document correctly compose begin read contain reliably classify document sentence read classifier classify classifier reading document additional four entirely action pick classify denote action denote see choose good action action high score minimize loss document action document illustrate stop action action set remain action repeat training document model
alternative cluster factor neighborhood one assign member cluster neighborhood rv value map cut influence neighbor away whose neighborhood learn par structure discover par together par involve appropriate par score structure application par algorithm graphical graphical par identical tie form graph par factor tie graphical next overview basic tie detailed reader case fully mle aim observe training maximize interested helpful undirected separately bayesian network consist probability value parent value observe undirected mle calculate close use parameter per expect current estimate equation extend tie done tie share count node share parent analogously equation modify share arise compute statistic database sufficient statistic computation cast discussion one dramatically optimize pseudo pseudo compute multiply reduce train fashion h possibly refinement set last skeleton set build correspond inductive logic view procedure parameterize potentially bias derive one start candidate trivial par proceed exist candidate score candidate carry refinement specific kind incremental par graphical literature several clause across relation learn path appropriately relational relational path crucial aspect relational hypergraph form cluster agglomerative intuitively entity cluster kind structure clause commonly occur pattern densely domain identify perform relational discover start relational perform walk entity thresholded include include clustered time cluster agglomerative via hypergraph analogous cluster long via depth adapt structure area approach structure modify methodology place fail focus discover methodology also follow clause observe way clause transfer goal representation source module source predicate constrain extremely structure target alternative approach general clique template logic clause quantification predicate take clique template possible mechanism template structure direct link extend analog pc causality graphical stage first skeleton second place skeleton consider orientation rule set orientation domain link graphical review par exist relational representation formalism algorithm include inference answer review come decade mix entity noisy hope synthesis idea provide point researcher field would early version ci fellowship nsf computing grant recommendation author reflect view entity type relation entity individual relate via biology frequently model chemical interact social medium user interact page reason word reasoning entity attribute unobserved entity web page improve categorization develop relational friend molecular biology domain researcher interact application learn field growth detailed overview define graphical omit discussion program impose probabilistic interpretation scope able focused discussion exist representation pass illustrate structured define recently available define generic template discuss particular aspect way relational study efficient reasoning uncertain machine setting entity entity iid relational entity type multi attribute entity entity among iid domain uncertain relation entity representation support essential need language dependency entity b reason noisy environment notation terminology rest survey logic describe word rv logical avoid example rv bold letter assign rv letter letter x logical entity x relational focus rest logic flexible expressive kind predicate describe entity entity act quantification g represent g relation category category predicate predicate operate entity convention name predicate capital letter argument call apply atom atom positive conjunction formula quantify follow typical specify understand express positive called contain whereas remain constitute clause variable replace possible type specify constraint present connect ground atom entity assertion friend reason logical parameterized rv parameterized rv alternative relation orient specific entity entity entity attribute commonly orient language refer category whereas author refer set category paper write author note author orient language mean write attribute chain par relational store self relation relational schema attribute entity type entity natural represent statement follow name name selection constraint heavily therefore introduction graphical reader describe store configuration assignment conditionally many redundancy explicitly general tuple factor typically draw consist rv necessary e define conditional assignment computing sum assignment necessary factor generalize graphical represent whose vertex distribution parent via network convert represent product automatically normalization sum network compute maximal argument function compute x feature capture characteristic evaluate clique variety learnable factor clique define advantage discuss bayesian network true hand beneficial separately undirected representation graphical representation structure language graphical second impose interpretation logical allow group convenient representation describe start par short parameterized define terminology analogous generalize treatment regardless undirected par factor consist par parameterized operate evaluate positive par par par specify par graph par together assignment set equation par par popular discuss view par group graphical undirected hybrid list highlight subsection model relational network orient section par par establish par take tuple select constitute thus appear return tuple markov par log tuple arbitrary however share par property language language generalization illustrate document present goal clique assignment document category document document statement clique function incorporate example page page category encourage page et favor clique tractable closely markov specify par logic par rule establish boolean value implicit truth clique potential feature weight human similar people capture order logic rule par par establish clique markov g entity clique par factor constraint implicit variable entity name want constrain rule predicate time suppose know inference infer people entity friend correspond trivially regardless assignment ignore extend predicate hybrid contain conjunction logic similarity logic object orient language par weight whose par par rv interval rule continuous value manner unlike support similarity entity set formula mixed term consider infer document similarity wikipedia attribute state text eq refer entity par rule rule define par assignment par done generalize logic due combined boolean interval rule distance joint assignment arbitrary pick par use functional type sub relation type variable novel link list par template determine par factor template rv par template class template template compatibility entity assign mention template mention def mention entity mention yield mention entity string canonical describe representation consist child par rv parent par equation specify care ensure would cycle restrict par atom parent pa pa implication probabilistic distinction ordinary logical clause clause logical restrict par factor probability pa example copy x dependency example whereas par complete provide combination aspect suppose predict quantity rule represent logical par rv atom par formula correspondence pa implicit evaluate pa formula indicator tuple logical variable combination boolean operation combination slight two formula evaluate sub mean learnable range undirecte discussed specify logical entity name background model take relational perspective orient language relational entity one citation attribute paper reference par attribute go chain specify par pa pa par topic topic direct parent par paper vary aggregate correspond example reasoning perform relation reason presence uncertainty extension deal link uncertainty well link known existence logic specify represent atom dependence par rv parent specify child model entity uniformly capture citation author title cite entity domain advance instead statement type drawn entity task advance standard poisson discuss define either direct undirected advantage graphical direct need express causal hand suit dependency care ensure direct fast typical dependency word parent parameter adjust parent undirecte adjust entire structure undirecte straightforward mechanism par share par rv simply multiply whereas direct require combine aggregation independent cause par causal exploit factor represent conditional normalize efficient direct undirected dependency variable parent immediate neighbor however dependency cycle necessarily coherent marginal dependency similar set lift relational represent child rv parent dependency represent effort undirected model provide advantage direct combine undirected draw analogous graphical type variable refer marginal map posteriori likely unknown variable state overview inference graphical inference strong emphasis place discuss graph model directly efficiently knowledge extent necessary adapt undirected property graph answer need conditionally independent variable briefly refer simple algorithm factor elimination would like particular rv par rv rv call proceed order establish factor contain factor contain multiply effectively remove algorithm heuristic normalize sum proceed contain value gave marginal propagation product bp marginal contain cycle frequently know bp operation send node message reader message send send node message send expression neighbor use message receive message product leaf neighbor message together variable send leaf end incoming message neighbor bp graph sequence iteration correct frequently converge happen typically variable elimination summation product viterbi inference popular randomly carefully pick one iteration iteration condition break ergodicity assignment visit ergodicity violate closely deterministic dependency slice auxiliary slice slice jump region slice sampling derive algorithm slice sampling mc identify set assignment appropriately clause factor select clause mc slice orthogonal concern efficiency care take keep independent one par linear program construct graph introduce program constraint enforce factor value share consistent find f general factor case graph integer rv whose formula simplify replace par whose par rv correspondence formula establish logical effectively rewrite formula convert form carry cast cone rv par rv corresponding correspond include constraint represent description page hard rv rather boolean variable similarity program integer np procedure inference formula convert reality procedure satisfy par rv par rv whereby yet rv relate cut cut plane optimize constraint proceed iteration solution bad case consider meta relational originally memory efficiency maintain formula maintain thus decrease requirement initially set assignment value activate active activate active carry inference technique describe par graph algorithm regardless exclude consideration need case low employ identify ignore take advantage particularly large problem prohibitive memory inference exploit obtain repeat multiple exploit gain technique would repeat perform approach variable elimination significantly extend ordinary obtain entire constrain rv par way result sum rv brief discussion several elimination summarize apply detailed treatment reader excellent introduction example operation auxiliary par par par apply par splitting operation contain par eliminate use essentially together rv eliminate facilitate par rv try inversion elimination completely inversion elimination
c vanish inequality since zero rhs hence general greedy suppose run nd degree false equivalent neighborhood characterize via likelihood generate least loss loss adapt yield threshold initialize break property small symmetric entail result satisfied third derivative dp nd parameter satisfy probability local mle lasso neighborhood analyze regularize mle probability sample greedy regularize impose minimum zero allow broad minimum neighborhood algorithm smoothness condition impose star node fig center node parameterize entry edge induce setup impose regularize entail equivalently regularization parameterized parameter correlation node check induce desire mle require case greedy depend impose fig let check induce graph mle equivalently allow since equivalently unlike previous impose dp figure greedy dp neighborhood suggest figure show neighborhood recover outline simulate achieve optimization relatively operation selection fast calculation entail close simplify single formal zero estimation learn global chain size greedy size structure performance support zero inverse use stop parameter well gaussian discuss implementation al lasso generalized optimally cross validation successfully control graph star illustrate greedy art definition theorem task pattern mean iid recover underlie sparse novel zero matrix series combine intermediate neighborhood principal rigorous consistency pattern surprisingly greedy graphical lasso require smoothness greedy weak extensive compare greedy well high increasingly field considerable dimensional specific vector inverse multivariate set associate set impose concentration require art line gaussian regularize entry result determinant program strong stagewise forward add backward removing greedy appear various community boost function pursuit context greedy show lasso forward general guarantee approach backward greedy iid structure require sufficient condition impose gaussian vector write problem recover determine drop abuse notation element correspond graph graphical include entry degree self vertex symmetric correspond underlie greedy begin gradually add remove basic forward find candidate add meet algorithm terminate
structure estimation term neighbor search separate markov compute empirical exceed threshold number low establish two common set mild assumption dimension graph successful graph complexity positive graph employ theoretic augment graph result various graph length potential consistent estimation establish relationship degree consistently estimate also infinite degree observation liu also guarantee family enyi enyi scale degree average recall sample tree learn parameter regime impose presence sparse parameter require potential certain establish effect path decay graph feasible conditioning ise criterion condition polynomial knowledge ise establish avoid tree ml efficiently liu ml maximum span complexity liu scales exponent liu acyclic maximum algorithm hence enyi graph selection approach approach incorporate term encourage logistic show algorithm incoherence incoherence np hard verify convex search rely propose spirit factor author greedy graph degree restrictive maximum albeit control rate consistent wang graph distribute degree state degree degree define relevant rest paper basic notion alphabet kullback leibler eq similar mutual countable know empirical define use empirical refer conditional testing family accordance undirected say vector mass hold say set ab ga markov positivity state positivity clique clique serve normalize important pairwise model ising take value know potential convention correspond say ise absolute uniformly bound potential ising define distance conditional replace distance ise belong ensemble graph regime grow growth emphasize probability denote distribution give graph draw asymptotically randomness assume every ensemble guarantee notion intuitively vanish ignore graph amenable estimation end characterize figure illustration denote subgraph span addition remove edge respect addition local separate length characterize def ensemble satisfy separation belong scale family satisfy enyi random world graph criterion separation novel graphical express separation width width asymptotic required exploit local separation potential supplementary explicit ensemble graph satisfie involve ensemble ensemble q set lebesgue also characterize specific remove edge supplementary edge potential relate marginally fail potential limit attractive model path material success generic potential potential lebesgue similar assumption direct underlie edge threshold conditional variation distance find conditioning minimum need size conditional inverse dependence increase establish distance decay moreover sample certain sample base variation asymptotic recovery graph eq supplementary material consistently recover graphical tend extend result guarantee family require free case scale alternatively prove hellinger kullback leibler mutual assumption term discrete consider variation analogous also applicable pairwise nature ise recovery uniqueness impose conditional hard complex low statistic graphical relevant high potential outline statistic base statistic similarly variation threshold version concentration bound provide insight ise threshold define guarantee threshold least number scale proof supplementary characterize distinguish variation detailed description family depend degree choose world subgraph corollary edge potential suitably pac majority ise logarithmic class deterministic local property neighborhood separate separation threshold potential also paper relax sequence maximum grow end structural bound path describe distinct disjoint local separation henceforth local word overlap cycle ensemble local path property length short satisfie separation property ensemble lead bipartite graph efficient propose short neighborhood cycle term locally graph node pair pair locally grow albeit tractable precise later supplementary enyi equivalently overlap scale free law relationship supplementary along ensemble regular denote ensemble cycle require graph supplementary establish simplifie obtain degree threshold fairly threshold incorporate ensemble represent natural node path threshold consistent degree learn path tree accordance successful recall liu enyi two occur property establish degree ensemble potential threshold requirement simplify practice separation former short cycle constant latter cycle degree short cycle property augment chapter graph union short cycle graph small graph cycle enyi local hybrid section supplementary grid ensemble threshold world p consistency small potential entail minimum sample provide threshold section improve world bound sample sparse graph regime complexity input assume ensemble without characterization bounded tailor exploit property underlie complexity incoherence algorithm guarantee improve sample enyi world random degree large strong moreover method selection logistic practice equip success np condition relate imply vice versa appear weak incoherence enyi require incoherence enyi similar weak wide parameter r enyi world algorithms ise model enyi graph previously bound loose ensemble enyi average degree sample enyi necessary estimation find supplementary material along line theorem remove converse depend sample expand albeit involve converse instead converse provide supplementary material thus higher intuitively grow learn converse tends merely converse maximum degree exist result mass enyi graph dependent field outline na I meaningful since realize another identify graph cardinality large lie condition probability typical tend typical set almost recovery belong various ensemble bind model markov family give necessary family follow ensemble g k ensemble local degree k material family family necessary ensemble slightly substitute graph characterize necessary synthetic implement variation different regularize logistic compare matlab regularize regression package http www di fr software www ac groups synthetic implement ise software order ise cycle enyi degree size penalize er cycle er cycle er er
categorization object vision web multiclass example later accurately classify input object feature many rely value predictor show linearity require recent year sparse sparse hardness norm compressed rely greedy g boost pursuit processing maintain class since class overall number possible vector row single equivalent non prove zero grows linearly demonstrate method selection approach column generalize rely object classify example image predefine feature patch equal inner learn object predictor parametrize follow map score maximal element maximizer break zero column pair norm respect surrogate base follow verify loss multi hinge support whereas result perform soft equality nice convex convex order order upper use know indeed close function approximately problem q among column parameter section unseen constraint hardness formal sparsity attractive multiclass engine efficiency centrality approach straightforward approach multiclass construction predictor eqn extensively great construction share pair mapping construction number exponentially e predictor mild tackle multiclass prediction like plausible go construct specific feature mapping adequate class form give train frobenius g point regularizers many non alternative mix like solve convex eqn advantageous optimization approximate goal mix yield optimum find solution error bound paper establish guarantee mix norm may seem strong strong particular rip contrast would generic correlate desire sparsity run easier whole eqn work large long extremely still contrast mix regularization hardness solve predictor multiclass sparsity group greedy analysis technique obtain greedy recently indeed similarity wide application objective cost norm rule toward square far seem predictor show learn constrain disadvantage pure subset rely heuristic boost allow real allow sharing class rich expressive identifie class rely also lastly finally share merely across propose share sharing solve eqn greedy column derivative eq rewrite maximized optimize far result example feature runtime step nesterov accelerate smooth accuracy runtime runtime minimize mixed smooth nesterov accelerate runtime chooses show sufficient decrease feature may lead decrease except runtime almost iteration incorporate prevent norm form verify adapt possible overcome define norm main rewrite replace version overall version e tradeoff quality smoothness predictor correspond non concrete mapping decision widely boost piece appropriate mix yield mix norm produce sparse let block two type uniformly zero second matrix denote equal note w expectation jensen mix norm prefer possible show block substantial mix tie way aforementioned well demonstrate scenario share demonstrate mild feature class follow experimental outperform predictor regularization dense show accurate experiment handwritten digit art runtime motivation derive multiclass slowly image digits letter vary class digit capital letter require class space describe rest binary namely select fig overall achieve set easily mild vs compare norm eqn surrogate section fair follow experimental code path regularization run original therefore expand product respectively much regularization preferable rather surprising controlling prevent overfitte one incorporate regularization experiment follow setup run implementation heuristic display decrease much rather surprising add runtime short discussion neural et error goal digit consist digit see example mnist extensively study multiclass handwritten digits rate advanced algorithm mistake challenge mnist nearest nn approach represent entry match x majority label naive produce advance naive run cost shape similarity introduce bipartite matching descriptor compute shape challenge therefore run top mnist feed translate million multiply well geometrically generate expand training million mac test summarize stand original example well svm expand training accumulate ten classifier describe corresponding produce center spatial patch pair pool template plus vector entry template fc response template template template converted maximum going template example template match digit top template produce encode horizontal expect digit fig misclassifie round round template dot product mac round mnist pool type descriptor achieve competitive mnist effort feature cc since fairly cc lead select feature correspond convolution mask might remove piece section mnist come gaussian anchor vector piece anchor anchor point together compare kernel framework subset corresponding point piece wise predictor share principle
intermediate fall break receive node expect link calculate vertical ratio segment contribute height illustration b degree illustrate process multiply since link actual become link generation hand randomly node weight take wise spike fig draw achieve take convolution also correspond isolate original choose ratio isolate rotation fix calculate scenario solid distribution red original logarithmic correspond delta accord deal line dash red logarithmic original setting obtain whereas frame rotation angle degree respectively plot distribution salient explain sect consist spike increase significant shift large fig wider well tendency much stay ratio isolate rapid tendency tendency display angle overall around degree point previously seem similar unique attribute area overall degree isolate change angle argue sect examine sect curve original affected rotation angle fig obtain slope curve curve slope low measuring obtain rotation angle comparison curve setting scenario rotation seem determination within summary investigate new dimensional determining projection question particular vast majority projection slight variation involve rotation rotation determine effect lack reasoning generalise related degree numerically fraction isolate original without science office research sciences h write furthermore line equation top line diagonal express eq parallel shift green write location calculate length area intersection box box triangular square intersect box fig either fig line three intersection four boundary intersect box remain position share fall opposite share intersection opposite intersection fig fall adjacent side fig change coordinate give skewed scenario accord slight rotation preserve degree almost completely rotation skewed nature slowly decrease tv os tool create embed isolated node relatively realistic dominant limit infinitely relation avoid rotation natural phenomenon become connection decade turn highly trivial distance combine average anomalous distribution modular structure play structure extremely understand principle test fine network past aspect newly due concept attract great interest hide systematic study entropy generate wide type coefficient distribution heart mapping measure square self graph become infinite parallel increase slight iteration grow infinity isolated si detail usually construct graph improvement light convergent sequence network size propose rotation sect continue propose avoid sect sect sect generator inspire early worker infinite dense graph adjacency unit square similar introduce used phase transition variety suppose continuous predict replace self define generating measure unit square identically rectangle normalise normalize probability measure integral advantage generate measure time analogy generate factor original convention stand generate factor term stand division analogous write simplify point uniformly interval link pair process correspondingly iteration standard limit e degree node appropriate follow area sized expression increase increase exponentially p iterate multiplying box centre obtain assign draw green construction make standard symmetric unit square although result kx preserve come setting increase degree obtain preserve probability degree natural way however multiplicative generating would column statistically degree topology compose individual row width row give ratio node accord sect fraction isolate depend measure equivalently projection projection belong shall small origin set equal sized box generate equal sized rest shape construction stand step generate evolution unit box length order generalised ref box inside box behave denominator rule isolate node crucial value curve point give occurrence link concentrate one majority isolate self govern expression agreement picture produce multiplication equal general value monotonically thus mean generating node modify construction way projection determine meanwhile kind angle long coincide generation construction set main cut square measure shown coincide measure point along side link pair examining adjust rotation link long arrange straight unique indexing inside triangle various line give angle node origin expression analogous
eps game walk landscape red line start middle black profile reflect white show representative landscape free landscape research intelligence dynamic system thus landscape www generality computer human blue http www game describe random player devise move contrast solve play optimally suggest applicability stochastic game game initially see reaction evaluation quantitative various factor material example material factor subtle form reaction htbp eps dash red sub optimal reaction game describe equilibrium solid red framework rescale coefficient unity reaction optimization position completely number transition direction game exponent dynamic scale sub dynamic project anti reaction alone reaction coordinate indeed poor reaction white subtle phase game exchange piece position computer perform extent maximize gain force optimize iteratively result high coordinate high fig indicate markovian reaction coordinate specify e reaction protein useful generic reasonably good reaction similar description sufficiently time markovian similar sub exponent time original scale distance dynamic equilibrium trajectory represent underlie fig emphasize robustness complementary markov formalism describe dynamic transition profile network agreement way game realization diffusion start continue either end reach profile initially switch many describe constructive game try opposite unlikely dynamic soon advantage barrier much piece roughly half barrier barrier win probability white w w kt agreement compute htbp probability white position play suboptimal reaction game compare win position profile reach protein calculate dynamic network coordinate notably neither naive win collect every position impractical configuration markovian coarse onto construct reaction coordinate simple showing win probability initially black analyze sophisticated make entire try optimal reaction move conclusion move game impossible game move reaction diffusion conclude reaction suggest blue indicator improve win good playing improve heuristic sophisticated treat phase game reaction tailor perhaps reaction game dimensional landscape game assume stochastic coordinate description free reaction coordinate alone specification allow analyze phenomena characteristic impractical play central role construction cancer landscape description construct indicate markovian sub essential missing fellowship appendix give reaction coordinate reaction coordinate trajectory profile energy transition assume constant obtain correct correction entire trajectory move significant net zero detail profile introduce equation steady obtain expand reaction coordinate coordinate diffusion unity numerically integrate free energy sub absolute scale cx different reaction cut value generalize transition minimum position reaction reaction different exhibit coordinate close coordinate maximize flexible reaction attain minimal reaction part another hx dx kt assign profile give langevin attain reaction reaction sum component weight evaluation position iteratively randomly construction reaction reproduce order absolute goodness describe construct partitioning reaction bin transition bin bin reaction convert equilibrium population connect component terminal discard protein reach diffusion energy directly trajectory bin end black ensemble trajectory game play phase game exchange piece accurately evaluation initially piece game evaluation evaluation position white fourth black coordinate avoid phase piece change coordinate threshold discard
library goodness formula ref pt pt pt pt program unweighted unnormalized chi test title computer pc
meta fraction cover clustering iii redundant clustering overlap maximize minimize fig clustering clustering order tell nothing bring novel content clustering way node clustering clustering clustering clustering clustering pairwise avoid triple overlap practice title etc physics article citation apply process paper edge ht article connect bag article connect author article article detail graph composite cluster compare clustered clustering block diagonal clustering clustering set considerably exhibit modularity imply cluster statistically quantum algebra scalar ad world dirac string statistically dirac challenging article attribute country mail country statistically clustering show separation various domain expert clustering vi large far quantitative inspection cluster graph represent influence many many bias may instance clustering clustering bring graph multiple set clustering sphere ht illustration multiplicative clustering raise question improve multiplicative oppose illustrated account cluster language produce modularity graph edge modularity significant modularity clustering modularity score edge pair display c single name modularity vi modularity vi section maximize cluster graph look type distinguish high quality distant clustering distinct distinguish almost clustering sufficiently distant clustering membership achieve choose representative drastically another always group invariant cluster together l address multiple type aggregation scheme finding represent find working prevent rich cluster exist cluster compactly explain show significantly significance improve look intersection increase due multiple algorithmic tractable draw research community challenge room growth nonlinear see section importantly work give chance well method cc edu type main motivation similarity paper author publish accurate describe similarity object metric information datum recover clustering become addition deal clustering graph multi remain good clustering efficiently clustering cluster distant clustering give ground couple community fundamental effort effort construct object represent two graph type graph object scientific base author keyword publish relationship people nature business communication phone etc file group name create graph relationship subject multivariate construct convenient lose aggregation crucial believe working edge accurate analysis importance community sample svd generalization identify factor ground recover aggregation efficiently cluster significantly clustering community challenge problem propose technique rely value detection edge metric quantify clustering result file system group project rest review paper variation clustering aggregate truth multiple type aggregation ground method address latent clustering introduce concept meta cluster discuss represent structure efficiently seek cluster edge type weight represent tuple iw kk multiple intuitively cluster break well mathematical formula algorithm pose significant quality continue lot challenge pose concept graph without get formulation study scope software et al top recursively agglomerative optimize modularity et core discussion similarity call distance clustering metric compare variation measure expectation learn mutual define variation intuition locate rewrite lose know instead second opposite clustering independent cluster multiple type aggregation scheme ground crucial finding community fundamental scheme truth available apply I formally cluster addition difficulty define good matching truth discuss problem find weight quality cluster problem scientific finite observation prediction since try compute random guess cluster clustering quality guess within close ground truth disadvantage reason consume graph aggregation maximize ground cluster quality individual instance quality move ground truth misclassifie truth vertex compute maximize vertex vertex belong large remain strongly proper hold power maximize hold change ex limit step encode hold positive routine benefit gradient gradient nod hold power lose cause hold near extent parameter function resemble position justify individual vertex overall maximize potentially use purpose measure cumulative edge internal edge set cluster cut edge objective rewrite trivial assign exclude yield within feasible modularity random setting modularity modularity score denote vertex graph correspond hypothesis cumulative edge optimization solve hybrid package national problem recover weight experiment aggregation justify et propose benchmark graph size edge perturb independently identically average edge weight three histogram hold power blue example perturb poor green optimal perturb preserve vertex bar correspond hold bar correspond hold presentation portion hold power mean red bar show hold power ground hold power result similarity none vertex hold c ground system classify file belong take similarity modification file file show sensitive correspond hence number positive hold power lot node far account take energy consider similarity citation link share author edge article mail uk proxy truth cluster author edge come distant intuitive article common link topic nearby text encode factor plan nonlinear aggregation state goal aggregation function position pareto objective axis cluster modularity percentage node hold modularity modularity number since aim modularity fig show trade objective axis range modularity modularity range power importantly look hold power preserve reason connection within cluster cluster complex network matter hold power file system aggregation clustering vi truth measure vi clustering objective second clustering experiment produce promise scalability propose aggregation graph operation synthetic graph present correspond runtime depend evaluation since function different run informative grow expect cause scalability linearly metric result omit space edge share redundant combine example connect text similarity cross etc edge document influence similarity redundant article share author tend field nearby say location attribute explicitly much relatively clustering document meaningful class illustration perfectly embed vertex weight euclidean visually nine cluster arrange along axis would along two physics mathematic west east example article datum set directly journal aspect provide partial view underlie structure geometric feature partial depicted fig green red projection provide diagonal partial column partial metric provide however ensemble picture goal view paper conceptual weight combination graph depend particular graph characterize space clustering homogeneous identifiable boundary clustering etc exhibit community present method
minimize remain instead perform minimization w internal external n n concavity exploit positive involve new result version difference present henceforth term counterpart lead threshold cluster associate non early convergent sequence guarantee characterize point beyond scope yield computationally algorithm deal identify cluster operation per cluster relatively data presence imaging application wish dna microarray rarely occur sample expression thousand avoid store process operation hence preferable offer dimensional identify separable next subsection soft algorithm modification entry pairwise update product lie alternatively involve induction every centroid q scale scaling depends latter readily compute tn thresholding prove include initialization column lie exist update n update summarize f exploit algebra stop actually ignore store need purpose input acquire cluster initialize update limitation shape cluster assume linearly separable share limitation mean mapping map term space mean separable partition feature partition able operate two mean trivially th entry term replace must even even reproduce hilbert typical datum non well string graph tailor input regime straightforward compute mn carry nonlinear regime high nonlinear recover input I facilitate dimensional input enable simplify presentation spherical gmm remain dimension implication updating entail become come second aware degenerate overcome limitation fix induce also setup mixture remain valid input property know product update n randomly via c alg whereas store reweighted reweighte introduce readily numerical dataset assess latter adjust rand ari partitioning search remark thank warm start comparable bivariate distribution belong lie outer outlier multiscale spherical outlier outli curve fig correspond initialization zero outlier assume exhibit identify value transition outlier value outlier outlier show slope indicating identify table square center average initialization contamination point approximately apart novel k em include robust mean em low weighted weighted rmse hard soft soft n robust plot point identify outlier proportional kernel circle radius proportional large test united states handwritten digit recognition corpus image intensity digit although label inconsistent digit classify digit digit combine sample image initialization common choose assess ari exclude vector ari table ari assignment already interestingly digit monte c kernel k polynomial cluster obtain yield outli image become nearly dataset cluster fig centroid fig identify identify outlier small trait similar homogeneous kernel ari robust score important observation outlier improve outlier fig use partition identify outlier social link adjacency identify group connection kernel mean spectral connection conventional spectral k specific partition poor local depend cluster initialize symmetric laplacian depict identify identify marginally outli structure yet outli interestingly degree gender dominate unobserved show share cluster college college division games node team link divide team game conference often spectral tune outlier ari coefficient yield outlier sort central three namely mid american conference many game conference game play mid american conference conference east mid american mid american conference american conference ari coefficient partition outlier outlier identify principled accounting outlier cluster exploit appear connection aware processing lead development efficient provably convergent version well suit among developed validate numerical contact author popularity conventional sensitive outlier meaningful structure outcome robust rely translate outli domain robust lie identify sparsity outli choose iterative algorithm robust algorithm develop numerical applicability block maximization lasso mixture robustness subset unlabele minimal challenging yet dna microarray bioinformatics mining label costly k gmm conventional relie thereby minimize cluster scatter fuzzy tailor identify belong gmm consider observe arise ml estimation gmm ml method separable cluster popularity gmm cluster inconsistent appear reading belong rarely phenomena cluster parameter motivate approach outlier robust cluster investigate cluster intend centroid assume respect outli sensitive output cluster approach robust statistic minimum ellipsoid huber contaminate extract cluster root robust linearly contribution outli vector fact rare outli vector compressive methodology deterministic second contribution work comprise cluster develop hard mean gmm coordinate descent close optimization variable particular group solution close operate counterpart application bioinformatic social call aware dimensional involve cluster accommodate probabilistic algorithm update section develop test handwritten digit recognition result letter column letter set expectation pp cluster contaminate develop probabilistic section vector exhaustive mutually exclusive vector euclidean assign history centroid introduce instead compare point centroid distance consider mean introduce membership valid membership coefficient apart satisfy empty pose centroid assignment even suboptimal drop constraint check set one iterate stationary motivate assume c easy c offer merely blind least square ls assignment constraint widely mixture pdf n controlling suggest minimize positive definite solving n membership conditional guarantee pose avoid spurious even matrix mixture grow unbounded g let possibility common common make conventional mle gmm unbounde derive outlier jointly nonconvex aforementione variable us devise minimize iteratively hold ls index however express show uniquely minimize strictly simply c n equation positively scale side yield compactly cost minimizer number exceed cluster point th minimize similar soft regard hard solving distance note randomly f computational resource next suppose cluster e g large mean per store scalar per operation store structure convergent increase sequence iterate follow boolean otherwise problem unconstraine continuous empty set per point
optimization correspond penalize recovery tractable approximation aside condition seem check synthesis validity may become condition purpose sense kn kn nh satisfy toward sufficient validity upper function satisfie provide follow entry equality satisfy recover way matrix augment satisfie condition satisfy efficiently tractable satisfy explicit system computable matrix simple efficient norm issue r tm simplicity associate satisfy explicit computable convex efficiently function context see tolerance difficult since compute relation slack component system tractable block derivation article symmetric matrix sufficient satisfy condition satisfied contrast sense essentially cf q sense range standard gaussian rip imply essentially range performance severe proposition mention validity cover role partition question provide quantity mutual incoherence compressed see sparse provide sense block incoherence contrast conclude pair satisfie suppose satisfy relation pt incoherence exceed pt satisfy assume provide block incoherence proposition imply gaussian norm enough condition fix k nh norm recovery norm block exactly ensure validity signal value consider correspond addition mutual incoherence block structure respectively candidate choose powerful recovery follow find large quantity sparsity matrix result contrast problem block sufficient recovery pt ccccc pt randomly submatrix hadamard h k entry entry take structure arise matrix four scale select norm display level penalize contrast goodness recover ax pt example block sparse recover latter experiment make two conclusion recovery level candidate experiment upper goodness h h r noiseless yield reflect run series simulation four sense block recovery test contain randomly recovery instance nonzero corrupt gaussian block tune actual signal instance apply recovery measure error divide error error routine routine good current rate rating close current simulation routine process experiment rating routine surprisingly second good routine describe guarantee recovery reflect table favorable guarantee purely model observation proportional plot figure recovery favorable performance lasso theorem business office nsf dms handle overlap block noise promise recovery verify condition emphasis recovery condition computable recovery respect oracle link recovery estimation utilize develop block close present linear unknown symmetric w random w k sparse sense estimate signal advance block refer advance representation transform appropriately implement block relate goal compress indeed sense nontrivial therein number band signal measurement share sparsity pattern plain recently relate lasso recovery euclidean block plain lasso lasso selector extend structure cite recovery concentration true nonzero sum magnitude magnitude typically block isometry introduce es block block recovery minor mapping precede handle identity minor nontrivial ax py mapping hand impulse correspond introduce add matter cost nothing concern however emphasis transform provide question note verification condition rip hard study theoretical exception incoherence time restrict restrict eigenvalue property efficient meaningful believe recovery bound error optimize method recovery consider case bound error third specifically utilize kind show tune attain addition condition designing possess usual main organize recovery sense magnitude sparse restrict relation validate isometry property version introduce norm block norm observation guess error guarantee transform everything contaminate notable norm condition among efficiently besides tractable mean recovery satisfy sufficient recovery signal validity build quasi also relate incoherence incoherence conservative latter limit performance alternative either require provide guarantee regular present nx short block magnitude tie vector block denote w sl interest q know advance approximated block observation compact origin know introduce instrumental subsequent construction sense n evident let satisfie know sparse whenever condition closely sparsity satisfy euclidean norm fix play crucial role selector bound eigenvalue assumption l recall sense restrict isometry recovery b define slightly replace weak condition let recovery pair let see thus small small regular good guarantee extract good component whenever say recovery build quantity independent assumption automatically quantity numerical support favor penalize disadvantage routine tune recovery guess tune loose rough grow infinity block block observation error well validity efficiently sense difficult give candidate pair satisfie fortunately become fully computationally necessary optimal contrast induced argument
ks n apply estimator recall converge proposition apply p I k go estimator select perform obtain argument n n inside integral convergence treat n nr rewrite integral trivial conclude every number globally result small observe hold consistency immediately ls suppose clearly p entail closeness ls prove second claim l l I I ls ls n ls ls kp kp b large k l np choose proposition appear far bound sn na hold complete similarly tight already subsequence argument case n arbitrary observe exist eventually par nm n n I view select par remain observation distribute n boundedness necessarily l hold select par observe n n h n identifying immediately calculation eq result follow elementary making distribute front indicator inspection cdf converge part move par unknown proof show atomic theorem move par continuous n convention real I respectively absolutely part see mx converge involve n iw select n continue converge nz nx furthermore n dominate n h n ls n I distribute note establish rewrite display p display n establish complete subsequence n normally use move reference l event tend p stochastically collection note par minimization norm bayesian procedure g shrinkage li penalize fan penalize nd ed york frank unimodal fu asymptotic selection estimator versus fact estimator return estimator penalize adaptive lasso zhang concave converge sense distribute variable freedom method converge variable mx since cdf associate unimodal true distribution lemma argument axiom theorem theorem criterion exercise notation em linear depend addition investigate estimate degree slowly tune perform tune conservative furthermore discuss subject classification e keyword phrase lasso variance high thresholde soft soft regression number orthogonal design soft frank adaptive see wavelet contribution concern thresholding fu tune act study estimator selection fan li fan asymptotic call smoothly absolute act selection fan li fan papers partial except fu paper parameter highly penalize maximum estimator sample penalize estimator adopt move post estimator paper variance contrast regressor depend case variance create trivial consider differ asymptotic variance degree result vanish limit tuning imply distribution thresholding slowly idea rough exposition variance hard parameter denote denote use instead consider variance thresholde conservative threshold estimator infeasible estimator case soft thresholding estimator move tune limit h convex absolutely infinity limit functional finite next tuning possible limit combination location depend arise infinity fast sufficiently slowly weights picture adaptive thresholding well infinity constant absolutely point consistent uniform soft light theoretical orthogonal sample latter non paper thresholding combination absolutely continuous normal component non parameter line expect limit behavior coincide infinity case conservative uniform convergence simplify simplified assumption numerical study thresholde qualitatively adaptive long follow treat consistency rate study l adaptive q vector allow support shall notation regression model situation element square least estimator shall infeasible infeasible counterpart adaptive soft thresholding l cc ls n l ls I ls n l infeasible specific factor interpretation value always infeasible version note hold estimator unique minimizer function exist diagonal reduce coincide thresholde lasso selector independent solution specific iii obviously hold infeasible version exist least event specific depend case hence case lasso soft thresholding therefore adaptive immediately correspond obviously thresholde nonetheless estimator p easily result spirit result thresholde principle zhang minimax mcp penalize apart tune mcp depend shape turn thresholde mcp cover mcp analyzed form mcp thresholding namely least estimator brevity exclude seem mean regressor get least subsequence wise impossible use endow usual topology shall cumulative probability degree centrality write convention selection estimator I consider drop probability infeasible version suffice probability study selection fix n c gives correctly detect nonzero correctly detect coefficient part part except little interest case slow cn n approach lemma kn kn bound least converge cn n c bound incorrectly unbounded converge p n product apply asymptotic depend sample shall concentrate pointwise shall case respectively light extended suppose satisfy I I ir imply limit properly capture move analysis limit uniformity arbitrarily neighborhood holding entail tuning go discussion essence approach speed cn cn convergence case slow remark variance finite allow replace chi random degree make perfectly enough necessary condition k e k conservative tuning know converge eventually pointwise unknown reason variable move eventually generality consider pass converge satisfy n km p n n r I n ir conservative furthermore variable selection case know freedom infinity eventually turn known variance case case expect rate fast slowly b average degree look average respectively variance later par apply see quantity sequence select along quantity proposition convergence theorem par one essentially free accumulation point discuss conservative get behavior tune case state consistent consequence hold little move par probability along purpose follow immediately sufficient every uniformly sense display neither display counterpart follow consistent I sharp conservative one precede tune thresholding matter pointwise equivalent immediately tight proof infeasible hold conclusion continue supremum replace display statement turn arbitrary h I equivalently h x I n multiple absolutely generally case independent hence entire particular cf remark scale apply cdf equivalently cdf n h variance indicator function lead function absolutely density soft thresholding indicator involve shape qualitatively finite entire conditioning provide lasso lasso cf remark estimator framework asymptotic lead conclusion regard cf version give satisfy true n converge reduce n n ir h weakly weakly measure b k n h weakly measure e e tx dx tx e I distribution b n n factor proposition conservative consistent uniform essentially distribution demonstrate parameter framework satisfactory correspond property parameter hard regardless soft observe discrepancy limit proposition uniformity convergence limit locate hard thresholding essentially mean tune component variability randomness two seem connect limit hard sequence recall issue soft thresholding problem high thresholde infeasible parameter variation supremum degree freedom infinity enough apart instrumental subsequent factor satisfy q eq conservative weak general remark none constitute show limit conservative k n cdf se se se se reduce hold weakly thresholding scaling n n n converge atomic normal soft enough factor true n n km weakly cdf measure se I n always reduce proposition conservative tuning limit discussion apply obtain limit limit limit limit essentially set property limit always soft thresholding finite especially ie consequence uniformity distribution degree freedom limit distribution move framework tuning thresholding I I I converge weakly ir n ir r n r consistent satisfy n n convention soft unknown hold tune enough factor km cdf cdf I jump height ms particular weakly converge weakly distribution sense show soft thresholding violate however hard violate limit different seem thresholding continuously fix limit case freedom limit absolutely component seem estimate thresholding soft parameter limit limit know thresholding soft result center scale correspond next asymptotic arise degenerate condition parameter soft fact thresholding estimator property mind long stochastically move cf theorem oracle call statistical extensive tune additionally inspection use I n z n I asymptotically chi distribute reduce I stochastically unbounde adaptive thresholding parameter conclusion apply infeasible simplification I illustrate moving limit n analogously proof subsequent analogous p suppose I I n ix x I ix I ir adaptive thresholding large enough I n ix f n w oracle proposition characterize mass bias proposition whenever save proposition fail different theorem I I fact consequently also parameter lead center identical proposition result n n immediately extend whenever extend fail proposition theorem free reason explain remark
bayes factor often differ bayes discuss sufficient behave abc without ideal achieve tolerance odd feature besides inconsistent choice case assess divergence measure beyond application summary rather reason storage dataset several hypothesis handle solely k relevant factor instance outside domain sample infinity similarly compare point statistic connection appear current paper condition either converge answer precise consistency statistic wrong except nest statistic summary rarely candidate formally state quite convergent abc computational checked summary statistic also summary connection whole bayes factor sufficient solely furthermore collection random conclusion necessarily true outside imply performance illustrate impact double mean equal sample median deviation statistic median absolute deviation statistic explain fourth expectation always artificial computation laplace bayes step show distribution normal oppose data outcome predictive base statistics abc deviation quite hence summary abc median satisfactory statistic fig outcome statistic abc solely fourth moment slowly fourth case occur use distance euclidean lead experimental fig term observation centre either summary abc abc table tolerance quantile ht illustrate fundamental convergence e bring validation abc realistic situation contain behaviour behaviour likelihood section result criterion evaluate summary discussion dd ng n dominating state result hold brief letter occurrence write denote number resp resp symbol converge distribution constant asymptotic mean satisfy define set compatible mean assumption claim mild easy check illustrate versus laplace realistic likelihood lemma marginal compatible equivalence g detail result corollary inference compatible provide yield lemma go tail follow help assumption compatible I n spirit criterion regular dimension non regular abc illustrate present implication relevance behaviour bayes drive value model study behaviour testing probability bayes generality actually enough compatible generality belong compatible irrespective true solely embed go bayes convergent behaviour neither compatible satisfie lead consistency compatible behaviour irrespective therefore select effective dimension compatible essential merely drive set bayes behaviour neither light summary rescale asymptotically leibl distribution variance pdf evaluate purely realistic formally statistic differ example statistic explain discriminate fig expectation differ hence occur must hold fix iid identical truly toy constitute exception case q bayes statistic merely bayes else belong phenomenon consistent illustrate section practical relevance sense assumption summary condition concentrate low g type like statistic chi square also even though control moderate deviation mean model mean typically weak set assumption behaviour often find condition usually vanish marginal become invertible near continuity condition deduce imply assumption term first secondly density close become check case point whole informative informed tail depend recall trivially consequence central verify address mean equal ball case mean address fourth expansion model equal belong bayes due special illustrate lack consistency quantile quantile quantile scale skewness well easy function abc statistic consider model perspective set sub abc tolerance quantile empirical quantile agreement bayes g irrespective factor statistic make prove hence bayes consistent quantile obviously fig one empirical quantile right quantile empirical level row quantile third abc algorithms proposal tolerance quantile distance examine carlo relate abc namely population population divergence effective population define behaviour loss genetic variation act proxy assume mutation occur repetition probability configuration term common single parameter mutation choose parameter abc number allele let copy past copy branch explain mutation adopt model mutation eq apply mutation genetic realistic relevant statistic satisfy statistic moreover couple poisson order theorem least g verify compatible satisfy compatible use uniformly argument bayes converge indeed expectation table second either distance factor expectation fig analysis medium indicate summary model abc software ht summary statistic algorithm tolerance operate consistent summary compatible propose run practical check relevance hypothesis compatible statistic equal relevance mean eq summary compact continuous nan relevance proximity indicate discriminant quantify produce sample posterior choose go infinity asymptotically even case convergent covariance adequate toy run three summary statistic lead respective model quantile distance approximation produce derive mean evaluate experiment base quite satisfactory approximately fall within nan hypothesis include
transition probability measure take respectively environment control probability event policy blind policy expect utility action define value process sequence denote policy handle mdps mdp lie outside thesis reinforcement use framework whereby distribution mdps represent mdps consider hybrid rare derive robust bayesian mdps mdps policy without mdps share state abuse optimal find bayes generally note policy necessarily optimal policy mdp draw computation intractable polynomial belief utility optimal policy tight optimal expect mdp bind refined branch technique employ policy belief variational propagation two moment none return stationary mention interesting belief mdps utility complete utility report condition belief tight mdp mdps belief significantly principled approach mdp optimistic policy expect problem tackle compute belief finite mdps induction arbitrary monte bayesian approach optimal restrict draw analysis probability mdps reason policy respect far belief condition history since policy measure small follow induction together policy utility aa ts ts r uncertain certain belief step utility future past easy alg correct expect bound gap bayes function belief small posterior let dominate final result geometric policy simplicity hoeffding imply policy combine achieve expect alg policy lem lem require belief reinforcement simple approach follow alg calculate return accord computationally via new mdp use rather policy stationary incur small loss htb discount mdp act accord policy mdp heuristic step mdp reward belief act bad htb b alg alg reward expect relative sampling curve alg curve act mdp reward mdp algorithm commonly reinforcement traditionally discount expect reward discount estimate generator transition dirichlet equal parameter reward total hand side difference reward optimal total reward monotonically detail behaviour distribution get sub environment explore less run remains run c percentile interval alg algorithm cite paper interval base report present alg reward enable alg surprisingly actually outperformed alg would bad alg long induction near policy mdps generalise expectation polynomially arbitrary reinforcement sample mdps regular principled calculate near interval experimentally perform make tight result perform surprisingly attribute learn multi account uncertainty dynamic application reinforcement loop employ planning tree tight hope consider wide
adaptively normalize available surveillance camera evolution frame static exploit code encode size bilinear encode scan encoding procedure common coding depict produce regard shannon encoding column laplacian estimate laplacian constant purpose precision short also expect redundancy dimension column also index predictive sequence ij v encode part together length column index non part code bernoulli code model encode row pixel frame location hence parameter use code decomposition decomposition sequence simple augment alm algorithm alm computing base repeat consist surveillance point pass frame stack background frame matrix appropriate long obvious summarize top figure two curve show bottom leave plot scale right recovered background change illumination combination eigen low combine low decomposition theory able promise description graph turn right figure rest capture second eigenvector matrix increase vision recommender theoretical establish portion measure hypothesis nature device regularity estimate address theoretic datum demonstrate complex extraction sequence robust success processing application dominate pca develop alternative review true value recover exactly power demonstrate modeling tool achieve goal want approximation balance estimate generalize adapt overfitte selection formulate balance work assess ability regularity regard practical implementation state short ability model result selection capable capture video surveillance illustrative bring theoretical perspective problem another information naturally incorporate assess matrix matrix iid entry pca vary introduce type I framework mean consider family use use denote description bit description shannon scheme value fractional shannon code naturally lie define assignment possible describe pair description term reduce decomposition rank discard description n safe magnitude encode arbitrary positive integer sum stop non add requirement uniquely diagonal map round virtue assumption unit encoding manner encode
consideration belong low I message jj bound involve potential subsequent iteration lower indeed belong guarantee upper quantity presentation primarily bound I associate compatibility matrix equation entry frobenius guarantee maximal eigenvector scale entry stochastic eigenvector guarantee value proposition simple often denoise potential edge tune smoothness type smoothness bp update quickly motivation consider illustration singular upper sufficient condition intuition node potential degree guarantee contraction inequality refine previously proof main result note ordinary write randomly mass state somewhat e immediate uniqueness bp iterate bp iterate hence uniqueness bp direct consequence accordingly unique iteration claim almost sure consistency claim global stacking rewrite representation side triangle via recursion martingale equation representation martingale martingale difference martingale equation conclude zero vector surely extend argument consequently deterministic iterate piece almost thereby complete part claim martingale concentration know hoeffding integrate turn inequality upper dx manner expectation substitute bp fix immediate realization obtain mapping index notation write compact suitable application conditional know past step apply need verify hypothesis constant lastly define averaged verify euclidean product message lipschitz strict conclude upper corresponding increment recall q side recall update recall obtain combine piece since hold recursion product equal upper product substitute dx fast rate algebra yield claim begin subtract equation denote recall substitute recursion yield note martingale namely cross product term vanish show move martingale expectation finally put piece inequality fact consequently chebyshev deviation specific thereby conclude simplify last obtain mass function depend generate rewrite bp block place q frobenius know corresponding equality representation upper somewhat complete variety confirm theoretical provide simulate denoise simulation edge potential node potential variable interval h cc path couple panel node potential topology collection bp run average run cc b curve examine chain structure particular instance algorithm step trace square versus number plot contain confirm strong give particular observe typical concentrated amount sample path next increase dimension simulation square edge potential trial chain guarantee iteration straight slope exhibit exactly vertical slope grid panel convergence predict theory convergence bp iteration bp c total graph top iteration bp total discuss previously update time algorithm dimension indicate scale linearly bp fair also measure either tolerance comparison fact bp iteration approximate nonetheless table chain graph significant running become large type ccc b gray study instance computer vision two node graph pixel base noisy scale enforce model potential respectively c illustrate square denoise despite jump reduction substantially low marginally job substantially shorter gray deviation experiment vision popular base original et enforce observation potential paper dissimilaritie maximum dissimilarity vision per however comparable bp since standard dissimilarity develop analyze low complexity dimension oppose bp oppose belief also require number usual distribute algorithm main contribution graph ordinary update contraction provide expectation suggest exploit generalization also operate product semi include reweighte bp bethe free bethe free minimization natural idea analyze stochastic posteriori graphical markov undirecte discrete reduce form decode useful variant directly general likely requirement could work approximation phase associate careful reading suggestion improve recall relation substituting bound algebra yield degree inequality constant claim construction convex conclude message particular jacobian derivative continuously continuous set sub direct denote integral value triangle upper eq diameter graph let indicator straightforward verify positive pair direct edge induction claim entry non direct path complete tree structure graph note bind logarithm claim state differentiable apply integral form value show suffice quantity vector bind inequality final defining hence final argument repeat valid take algebra piece definition find bind conclude lemma ccc department statistics electrical computer california berkeley propagation widely message graphical core bp update apply transmission dimensional vector involve dimension complexity bp belief propagation pass neighbor complexity quadratic linear without assume per establish theoretical guarantee performance converge graph condition provide asymptotic decay slow yield communication various graphical belief provide among collection field bioinformatic formalism mean computing distribution subset marginalization computationally intractable efficient graph without marginalization solve product belief propagation bp computation neighbor form message cycle long nonetheless extremely effective marginalization reader e nod discretize estimation sensor network store message scale researcher bp therein pass multiplication graphical exploit check code complexity perform use fourier reduce linear arise computer involve fast transform computation accelerate property absence researcher quantization bp update non propagation certain method consistency particle particle negligible proof researcher technique decode encode message bernoulli lead decode hardware bp novel refer adaptively practically order potential communication require transmission oppose bp even converge stochastic bp quantitative rate provable computing point tolerance precise structured meaning converge asymptotic upper maximum cycle bp update contraction condition rate around simulation showing reduction begin background bp turn statement theoretical consequence devote proof aspect proof defer correspondence prediction practical background belief graphical define joint vertex collection edge among particular clique edge graph clique graph index eq clique dimensional show clique vertex edge take factorization form pairwise discrete variable convert suitably product message easily translate vice ccc unobserve inference basis observe conditional probability write control code value additive white denoise vector corrupt bp incurs require real number pick column message probability costly ready precise potential ji pre make sequence quantity vector message complexity calculate mass weighted operation operation find multiplication regular degree reduce significant message communication bp gain require dimensional edge summarize feature appeal practical edge dimensional bp remainder paper understand message intuition behavior index expectation effect recall perform algebra equivalent lie show despite update bp point stepsize precisely stochastic variant bp update randomness raise ordinary per significantly incur answer certain provable gain tree choice message update bp uniqueness develop working decay high iteration concentrate around potential graph structure provably correct substantial gain obtained begin case markov figure structured integer say degree refer background definition graph diameter state let notation ready
part understand difference area survival problem decomposition class life bias regularization deal survival event indicator event occur censor survival analysis understand accept modeling index measure interpretation censor observation failure order proportion order pair tie rare proportion index allow survival classification commonly ec bias x py py set class bias closeness class variable estimate decision set sensitive shall function especially case high depend size function many algorithm associate sir david cox concentrate hazard conditional survival individual cox strong hazard assumption imply individual proportional modeling actually dependent hazard advanced survival make survey regularization penalize cox attractive produce interpretable path regularization select performance regularization variance variation intend bias step overall describe evaluate increase life directly sum evaluated sample aside size evaluate randomly test average training evaluate experiment conduct conduct characterize dataset hierarchical associate identify cluster single gene gene feature group patient expression signature aggregated figure method figure ph dataset opposite offset additional due dataset regularize cox ph fix may recommendation preferable please file size letter example option file year reader available machine recommendation pdf file acceptable please http www pdf files figure otherwise please check contain type shape implement solid shape complex sometimes problematic include files eps eps clean figure black user file microsoft save office www microsoft com en b ed save office file os drop box save user file computer file http www ps click advanced font option font outline select click ok file ps create file ps file
daily leave change apparent forecast model pick risk bind square give select strategy use risk pick high small dramatically ar truncate daily ever none include mis specification see reference therein control cover autoregressive family believe elaborate family average conditionally restricted stationarity strong variant particular specification theorem theorem proposition edu department statistics university edu mark pa edu univariate autoregressive impose stationarity enough risk demonstrate variable wish another time predict measurable measurable go infinity building learn past forecast ideally function minimize estimate minimization train optimistic risk grow tend prediction strategy restrict risk pac confidence confidence bound time prediction fairly development extend important problem set series stationary decompose predictor finite part memory index oracle finite complexity amount incur due loss approximate infinite finite provide pac non regularize learn sort weak serial square close strictly generalize apply stable sub iid generalization iid generalization learn literature svms complexity risk autoregressive conditional model keep use stationarity autoregressive ar allow application bound without system series theory explicit applicability forecasting interest direction develop need explain idea effective related dependence mix technique risk must source effective investigate reader notion strong stationarity finite vector one imply distribution definition event variable restriction restriction norm one make mix increasingly approach probability typically supremum unnecessary use complexity e g think measure seem white gaussian necessarily everything supremum nz take risk intuitively seem fit nz f ergodic complexity slowly unless flexible nothing predict gaussian base bound present datum predictor space loss loss stationary coefficient bound straightforward control first describe complicated increase small calculate expect tight third term point train reduced process accomplish spaced block asymptotic independence quantify treat independent frequently economic lie interpretability combination value evolve finite ar amount estimate square generalization characterize complexity mix truncate ar bound jj slight ar least norm ol solution despite autoregressive stationary check complex root
term since discrepancy affect hold suffice failure repeat runtime construction p topic fix integral usually shift kernel bandwidth bandwidth write arbitrarily allow www nx n nx n hold bandwidth affect discrepancy imply completeness shift via set point locate locate range small point th would great ball leverage bound instance exist point dp translate size outline absolute reveal hence reveal assumption instead point matter sample subset kernel convolution e input input represent range simplicity focus smooth notion space turn benefit greatly kernel slope bound accomplish study space improvement plane ball set kernel formally start pp define kx kx broad example kernel gaussian normalize kernel pt geometry kernel binary seem heavily result adapt discrepancy imagine space ball range pd kx xx drop apparent book pp range binary range recently show kernel super level approximate proxy space constant vc achieve approach outline book base idea discrepancy repeat point leave colored much size tie classic range resp although intended range discrepancy cost minimize sum point match color discrepancy p algorithm construct know factor appendix combinatorial result pair color long result constructive simple still cost difficulty prove single kernel least section slope space range space boundary slope lot boundary range slope chernoff bind discrepancy analyze kernel j ball extend section extend generalize distance th relate reasonable admit sample sorting sort range align variant fix necessarily vc super increase provide distinction important small quite rarely many simplification family range near net improve low bound question regard improvement logarithmic mainly approximation choice focus determine original highlight book error book generally kernel typically first chen kernel random require require binary completely completely last year although similar literature focus sampling show ball sample space open go range answer size space correspond super raise space simplify analysis range possible quite bound focus write rotation invariant would presentation assumption generalize slope kx simplicity since instance ball volume radius dd dr let pair match main volume ball contain length associate edge volume shape edge edge proportional volume ball minimum sum length shape disk apply ball fit inside length back assume sum handle appropriately next long angle differ edge point towards geometric fact low swap pair type contribute construct cost matching color discrepancy insight apply say bind straight forward point apply attain bind eq start kx slope replace since jensen dd kernel kernel application kx kx p b dy I would replicate bound volume width th edge situation count towards volume could point close close edge either entirely entirely contribute respectively
initialization explain lack accuracy situation miss algorithm remark point outside discriminant group c gmm com diag pca accuracies deviation uci c mean accuracy percentage uci experimental experimental em apply detection mass technique useful disease assess tumor evaluating drug treatment particular detection cancer cause cancer year estimate control spectra spectra spectra patient hereafter spectra control cover read supervise experimental protocol da spectra fisher cluster em pca mixture apply ask remark cluster among deal cancer l cancer control table em fisher em confusion compute fisher principal axis axis em appear em significant negative classification symmetric conversely em point medical negative acceptable false positive em provide understand indeed load high highlight fisher correlation arbitrary peak discriminative axis plot spectra cancer red triangle variable surprising cancer big discriminative surprisingly cancer spectra value extract unsupervised framework power original original subspace spectra indicate triangle discriminative subspace intrinsic dimension parsimonious group estimation estimate discriminative determination procedure discriminative unsupervise furthermore discriminative em em cluster dimensional mass give prove least work visualize cluster estimate discriminative version gram kernel finally could ease discriminative axis author suggestion greatly article index fisher indicate matrix first complete expectation well group th mixture previous cost reformulate us notation definition rewrite operator projection reformulate associate complementary py iteration orientation write ts reformulate tw ty scalar quantity point k column provide maximization follow equivalent lagrange lagrange multiplier consequently th group maximize log partial ty partial formula logarithm determinant td leave covariance conditionally imply du j rewrite q derivative estimation equation follow partial derivative tc already expectation complete partial respect du tc equation partial du equation proposition universit paris paris france universit france high recurrent many fit datum orthonormal intrinsic original parsimonious algorithm propose mixture discriminative subspace simulate dataset provide method visualization parsimonious scientific popular show behavior suffer know mass intrinsic dimension traditionally extraction popular could method fisher discriminant framework powerful span discriminative subspace past year method practice specific provide cluster helpful result difficult visualization cluster addition scientific economic actual accord overcome curse model classify discriminative combine cluster goal discriminative introduce name objective firstly subspace secondly discriminative cluster review mixture link discuss base estimating also algorithm exist simulate present fisher real world conclude give statistical aim observation homogeneous group review scientific field require task conversely refer live efficiently classify firstly review deal widely model group time consider divide homogeneous realization vector density often parametrize unfortunately parameter function particular dimension numerical ill furthermore without restrictive dataset small overcome several strategy subspace dimension tool reduction certainly project axis maximize project projection refer alternative similar spirit find map grid relevant variable selection cluster approach past new approach group two category search subspace bottom discriminate reference algorithm iterative start original variable remove heuristic conversely probabilistic group live data generative space linear relationship recently well mixture family parsimonious partially technique turn consider enough account visualization fisher pose discrimination goal fisher linear separate assume fisher look project observation class small discriminative subspace different criterion constraint traditionally ts kn km vector column maximization eigenvalue optimization problem solve determine discriminant analysis classify solution sensitive noisy focus discriminative subspace compute subspace visualization introduce call latent aim find parsimonious fit clustering visualization idea firstly actual live dimension observe discriminate want group observe realization independent realization discriminative dimension strictly describe dimension latent link transformation orthogonal group center noise follow assume density latent space eq therefore gaussian space complement finally sequel summarize parametrize proportion orientation basis subspace notation model model generate constraint matrix instance similarly diagonal k discriminant view noise variance acquisition process introduce refer variance within latent outside parameter rise independent mixture model nb com diag gmm gmm text parameter gaussian model com diag give number specific decompose number term classical parametrize conversely diag gmm gmm parsimonious model require com gmm intermediate diag gmm prefer turn whereas comparable model establish exist literature close flexible refer parsimonious isotropic principal parsimonious model year well parsimonious loading isotropic remark family parsimonious parsimonious loading loading variance despite difference parsimonious model common orientation close loading subspace orientation good visualization particular axis great parallel discriminative illustrate introduce model probability group orientation probability parameter hereafter fisher sir r discrimination simple fisher combination classification version conditionally current come let observation belong eq space u interpretation mainly mainly depend discriminant complementary simulate initialization frequent datum number parametrization parametrization one penalize likelihood criterion bic criterion certainly popular selecting criterion favor model section criterion context stop computational em discuss condition orientation maximize em view demonstrate supervise fisher criterion covariance expectation assumption maximize u u step em algorithm condition guarantee em procedure rarely fail converge initialize decide algorithm converge detect advance em sometimes slow practice necessary estimate converge user criterion maximum provide stop check whether maximum experiment fisher check procedure somewhat big ordinary require schmidt suppose small notice observation propose consume usual actually pc scale problem fisher appear slow average second necessary fisher em fisher estimation procedure practical interest among visualize axes deal problem discriminative theoretically equal strictly dataset possible extract besides axis may certainly visualization cluster indeed operator visualize actual look equal visualize discriminative visualization obviously axis discriminative one axe visualization relate indeed visualization fisher may bad mini strategy discriminative interest visualization point interpret axis notation look matrix column loading axis discriminative original select high loading highlight relevant loading set remark interest application economic frequent sample problem overview larger available frequently modern mass generative impossible indeed ill well bad overcome step require determination last deal partition column orthogonality fu tu tu theory theorem maximization space reduce maximize orthogonality gram reduce procedure axis dataset sir apply fisher illustration discriminant collect correspond specie discriminate consist specie length fisher community datum latent em datum unsupervise width axis supervise fisher apply course method panel projection show perfectly three reach model secondly show monotonicity stationary present matrix partition provide result confirm within loading stand discriminative axis estimate case case first axis em discriminative subspace axis sufficient discriminate besides turn accordance variable discriminant via visualization simulate right gmm diag gmm fisher ccc ccc gmm em com diag gmm gmm experiment compare efficiency fisher fisher standard model diag gmm gmm gmm simulate simulate unbalanced group group model space complete noise experience whereas axis observe
time approach invariance provide lyapunov framework element work realize relation consecutive allocation reward among player well aforementione mainly uncertain element add robustness capture design allocation base observation allocation specific core game priori direction design rule partial extra cone convergence prove via lyapunov connection lyapunov theory contribution delay flow turn robust control aspect lie fact lyapunov stability idea turn tu theoretic novel represent far main organize conclude denote transpose denote th scalar boundary nonempty utility tu characteristic nonempty tu core denote integral formulate elaborate role among characteristic ergodic set value coincide average dynamic tu player tu games whereby tu nonempty game game dynamic instantaneous game tu vary core nonempty assumption average nonempty introduce steady average game subject fluctuation instantaneous game instantaneous tu player instantaneous assume budget allocation within eq priori starts central know bound know function know observe give line paper goal difference integral reward excess give answer question allocation yes core game nominal rule converge priori say dimensional motivate think stability sense provide statement henceforth symbol f r ft cv look require full know sign relaxed network flow material resource different production serve one demand value realize demand hold therefore also particular play individually cost single serve dash cycle demand ht play agree equal total among serve reference comment apply cycle v topology figure describe clear look subgraph vertex except subgraph connect e player share part conversely subgraph serve represent capture player nonempty reflect nonempty q derivation player also st bound tu model generic dynamic description game discuss encourage play core run occur edge hypergraph player game edge per player generic player associate incidence describe arise naturally allocation demand correspond vertex translate satisfy specifically last satisfied sign condition admit limit driving reach note tb dynamic xt state capture characterize control induce previous full rule observe detail subsection core study allocation satisfy probability solution structure generic pseudo inverse allocation also depend control obtain converge core average subsection elaborate structure highlight feedback review sign since xt xt threshold generality reference component tell choose every accumulate return yes actual ellipsoid optimization cut feasibility us comment condition certainly excess imply content generic development full case know intuitive last positively speed nominal nominal reflect aspect demand stay fulfil theorem next principle share similarity section let close euclidean discrete time analog excess dynamic consider result controller strength result light convergence one let version continuous translate principle constitute guarantee player repeat game payoff root production application player cumulative payoff sup vector payoff player share notion refer describe set proper control independently notion follow sketch formal proof aforementione notion reader continuous differentiable condition strictly formulate prove control set condition payoff instantaneous payoff player h cumulative payoff condition vx towards interior use lyapunov stability start lyapunov establish condition martingale surely tt tt b tt consequence assumption condition imply therefore far integrate last yield u u claim lyapunov true tt xt b tt x tt also conclude cv k cv controller invoke condition evident kx tx conclude invoke condition attain evident thus equivalent tu game value interval convex interval know long run e balanced simulation instantaneous game behavior random r r tp else go go construction interior generate game probability pair simulation run pair take size nominal ensure min u instantaneous allocation allow allow game threshold robust illustrate far law ensure illustrate crucial ensure translate instantaneous allow great next probability behavior corollary law converge core instantaneous lie neighborhood nominal uncertainty nominal study instant unknown robust use lyapunov lyapunov law scheme gd know gd allocation design average capture expectation question like thank explore laboratory university usa mail consider dynamic central continuously instantaneous subject game know bound mean ergodic long run hand generate allocation reward convergence average drive priori cone allocation vary nature highlight extra core converge priori allocation observation
prior imply eq g denote denominator prior write infinite bernstein density mixture normal evaluate side mixing construct alternate augmentation difficulty symmetric build classic represent make proper mix crucially mix monotone moreover completely bernstein return bridge positive focus refer usage west robust densitie connection kernel symmetric integrable let line kernel p constant mixture normalize constant mixture drop originally work stable quite bridge analogy slice form likelihood prior normalization example slice auxiliary interest marginal able already monotone invert slice uniquely course calculation simply case invert wider well know variable latent else choose kernel mix wide bridge especially look like additionally match fact component beyond pose become representation turn case may simulate extended case strategy important bernstein monotonic sx measure implicitly evaluate mixture limit inversion formula expression give discussion lead main could mix bayesian bridge recommend default package support recommendation describe study briefly conclusion exhibit well interaction substantial design matrix orthogonal representation lead roughly principal covariate expand arise discuss finally power conditional orthogonal alpha place appeal omit long normal lead simple introduce far implicitly apply slice latent bridge region define sampler start iterate step generate update truncate normal least simulate e p come invert define back fact center usual naturally account case fact component represent mode run triangle identify try show recent draw marginal draw far plot recent two repeat identification mode parameter directly favorable evident give take hyperparameter draw gamma distribution transformation consider useful proposal approximate posterior introduction show ability local scale crucial ahead reflect shape beta walk level diabetes patient available lar predictor amount lasso concavity diabetes random sampler autocorrelation posterior draw fit bridge lr qr outcome encode desire within introduce term likelihood logit mixture gaussian detail diabetes solid penalize validation dash line predictor standardize bridge diabetes bridge default prior bridge generalize em predictor response center step mcmc relatively difference bayesian classical fit density notable joint bayesian bridge distinct mode coefficient extent predictor case satisfactory summarize classical force coincide attribute ignore classical fundamentally posterior difference mode posterior tc bayesian coefficient say predictor sign objective sense tc copy mark mode mode conclusion posterior different clearly involve median house census plus interaction quantitative concentration package predict environmental predictor use create train split split estimated square bridge bayesian bridge method case standardize bridge regularization bridge jeffreys bridge set implement experiment file compute test case bridge estimator choice nearly square error different bridge bayesian estimate involve construct coefficient simulate correlated residual loading typical simulated bridge bridge assess square error convergence algorithm assess identical machine carlo bridge estimator jeffreys gamma square estimating square classical bayes classical factor tool bridge bridge posterior model summary importance predictor substantial estimate squared loss existence generalization mention scheme bridge virtue directly capable orthogonal mix strongly limit normal method work stable perform global model study file concavity incorporate naturally implement r available explore bridge across wide commonly situation clearly density decrease continuity q fx gs beta give conclude mix regression key bridge normal stable component turn complementary efficiency representation orthogonal problem avoid need exponentially notably wide explicit slice normal explore classical variety datum simulate fitting exhibit excellent mix parameter favorable analogous algorithm sparse bayesian package extensive provide file analogue unknown vector bridge minimizer shrinkage lasso find recent focus asymptotic strategy compute work adopt bayesian perspective bridge arise product power proceed markov stationary bridge group exponential yet penalty crucial concave prior meet bridge dominate although oracle property relevance correspond distribution important yield desirable situation avoid coefficient oracle bridge explore multimodal surface view argument full thing summarize multimodal surface matter serious computational target convex approach seem explore good augmentation strategy mcmc behave anneal never addition present cross validation bridge orthogonal supplement iteration start difference rate control sparsity equally difference bridge form account vertical axis model object comparison rich well uncertainty quantification tail mcmc compare heavy prior advantage free bayesian model
chinese avoid completely world illustrate e facebook information typically available know technique visit challenge depict connect walk rw proportional simplicity illustration rw sample population face resource sampling edge graph setting might poorly mix shown determine fast walk practical heuristic balance relevant part example visit facebook irrelevant resolution category contribution weight relevant measurement facebook time simple walk outline rest paper follow graph sampling combine present unified take account various trade simulation college facebook present conclude scope node list neighbor weight volume weight q volume collect sample contain copy briefly weight build uniformly random node independently frame publicly allocate nevertheless comparison walk possible connect metropolis hasting transition probability desired randomly move rw outperform comparing rw rw choose proportional weight stationary basic design next family sec rw weighting walk sample estimate whenever china take china bad naturally survey partition overlap next select budget prop q another opt minimize every carry may application variance category mean average allocation sometimes give allocation n easy variance translate length opt prop prop long opt order gain gain evaluation section efficiency equally translate interested alternative maximize precision comparison lagrange multiplier gain assume know size application allocation gain many want interested node white similarly facebook interested student account cover prop optimally become gain calculate compose category gain allocation let look work many typically simplification simplify compare node degree category rather entire section allocation independence typically impossible graph available primitive ng minimize although analytically specific topology address limited paper arise graph every undirected edge edge fine collect sample e individual simplification category ideally enforce strictly sample exactly category node discard natural mass equilibrium weight draw try many count category edge weight advantage weight evenly across central factor probability modify arise graph c ic sec resolution fortunately easy category volume information enough run pilot rw derive category weight plug toy interested find red dark middle distinguish type eq white similar show appendix fig due irrelevant benefit contribute estimation need fast goal lt optical system fundamental exception toy clique node weight sec relative category behave black category suggest sample category small fig length goal replace formulation pilot rw sec modifications weight inter edge weight end edge node weight intra category edge inter opposite assign enter stay short intuition induce draw assignment take geometric mixing alone avoid effect hybrid edge assignment pilot rw category fortunately additional especially page sec college neighbor friend pilot rw consider node neighbor facebook sec robustness sec pilot rw facebook set black natural interpret exploit pilot appear poorly node relatively high extreme however induce node connect recommend use facebook maximal category relatively sec extreme grow graph skewness degree rw rw almost equally cluster rw advantage node nevertheless later sec affect significantly fig e course grow weight likely visit rw drop monotonically u shape confirm indeed plug optimal w ec ec minimize already increase high demonstrate black discuss presence tight course vanish relevant control obtain consequently gain achieve exploration reduce factor much length cluster rw perform nevertheless show significantly sometimes significantly outperform walk improve far focus set gain equivalently fortunately line drastically nevertheless drop address category volume estimation sec treat volume affect weight sec unlikely even bring benefit avoid ii size still category form tight community effect choose translate community rw hybrid arithmetic hybrid c college f concrete facebook motivate purpose facebook member college membership publicly available interesting question college college college college evenly default facebook name friend together membership facebook interface inform pilot rw sec pilot visit volume among neighbor greatly outperform volume cover several resolution college student pilot rw fraction irrelevant small phase collect edge resolution hybrid rw baseline table user time bandwidth cost perform rw hybrid geometric arithmetic total college rw come vast majority effort collect geometric arithmetic sample value low target suggest hybrid reach agreement sec finally discover unique rw seem rw facebook look facebook friend user rw advantage rw avoid node irrelevant category one feature important differ facebook college exhibit heterogeneity fair discover rw actually rw span order heavily confirm small per college around rw college rw college well size facebook medium college come walk three rw fail mit college majority middle sized small number rw walk contain rw agreement fig produce reliable estimate finally aggregate gain error ground collect type rw length expect gap rw version rw l pilot rw naive relative size sample base plain dash collect effect recall describe resolution choice try shorter run ten sample small college rw facebook member greatly range member aggregate sampling rw perform rw collect college average sample typically per college length per college outperform rw overhead pilot rw attribute college compare rw factor important robust way resolve implementation minor bring benefit www variant however introduce towards degree impossible later walk rw rw remove bias walk alternatively rw rw rw web therefore rw walk outside walk efficiency rw walks walk large equal orthogonal side fast mixing target applicable knowledge estimate sec perfect assign loop likely importantly face poor sec statistic relate flow random walk equivalently reversible markov state knowledge design explicitly close discuss online employ edge link sample approximately uniform population paper extract something unknown world wide web technique follow page specify interest avoid instead queue queue suffer regular strongly able sample
balanced subtree require inner product comparable product trajectory cost negligible define deterministic element u preserve include exclude u r u long exclude generate equation build stop either final exclude since start satisfied full build stop leave meet state subtree j u u j r j build r building resample initial first explicitly store momentum slice momentum visit indicate meet subtree subtree repeatedly call stop new return satisfied position union return clearly leave leave r u resample momentum simulate hamiltonian begin low encounter slice choosing position height subtree introduce boost jump require operation stop practice dominate algorithm however store momentum memory transition large jump stop easily soon stop correctness save use transition another leave uniform kernel require momentum position disjoint hasting satisfie I leave empty set iteration return final old replace sampling uniform invariant propose half invariant build jump consider reverse try jump fail try jump rejection avoid associate able uniformly return contain maintain exploit subtree explore leaf subtree multiply choose observation build subtree layer build sample subtree two give pair represent subtree subtree continue complete subtree return weight encoding store momentum usually algorithm implement improvement matlab implement part package step attention parameter propose adaptation specifically adaptation base implicitly vanish stochastic statistic aspect behavior metropolis average acceptance boundedness iterate meet converge schedule satisfie satisfy long impact adaptation unchanged say enough stationary therefore burn phase burn quite ideally quickly shift sampler initial regime size iteration issue algorithm nonsmooth find subgradient straightforward adapt mcmc adaptation want towards schedule define clearly go converge update slightly elaborate introduce issue mcmc adaptation conventional optimization setting improve stability iteration prevent computation simulation length iterate forget produce burn apparent setting originally nesterov computation allow affect eventually rewrite feature need try hmc specify produce reasonable well consistently hmc test entirely setting may work neither small computation high rejection rate tune probability strong simulation produce metropolis acceptance probability position momentum th proposal equilibrium accept reject acceptance iteration reach state chain momentum chain understand acceptance hmc momentum explore mr r r r j r base r recursion implicitly build leave r j n r r averaging scheme target convergence recommend heuristic english value langevin result enough amount recommend dual algorithm preference less value save algorithm hmc specify incorporate averaging derive initialization scheme acceptance require implementation package examine effectiveness outline average sampler hmc hmc distribution allow adapt averaging update first iteration hmc space test successive large try evenly space evenly hmc simulation length target iteration different seed hmc term sample ess gradient ess distribution precision ess worth purpose give large ess computation evaluation proxy overhead hmc gradient estimate experiment discard burn ess ess inherently test multivariate ess sampler dimension ess central moment take long reporting size well length hmc anti yield low variance ess evaluate hmc precision matrix wishart target correlation uci repository customer customer receive regression normalize give variance predictor datum customer parameter exponential expand vector make challenging dimension remain customer weak relatively stochastic volatility day return generate refer integrate speed mixing versus target statistic post burn job somewhat attribute regime stochastic algorithm appropriate value since rejection cause lead slow convergence iterate volatility burn converge histogram trajectory length power turn equation complete intermediate desirable half trajectory balance state visit rate turn back compare efficiency length choose length automatically tune evaluation seem occur suggest occur seem seem reasonable problem produce hmc volatility hmc well ess expected test good simulation hmc varied factor optimal hmc usually preliminary figure efficiently preliminary tune hmc section hmc demonstrate advantage rwm gibbs run rwm section run first burn rwm normal whose produce theoretically cost per rwm effectively identical cost algorithm run gibbs long rwm multivariate cost evaluation nonetheless rwm visually independent mcmc algorithms rwm leave independent visualize relative rwm moderately correlation rwm rotation appropriate rotation expensive rwm gibbs highly optimistic parameter multiplication rwm result require time per expensive transformation rwm efficiently tune present sampler hamiltonian monte hmc hmc ability effectively parameter hmc make run place mcmc stochastic paper extension several review consider hmc introduce covariance bad hmc window simply trajectory step size expense window tune lack accept responsible gain introduce manifold hmc hamiltonian space effectively mass although bad inversion expensive dimension function stand ability adapt matrix explore hybrid seem hmc unconstraine value target everywhere handle trajectory make hmc present since tie hard eliminate change probability hmc stop momentum vector region short point visit make progress dual make hand desirable hmc valuable experience particular without intervention suit largely much currently develop core value able effectively sample posterior order magnitude fast summary inference minimal intervention allow researcher datum target markov chain mcmc produce set number would lag estimate ess compute eq estimate precision separate chain try autocorrelation serious fair necessarily lag bad yield ess give interval autoregressive come expense costly quality cutoff find precise cutoff hamiltonian monte many inform feature converge dimensional metropolis sensitive step small undesirable walk behavior turn hmc step recursive algorithm start perform efficiently sometimes tune hmc method require intervention costly adapt dual averaging automatic inference efficient sampling chain monte carlo hamiltonian carlo monte model machine rarely researcher practitioner resort inference review rather series sometimes deterministic counterpart method walk metropolis gibbs require long converge tendency via inefficient walk continuous discrete hamiltonian monte mean scheme transform target simulate hamiltonian cost hmc roughly cost hmc come hmc posterior impossible hmc step simulate hamiltonian system literature set costly tuning hmc challenge hmc general inference http net contribution sampler resemble hmc choose problematic tuning hmc make tuning version sometimes cost find tuning hmc bring generic unable mcmc hmc momentum usual variable standard unnormalized augment system position dimensional denote momentum particle energy simulate update depend coordinate preserve volume region remain map hamiltonian monte describe identity denote covariance momentum generating proposal position necessary simplify addition analogous slice seem present trace process build leaf correspond figure subtree overall start double e start make u stop simulation care trajectory implement derivation follow motivate build intuition use
heart gmm base decoder gmm describe mixture gaussian simplify mean gmm gaussians function decoder posteriori map decoder posteriori decoder heart therein understand concentrate gmm involve gaussian orientation e distribution sort follow next us selection oracle situation signal generality decay dimension understand kl second compare monotonic analyzing help understand lead cyclic maximize direction increase gaussian basis common second give ratio increase check maximize orthogonal writing observation angle principal component gaussian go eigenvalue indicate large lead check anti diagonal zero elsewhere give bottom decrease take eigenvalue correct carlo simulation figure principal gaussian go illustrate behavior similar divergence increase increase roughly rapidly quickly towards cc gaussians go correct oracle hold orthogonal plot give increase eigenvalue decay fast gaussians rapidly show gaussian high htbp two decay gaussian one signal follow via monte expectation draw investigate result number sense correct selection reconstruction going assume gaussian eigenvalue decay random go random dimension even remain high rapidly increase converge stand certain note close far fix indicate accurate low energy concentrate principal interested error energy go obtain reach value cc measurement b normalize plot correct mse normalize ideal energy eigenvalue decay mse respectively converge go htbp cc different eigenvalue signal gmm influence geometry sense selection gaussians higher concentrated assume covariance j real posteriori maximization em iteratively signal gmm map apply sense conventional cs iterative two assume j following assume signal signal code estimate index use assign gaussian well estimate complexity map cholesky map describe iterate map observe coordinate natural image geometry motivate gmm conventional practice decompose patch patch consider follow patch decoder em initialize geometry capture algorithm parameter standard berkeley database contain image dictionary show decoder calculate image house sense slide regard extract subsampling realization sampling outperform sc gain db gmm measure use ideal improve db high subsampling low rate sense gain learn dictionary compare slide regard patch house patch sampling sc gain db rate substantial illustrate truth show first outperform cs contour rd htbp reconstruct image individual aggregate whole illustrate patch remove considerably image sense patch dramatically increase rate nevertheless reconstruct computable sense operator matrix diagonal operator figure typical far remove improves reconstruct generate subsampling reconstruction rate former db rate cost rate compress oppose aim sense reconstruct collection cs depth sense measurement small conventional cs optimal implement pursuit bound constant failure rip upper term efficiently piecewise estimator selection heart decoder term sense compressed present gmm compare considerably compress formal compressed study compress sensing generate nsf author thank conjecture definition compressed aim introduce depth follow sense require conventional cs implement via filter fast pursuit orient yet result calculate multiple gaussian unknown heart decoder analyze sense expectation maximization iteratively signal application show conventional considerably sensing aim rate small interest consist encoder reconstruction reconstruct ill pose require signal frequency classic shannon theory conventional whose column approximation amplitude random minimization greedy sparsity reconstruction obtain approximation upper achieve dictionary basis investigate sparse well novel framework cs oppose cs aim efficiently collection signal reconstruction instead restrict work general bayesian pdf encoder decoder error bound relative conventional signal implement moreover mixture describe decoder motivation first control average real effective example signal overlap short signal signal signal signal real signal art image miss mathematical adopt conventional cs reduce optimal linear significantly cs error cs sensing time bind extend gmm introduce accuracy heart analyze property sense measurement general selection compress presents posteriori gmm calculate map algorithm apply improve cs cost reconstruction discuss rest eigenvalue error orthonormal pca pca vector generality pca bernoulli universal analyze canonical cs perfectly kk belong condition perfect must exist possible construct size requirement mm compress sense reconstruction measurement signal reconstruct position non measurement thus suffice concentrate signal gaussian full eigenvalue decay analogy conventional mention simplify without generality gaussian one always center signal prior e e mse pursuit calculate signal fast via filtering store since decoder seed change equivalent nan index condition decoder optimality hold hold q nan constant nan decoder follow choice decoder necessity let otherwise split vector deduce proceed mse comparing require good requirement thank linearity good prove decoder nan consequence theorems gaussian mse constant optimal instance follow mae mae decoder consider instance optimality isometry rip measure preserve conventional rip order requirement rip support positive integer let index rip rip block rip sparsity consecutive relate distribution let satisfie decoder assume e rip something conventional sense signal index inequality rip fact realization verify rip address concentration eq rip outside great rip require next multiply rip fail subspace let draw accord rip great prescribe matrix rip fail whenever exponent exponential enough ensure prove nan rip linear rip coefficient pre consequence measurement gaussian bernoulli rip rip improvement signal gaussian sense example
also maximum likelihood two average svd mle via distribution near page near object http bin fit fisher analysis near focus direction direction direction close direction object pair lt unit lt lt meaningful identify treat plane sign preserve svd q analyze either statistic preliminary size hypothesis uniformity nan uniformity statistic chi let column similarly uniformity almost fisher two clarity fisher expansion add mle normalize truncate maximize implementation bfgs start gradient find reject reject normalizing series next gradient direction add descent respectively agree mle mle improve mle expansion aic criterion minimize statistic degree ccc aic show numerator check nk generator generator look coincide module theoretic nk derive let long odd number respect lebesgue value eq odd mm mm langevin rotation group gradient descent expansion normalize constant estimate compare manifold illustrate keyword algebraic gradient method series evaluate normalize constant rotation manifold exponential family manifold family nice normalize I generate calculation however define expansion integration monte iterative recently introduce likelihood normalize integral involve module equation constant simple normalizing von respect assume simplicity maximum method update accord dc gradient modify know give point obtain numerically key give compute numerical repeat converge determine need direct normalize partial contain let numerical paper apply gradient descent rotation orthonormal manifold study number orthogonal illustrate develop integration distribution normalize normalize organization rest section fact group manifold section equation satisfied normalizing section series normalize constant datum notation summarize preliminary fact interested orthonormal denote matrix transpose orthogonal group volume probability call denote two prove similarly qx qx ie rr value decomposition svd let svd also preserve singular fisher distribution fisher fact chapter distribution r matrix argument group correspondence projective show one dimensional integral normalizing constant argument section derive differential column question distribution fisher fisher distribution redundant however fisher state fisher dimensional smooth measure open neighborhood function hull sufficient see zero p ie ie ie e ie ie ie l e x v r let svd solution preserve explicitly remark normalize sample preserve ia ij jk il ij cd polynomial denote ideal diagonal diag operator rational proposition dimensionality gr find ode apply maximum ideal ideal dimensional prove utilize gr generator function representation generator ideal span field rational function proposition large gr basis program website conjecture dimensional operator respectively denote generate matrix rr differential rational equal span proved computation program conjecture consequently differential well differential equation differential differential polynomial differential operator equation show extra differential put normalize satisfie equation utilize expansion normalize next subsection computation explain auxiliary operator ideal normalize find follow q e find ij obtain coefficient polynomial solve candidate also ij n analogously eq normally order system collect ij put analogously method q straightforward calculation theorem explain evaluate derivative truncate series expansion extend equation ode constant restrict differential satisfied apply broad guess search point mle guess system become current heuristic grid make exhaustive
applying alternatively verify choice satisfie radius appendix fix high extend program recall expanding imply triangle ii schwarz h old conclude follow apply cm lemma assumption ccc department berkeley berkeley version technical report berkeley solve matrix noisy transformation sum matrix complementary statistical share give bound pair norm impose incoherence result study plus plus frobenius stochastic noise matrix approximately low approximately identity operator establish show result theoretical confirm decomposition problem suppose recover pair course ill pose necessary denote allow form sparsity matrix motivated classical reduction matrix problem robust covariance describe plus arise decompose decomposition arise collection task common weighting preserve model discussion motivate study use linear operator simply map task linear operator deterministic stochastic exactly assume instance observation version involve noise matrix form share across across observation decomposition analyze past noiseless case entry give sufficient adversarial zero analyze uniformly xu analyze aware detail yield oracle structural impose decomposable decomposable regularizer theorem solve class convex program composite regularizer frobenius sparse asymptotic sparse corollary corollary broad class model deterministic rank observation addition general identity operator error estimator see feature impose incoherence condition norm sparsity dual bind proven set identifiability exact noiseless provide degree identifiability arise devoted corollary decomposition bind devoted proof technical aspect defer discussion convenience ordinary family form act reference therein applicable regularizer satisfy property particular work motivate factor random generate I load project onto guarantee need span nonetheless ty l relatively constraint via see give problem j write multi application share across type share model impose extreme enforce however learn complicated subset task subset substantially task amazon book include user numerical mean category kind namely differ significantly baseline instead row zero row discuss appropriate enforce column sparsity general notation obtain linear corruption straightforward perform index subset corrupt corruption z sample covariance center wishart remain write column write column entry sparse example observation consider estimator solving regularize negative regularizer guarantee property estimator although reasonable yield attractive property practice additional noiseless setting indeed recover unless incoherent position decomposition low sufficient exact impose condition introduce past common statistical realistic observation contribution mean substantially regularizer quantity moreover norm estimator intuition sparsity sparse motivating include factor non sparse formulation robust pca gauss sparsity regularizer choice general take form involve serve type control prove relaxation appropriate signal maximally position lie decomposition impose low rank component singular vector quantity coherence singular canonical remarkable make exact goal noisy concern recover singular impose moreover bind incoherence poorly behave small number small include covariance xu et norm regularizer analogous verify constraint serve component manner natural noise ratio stay fix extreme maximally zero column lie discuss consequence apply belong decomposable regularizer restrict norm decomposable regularizer strong appropriate result devoted consequence complement convex show special operator show fail notion term subspace need special example subspace think choose complement guarantee deviation penalize decomposable subspace pair relevance example decomposable appropriately subspace kk decomposable define compatibility compatibility frobenius regularizer subspace yield convexity establish quadratic loss function convexity lower strong convexity restrict convexity weak condition norm regularizer weight correspond minimum well convex see strong respect norm operator choice non choice rsc establish error error estimating identifiability rsc state observation later subsection result result use across matrix notation model low frobenius consist three rsc rsc curvature exist regularization universal remark clear theorem deterministic statement apply convex program whole bound choice optimize obtain upper condition multi rsc hold correspond complexity sub associate correspond represent similar interpretation correspond index adaptively simple rsc condition low lie within decomposable subspace vanish theorem guarantee square frobenius error regularizer decomposable respect note decomposable regularizer decomposable program constant matrix index note theorem decomposable verify jk claim index subset integer far low natural approximation vanish guarantee noiseless approximate namely eq guarantee incoherence singular singular factor identifiability model operator consider fact lower precise improved impose restriction incoherence impose exclude attain bound however estimate soft entry component interestingly understand first step co convex step component minimize sparse constant observation regularization solve optimality descent method operator rely inequality hold trivially first introduce example show return consider one give small singular let consequently good mean exploit fact reasonable tail concentrate expectation large consequence method decomposable keep presentation relatively brief regularization arbitrary cardinality directly previously example norm discuss decomposable subspace rank column vanish bind eq slight modification exploit much small difficult compare xu exact concrete matrix suppose satisfie program great corollary upper bind reduce corollary interpret freedom somewhat subtle estimating column embed within frobenius selection parameter sub involve multiply usual discuss consequence column wise consequence theorem consider covariance sample satisfy column solve sdp great comment motivation concrete parameter involve operator factor case achievable namely achieve frobenius error turn complementary fundamental algorithmic independent limit question analyze infimum observation model give jk interest family corollary minimax risk family observation q agreement guarantee corollary respectively discuss corollary logarithmic careful relaxation worth involve noiseless analogous column program adapt excellent agreement square vector form sparse choosing position zero uniformly nuclear motivated matrix study study scaling predict universal fix square complementary scaling sparsity vary since plot agreement theoretical ccc frobenius corrupt entry growth square theory error e theory neighborhood around ccc plot frobenius error plot error predict curve dimension reach sparse matrix dependence interest different phenomenon report vary dimension zero plot dimension choice decrease scale error produce linear plot panel b moreover main proof defer assumption regularizer decomposable throughout convenient shorthand deal weight decomposition q upper result restrict algebra show slack frobenius convexity taylor complete algebra h triangle allow upper condition obtain inequality side yield combine lemma rearrange special corollary regularizer requirement pair little z w j requirement column gaussian variance bind union state turn rsc multivariate show rsc careful bounding reader wishart standard hence tail begin recall define error proof lemma component proposition optimal writing inequality side obtain choice claim addition low bound somewhat appendix bind order condition early upper noting concentration measure conclude probability replace refined detail noise singular wishart high quantity bound claim proof reduction packing collection pack q packing consequence component bind choice different bound technique begin prove matrix decomposition involve radius non identifiability namely scale construct packing matrix low verify matrix distribute pack pair pack equal piece imply eq pack subtle one component modify moreover matrix modification set interest packing integer conclude exist cardinality apply thereby minimax suffice divergence suffice exclude degenerate suitably sufficient thereby follow argument packing set bad denote construction pack verify distinct frobenius packing ensure control guarantee set characterize packing set sparse pack integer k n cardinality satisfy suitably adapt rate lemma apply state packing claim theorem analyze relaxation general class decomposition goal
distribution convergent motivation come shape attempt use shape distance explicit shape space next space linear functional word denote simple discrete row fix particularly form simple let suppose verify satisfie result dual us norm case large merely onto return norm yield metric return maximize project actual mind build take eq value linear linear interesting pick particular subset know hilbert distance rewrite simplification familiar us kernel section compare specifically surface viewpoint treat shape easy surface discretization yield invoke shape surface information location point hausdorff curve require continue surface geometric measure tool geometric surface encode orientation surface generalize form whose cross product vector imply simple combination vector generalize tangent capture one expression differential understand linear form use denote appropriate tangent vector surface action understand integrating point capture action definition continuous generalize orient manifold include relation orient manifold relationship schwarz distribution space back variation total variation variation manner example induce irrespective curve distance disjoint ball retrieve kernel instance vector expand represent transpose precisely lead alternate current introduction distance review provide tailor reader background computer exposure analysis geometric aspect shape g surface space rkhs structure elegant mathematical recent decrease simple kernel use root symmetry transformation transformation general induced multiplicative need construction theoretically symmetric difference set expression cardinality play kernel kp kp p act square view distant similarity similarity point another kernel distance naive would sharp similarity uncertainty feature kernel beyond set surface merely generalization intuition deep mathematic weight merely sum cross integration distribution usual let consider convenience continuous entirely rigorous think merely space vector whose coordinate entry interpret generalize look eigenfunction generalization definite positive kernel induce hilbert useful operator operate kernel take virtue integral analog decomposition positive positive definite eigenfunction coordinate eigenvector describe span euclidean instead euclidean capture definite euclidean map
frequency classification build shape fourier coefficient reveal sub set neuron unique spike unique word know neuron vice versa micro record neuron gmm detect neuron every spike necessary window tt detect spike entire micro long far spike since signal local window spike thus know tell look characterize spike visually quantitative automated rule spike spike detect outlier bias spike favor spike neuron exceed spike fire neuron characterize spike spike rule move spike identically iid signal slide window reveal spike histogram fig simulate play reason assume brain background noise neighborhood micro iid histogram slide background iid signal due lower indeed though know slow know clearly bottom move spike spike concern separate boundary extract spike spike fire firing interval lower good call neuron spike unit extract possible spike step take avoid slight spike sort adjust position show apply slide exclude visually spike seem noise low spike signal dynamic structure induce yield density series define sense process decomposition hence indicate important variance peak contribute integral spectrum fourier frequency frequent generate variance representative system correspond nonparametric raw series log signal stationary large sub system thus plot log minimum wave minimum figure separation match close overlap raw fit show axis represent fouri frequency variation series frequency cycle cycle year generation short business cycle two show growth center right code red green intensity equal n connectivity cluster confirm separate three major highly persistent slow business highly persistent affect business peak flat frequency short cycle slightly business red matter global affected rely heavily production technology affect global happen map effective policy boost effective public affect global try fit spike summarize greatly accelerate computation fit mixture gmm logarithm spike histogram peak correspond differently shape spike assign spike posteriori model accord high bic choose run gmm spike show conservative shape represent gmm neuron domain advantage domain signal fit monotonically separate spike relevant spike behavior simulation prominent occur shift easy reveal five spike domain spike introduce novel detect classify dynamic signal dynamic shape stationary signal auto stationary spectra research area process sort pattern economic demonstrate usefulness present also introduce neuron spike differently shape literature robust threshold chemical band hz dataset average entire rate remove overall us baseline rv finite alphabet kullback leibler kl particular sample dirac delta rv estimator mle data divergence intuitive empirical plugging kl log maximization conversely em spectra allow compute model em algorithm claim theorem exercise theorem summary department maximization em group time class dynamic magnitude fourier dft spectrum pmf similar pdfs pdfs parametric robust model shape stationary auto correlation spike sort stationary pattern economic usefulness stationary k example brain speech stationary economic researcher study send understand fast brain neuron necessary send neuron economic public market characterize country recover region formally entity economic divide homogeneous dynamic different neuron economic market need many dimension reduction correlation show carefully distinguish series extremely impractical observation fit average study test distinguish two necessarily independent datum see detailed although small suffer model wrong sense hard metric clustering distance similarity spectra treat signal pmf circle algorithm avoid cluster series macro economic decade law economic country us state economic help provide economic support overcome certain state show year business heavily analyze auto order adjust growth select base unlikely different dynamic particular model business dynamic spectra growth adjust store environment must visual back brain neuron front aside neuron measure top signal brain spike two spike visible macro measurement help
ccccc space decomposition mixed piecewise linearity comes induce link strategy consider internal bi piecewise linearity write belong additional assumption thus mapping partial mapping polytope inverse piecewise subset chapter vertex state lemma fix belong simplex belong possible decomposition associate convex get linearity conclude proof piecewise lipschitz norm contribution rely bi piecewise mean decomposition thus exist piecewise pp lemma bi provide show satisfied one p p argument monitor strategy compute mapping compute set ap estimate distribution exploration usual monitor ingredient unbiased extract I take conditional expectation equal first take argument precisely sake simplicity provide fix next simply proceed regime time round consider depend satisfied bi fed eq fed efficient turn respective indicate finitely round possibly successively notation th statement immediate linearity definition assertion direct element remain assertion q forecaster th block hoeffding sum hilbert space martingale get assertion q since lipschitz norm union block application assertion get put thank triangle length exploration need average result action pure setting rely parameter play form compute projection set ensure q slight modification one square third boundedness state inequality follow compare sum integral therefore conclude extension polynomially obtain indicate polynomially average consider variant block polynomially weight average call hence version ensure depend proof closely resort hilbert martingale translate begin lemma regime union regime substitute suffice original convergence robust guarantee get put thing obtain end constant ensure result minimization game result vector game monitor bi linearity satisfied consider sometimes scalar payoff player section q action second round regime external player round define maximum regret indeed first bad player play action induce law action actually play theorem explain however achieve question could strategy game hand proof partial monitoring strategy external latter case payoff consider j actually mapping fix coordinate true convex hull linear piecewise function exhibit consider therein end lipschitz denote soon belong latter norm define claim see follow indicate indicate statement component entail argument piecewise equation refer rp evolve way piecewise piecewise equation onto hence internal monitoring follow internal stage action play action player stage partial evaluate payoff pure element stage pure play convenient first mixed thin g measure round stage internal first player measure minimization swap regret definition swap regret easily construct direct swap internal two present strategy strategy grid complexity exponential idea reference provide explicit strategy player corollary idea role player proceed show assumption swap fourth final possess bind cone g q q suffice piecewise piecewise accord separately project equality bind swap follow p rp g r rp g rp prove norm bound j convergence conclude claim statement prove end l triangle inequality fact probability conclude simply resort write triangle schwarz final conclude exist game monitor follow bi mixed action play play marginal show set game dirac denote concavity strict side product also bi piecewise linearity show exist per linear consider conclude thank make argument notation therein payoff mapping mapping refer substitute satisfy interpretations hold note component construction definition show work positivity component payoff eq extend without piecewise linearity strategy rate complexity bi piecewise game devote scalar payoff piecewise linear equation replace extension polytope intersection original necessarily bi piecewise game payoff define eq equivalent sense translate hold lemma least cardinality vertex increase quality playing regime close strategy round fine around grid bb nan q form payoff element locate position close resort characterization pure necessary q rewrite correspond action form positive square equal schwarz last signal probability put signal indicate martingale whose generate since show sum variance sum get union onto convex proof paper appear conference page paris paris paris france paris en condition become adversarial setup game ambiguity belong rather tackle game partial monitoring algorithm regret partial monitoring strategy theory tool learning algorithm game numerous theory geometric situation easily quantity regret minimize game whose duration fix analyze perspective monitoring monitoring maker get maker setup include reward setup tell decision nothing obtain repeat situation gain situation one signal reward statistic maker regret minimization paper learn bad maker thank extra vanish procedure case base regret minimization extension partial value importantly concrete problem work constant seminal theoretical study concrete matter recall fact theory decision maker opponent propose setup term value decision maker obtain represent ambiguity concern reward linear maker robust value monitoring type property bi exploit game rich enough useful section constructive constructive highly inefficient sort probability mix require refined number external partial simple similar simple mention explain special efficiently convex although recall value payoff consider two player maker player refer reward payoff mapping whether pure action denote strategy action obtain accord denote monitor take opponent round monitoring indicate reveal action player strategy player ensure value payoff almost uniformly form often strategie player approach soon sometimes player strategy soon sequentially action monitor bandit monitoring reveal exist deterministic close simple consequence two pure mixed action observe require monitor play depend take projection norm solve equation inner case minimax determine easily zero associate enjoy follow theorem mix take pure strategy least imply monitor modify lemma constructive probability since equivalent equivalence indeed state mp e one arise former pure action mix include p repeat lemma indicate translate rate rate instance robust projection favorable g minimization external monitoring play mixed action play pure obvious bandit monitoring play ta previous value pay loss efficiency function uniformly x yx norm differently continuity q lemma indicate boundedness argument entail uniformly continuous argument close extend consider hoc contradiction assume satisfied necessity apply suffice follow concavity robust mixed action approximately player introduce calibrate first consider constraint denote ball choose forecast l choose denote nonnegative player score consider consist ensure probability existence calibrate base calibration illustrate proceeding regime dyadic round carefully term length modulus continuity argument calibrate associate element
process replace absolutely continuous lebesgue change specifically index whereas induce dynamic adapt satisfied integrable h log ratio version sufficiently understand locally additionally log admit suffice due end shift clearly remain martingale theorem pre follow vanish condition integral sense definition stochastic delay measure term pre optimality continuous generalizing sufficient brownian satisfied fractional brownian fractional drift brownian motion exponent way drift brownian motion section corollary proposition rgb sequentially test version special optimize criterion fractional index exponent quick problem application area track surveillance security rule raise soon occur detection small false formulation change detection trade goal develop change consider formulation exhaustive sequential book poor cumulative page quantify delay change criterion subject upper identically criterion attain prove optimization optimality continuous brownian drift case signal noise product obtain optimizes modify version diffusion satisfy criterion diffusion path process admit thus clearly exist optimality particular optimal detecting change fractional brownian fractional detect process polynomial term exponent rare recover point view presence application phenomenon characterize long memory result build traffic finance diverse great fractional calculus study drift coefficient process rao study version mle rao optimality process vanish coordinate every tt mutually restrict algebra denote change deterministic mutually stop follow consider constrain quadratic problem coincide explain inclusion redundant proportional eq formulation detection delay period actual accumulate variation latter appealing leibl detection dynamic detection define solve define satisfy correspond false alarm introduction detection diffusion full condition additionally diffusion ratio optimal prove reveal vanishing quadratic pg u stop event notion leibl consequence close expression function let introduce statistic consequently clear give definition u consequently moreover condition side monotone increase condition tt iterate p otherwise become infinite due contradiction go back simultaneously monotone convergence right side interesting terminate surely sure alarm explain constraint impose false alarm occur continuity increase word bad scenario alarm threshold generality time false constraint stop stop time second complete side suffice way standard motion random brownian motion two equivalent ds reduce constant proportional
reach event q convergence proportion imply define eq update eq without markov irreducible sigma rational q counting come detailed rely require satisfying proving seem finite irreducible proportion converge vector since mean visit frequency number x k proposal accept make leave use exact contradict go infinity toy toy consequence walk arbitrarily split desire frequency result wang use figure proportion bin use dotted line indicate check observe converge toward figure similar plot side equation theoretical limit example occur update convergence frequency ptc ptc vertical left proportion bin update update dotted represent toward consequently meet theorem allow irreducible measure require equivalent walk know precisely transition differ hasting generate different bin position metropolis hasting weak recurrence property fix flat reach frequency criterion note eq measure irreducible lebesgue time strictly lebesgue process visit time mh formally denote nonempty go statement chain choose proof go come let go define path reason wish construct put together prove point reach author thank lee p author theorem research science aim sampling region state use wang reach flat never reach variation bound context consider sampling suppose density constant proposal mh mh let denote indicator obtain ergodic algorithm wang eq choose multimodal distribution region attempt choice monte algorithms proportion choose might refer desire set typical bin guess frequency wang introduce penalty time act define distribution return index bin x jx mh wang generate past update widely physics practitioner flat convergence property criterion finite variation miss section wang penalty argue convenience section certain flat criterion might relax several algorithm schedule schedule temperature q typical wang pseudo code kernel I penalty visit chain decrease otherwise obvious converge use since update practice shall schedule along kernel fall mcmc hold literature wang schedule shall flat decrease schedule predefine threshold iteration intuitively criterion meet observe desire histogram proportion always finer flat decrease step meet know meet counter iteration meet counter something else case update give indeed need wang widely physics show meet one meet hold know meet occurrence schedule share invert plug limit proportion hence update proof go remainder define eq eq consequence run necessarily update corollary consequence validity proportion desire flat make simplification make bin empty compact mh acceptance side eq assumption verify proposal compact wang four propose proof propose increment bin increment depend whereas update prove go strong leave interval probability start decrease keep process event k dy qx qx dy qx dy exist mh ratio qx dy qx dy I define take strictly probability start increase u b increment increment return imply leave visualize process horizontal go time stay integer let take z first probability indeed markov state hence u
powerful gmm ability growth framework attempt er binomial process generate coin form random binomial growth need fix fact classic fix number rule er growth base unit interval node pseudo er gmm classic gmm growth rule grey bar density randomly er gmm pattern experiment classic er random build software network evaluate gmm simulation follow result describe purely create edge degree distribution binomial minus loop subsequent er model binomial distribution fit er top grey bar binomial classic goodness regression calculate wherein densitie approach count chi square case due abundance distribution assess error rmse well std rmse aic gmm er gmm gmm report measure quality fit aic quality level simulation basis er gmm successfully recover er er cumulative comparison parameterization encourage gmm easily er say er describe network people gmm classic attempt mention short average path node localize short length localize cluster clique densely lead hence produce lattice base additional incorporated lattice lattice iterate achieve desire localize graph measure comparable paper large theoretical attempt enter argue describe network regardless remain degree connect component pseudo report graph compare representative world lot specification benchmark compare gmm classic er variation experiment represent simulation line classic gmm simulation dot dot characteristic length low world gmm simulation gradually increase wherein flat short gmm may represent ba gmm vary base upper panel highlight strength framework panel gmm ba gmm gmm panel produce varied scaling seem gmm affect interesting variation produce well result parameter graphical rule produce scale average produce network graphical however undesirable show classic ba piece produce systematically precise maximum fitting law immediate considerable reduction simulation moreover mle ba estimate produce scaling figure finally size affect panel gmm encourage concept exercise gmm able successfully recover structural classic reveal gmm introduce alternative evolution assumption gmm assume evolve actor enter network growth current rely gmm count subgraph structure computational simulate basic gmm package programming rely scientific package specification near gmm test gmm classic small successfully outperform encourage gmm many many political science state much model often model lack flexible human interaction subtle complexity gmm able capture affect social outcome note technique limitation draw gmm sensitive conduct nature growth termination treat fix construction interpretation initial rule rule furthermore potential poorly suit human many growth contradiction biological poorly protein network gmm primary establish evolution human social limitation limitation order belief network structure problem therefore scale poorly complexity relatively result speed beyond research technology improve scale could remain static mean subgraph need count compute version improvement account growth base simulation run utilize consideration quality large ability compare fitness comparison constitute large portion future appendix termination si binomial section subsequent technique social one largely political discover primary relevant science represent actor interaction macro include micro study comparative party american finance contribution political science application variable measure centrality whereby relative actor type political science international relation inter key actor central actor vary etc similarity among international pattern american method political behavior outcome centrality member analysis causality static aforementione co difficult influence co behavior affect influence outside political insight process alternative viewpoint preference actor actor attribute structure simple one member tie technique coefficient group appropriate treat innovation highly unbiased network application political go suggest correct international relation causality wherein determine systematic science remain paper research trend continue advantage separate coefficient centrality co difficult affect relationship analyze longitudinal research much focus mle panel assumption determine relationship mle specify extremely envelope incorporate dominate give availability present exist network dynamic mle introduce tool method heterogeneous remain valuable social science especially political interested organization historical position actor simulate actor role attempt possibly something organization actor join function attribute always inherent security concern reasonable occur actor assign role network localize network leverage way political collective relationship two inherent collective action gain may make structure context consideration network collective much experimental game actor match color choose treat type reach equilibrium study variation vote collective act color structure vote evolution precisely base propose modeling tool remainder proceed graph modeling growth formal specification software package recover classic binomial world simulation characterize phenomenon social seminal decade social phenomenon technology improve ability complex mean vast majority heavy late structure model mention preferred network networks monte estimate model address closely gap concept social alternative level structural occur random graph provide insight structural remain specifically mean model design dynamic adequate modeling understand dynamic countable constitute practice generate achieve degeneracy implication limit interest complete connect empty model vast majority social actor model enter structure except simple actor actor structure immediate growth network rich complex social people meet change social create social friend worker simply create exist increase bridge relationship visually difference concept considerable ambiguity figure dyadic binary dependency metric include diameter likewise level rigorous yet assumption ignore inherent complexity self evident reality natural interaction provide modeling leverage graph overcome limitation random graph social network graph gmm new two key actor network actor bring build network process enter observable structure force strict requirement share random derive belief model one random base constitute despite degenerate base structure similarity note exhibit increase first mention preferable network relevant unweighted describe graph set describing may gmm however applicability unclear hypergraph wherein limit graph great account singleton entire gmm node ordering become critical describe let tuple increase node correspondence among subgraph put another subgraph np certain approximations belief structure order generate belief aspect growth specify evolve subgraph calculate count generate evolution simply define discrete count give mass element tuple element element dependent pmf purpose specify pmf exclusive example pmf satisfy complete exclude set pmf enter subgraph wherein zero mass subgraph base specify probability pmf gmm explicit latter pmf subgraph provide discrete structural proportion subgraph provide belief new structure subgraph base never enter possible pmf limitation structure exist maintain bipartite structure assign utilize element structural specifically alternative mean natural motivation occurrence increasingly event likewise enter graph around base assumption reflect generate however specification specification pmf pmf direct gmm require subgraph limiting define pmf distribution assumption probability gmm structure graph take form draw add growth rule construct assume decision applicable fundamental subgraph design growth rule form belief belief mass likely grow dynamic calculation continue termination gmm growth beyond restriction termination strict model gmm framework flexibility technique detail one gmm specify proceed graph describe attempt current propose wherein rich describe evolve assumption base upon form belief model base graph represent useful proxy model order tuple define subgraph pmf growth repeat termination gmm
vote give strong weak strong perfect next subsection first great ensure dictionary weak weak opposite opposite rule counterpart opposite decision ensure great classifier index formulation imagine space condition specify q pair final group opposite classification condition set cover output interpretation illustrate entire sort value weak classifier fraction vector th minus classifier fraction incorrectly classify illustrate minimal consider one suffice weak satisfy possible situation like shall minimal overlap condition arrive term illustrate pair show region incorrect line top pair provide classification incorrectly vote vote classification accurate vote bottom weak vote neutral vote majority vote entire vote term combine vote vote correctly make third overlap overlap amount vote substitute condition yet satisfy satisfy modify weak classifier condition interpretation condition classifier subscript add overlap separately completely accurate majority vote vote first element fourth pair fourth omit extra vote remain correct strong weak classifier similarly correctness overlap element pair accurate sum correctness everywhere eq select evaluate vote give classifier comprise give classifier case output vector strong classifier weak belong classifier comprise weak satisfying suffice correctness namely equality ns j set event great randomly correctness sum least tell correctness sum classifier sum equal correctness comprise solely correctness condition namely q proceed great imply sum three guarantee must violate way could replace minimum overlap weak q output overlap prove classifier everywhere namely eq select imply theorem replace weaker choose uniformly correctly comprise classifier satisfy perfect suffice correctness overlap condition definition correctness correctness sum exclude pair sum classifier partially exclude virtue tell always cover majority vote calculation alternatively could condition way weak correctness pair correctness include weak correctness thus conclude great probability correctness turn probability perfect classifier tradeoff turn weak subsection correctness classifier majority vote replace weak classifier weight know shall ideal determine weak classifier new write second term double sum solution weak classifier define correctness classify incorrectly member classify penalty classify incorrectly accomplish assigning pair alternate translate ise hamiltonian section variable yielding omit direct translation hamiltonian represent rather assign combination classifier pair set strong find shall henceforth input decide quantum speedup testing exponentially input output regard formulate space indeed perform parallel return state state recall concerned vector behave whether ideally behave combination eq vector use replacement purpose strong classifier specification construct training label produce output implement result task identify implement stress relax positive negative eq unfortunately relaxation trick classifier vote function combination gain receive exist condition part ideal program likely specification positive low lie input portion care specification enough program ever occur fall outside implementation try case classifier opposite definition receive examine formulate detection minimization need amount behave incorrectly program implementation insufficient relaxation energy hamiltonian hilbert span state candidate procedure never negative would positive input either find output boltzmann boltzmann contribution thus expect lie state return nearby implemented detect undesirable low probability small provide member identify classifier set member beyond scope write opt ic opt yet weak classifier p function logic intermediate I z z z z I I I I x x x I z z z z I I x z x I form subscript column e relationship output evaluate bit input measure classifier motivate correspond interaction weak boolean boolean concern input bit wish quantum dictionary implementation three perturbation bit local include devise product logical weak relationship specify form I x x follow body intermediate bit tie intermediate introduce penalty modify translate value modify amenable note act intermediate bit directly dictionary three bit function number need implement processor correlation relevant classification intermediate function sum add penalty equal ax intend behave f interaction quantum computer hamiltonian term implementation classified classifier whole weak classifier bit case hamiltonian inclusion tie product associate penalty appear hamiltonian intend outside instance nearly two simultaneously verification region limited classifier fit intermediate involve four bit bit weak classifier keep instead choice boolean become third index possible tie ground optimal derive detailed subsection consequence interpolation state superposition h implement starts grind boltzmann distribution state design find contain logical assume quantum processor preliminary achievable classifier implement toy design critical system redundant could particularly important different system indicate consistent redundant thus suppose implement majority vote simple explain implementation monitor routine variable frame implement shall logical tracking frame fail frame variable facilitate quantum program essential quantum three redundant frame per snapshot whether reflect logical bit whether redundant routine nine bit fail fail fail variable variable incorrect previous frame challenge recognize specification implementation classifier find objective hybrid quantum quantum technique classical resource example classifier computational study routine quantum set classifier appropriate set evaluate weak typically discard feed ideally quantum optimization hamiltonian place classifier reality quantum come classifier accurate dictionary address ground discard space weak classifier yet consider classifier include optimization weight weak classifier dictionary clearly strategy combine subset weak classifier could genetic eq precede eqs classifiers eqs create final spectrum portion preprocesse order small bit output cost ground simulation effort well parallel classifier simplification implementation frame result fig subsection produce comprise weak member figure perform amount time computer algorithmic perform error achieve quantity total incorrect classification eqs classifier analyze implementation classifier plot fig fig classifier consistently specification thing outcome limit note dependency simulation address describe increase far perform number simulation vertical horizontal axis arise incorrectly respectively classification determine observed iteration intensive computing plotted assign uniformity simulation weak classifier even misclassification classifier place wrong neither rather application datum quantify error iterate encourage generation challenge improve break intermediate light ask satisfy sort expect yield relate light pixel output horizontal red divide represent give weak black incorrect classification problematic aspect bar white accurate detailed completely fall classify incorrectly impossible bar classification span height weak incorrectly weak use correctly classify overlap correctness fraction weak arrive classifier overlap eq classified correctly minimum weak otherwise weak correctness force classification see fig correct vote pair weak show correctness overlap weak minimum weak misclassifie weak minimum classifier apparent short ideal weak classifier overlap horizontal correctness maximum order come estimate accuracy achievable classifier correctness overlap minimum correctness ideal weak overlap completely classifier cover space overlap weak extra vote fraction three minimum overlap relationship significant large classifier fraction input incorrectly currently produce classifier quantum apply classifier quantum parallel input select classifier first quantum quantum hamiltonian first inspire classifier weak selection optimal condition second quantum strong entire output quantum intermediate track execution tune weak classifier overcome limitation restrict state path initial hamiltonian world development characterization strong involve proposition machine learn anomaly via quantum training classifier form classifier superposition certain phase execute quantum evolution software verification validation computational range complex stock market concern address speed concept quantum computer show quantum require much classical handling classical binary assigning example whereby combine alone separation distinguish two specie pick letter letter alphabet base effort quantum advantage boost formulate use detection trading change verification classical software anomaly anomaly detection intractable exhaustive piece set input give software variable although exhaustive lost testing resource widely effort focus considerable new testing attempt available combination cause considerable software parameter formal verification phase software absolute software implement exhaustive consuming checking validity state solve np require correction provable check verification validation processing quantum use test quantum boost apply consist sort sort classical testing step turn classify likely error translate hamiltonian allow potential examine return candidate test use quantum encodes ground computation perform simple slowly system evolution hamiltonian universal overhead protocol degree begin problem define eliminate resource section establish implementation learn develop ideal alternate quantum section detailed result future formalize vector occurrence consider ideal mean perfect instead real refer wish verify operation eq generality think finite without generality limit length string within move specify specify input output binary string space output vector program output space implement program ideal element fix input trivially ideal map correct q every course implement ideally reality may mind simple software norm program purpose existence error program order incorrectly pair count number incorrect classification strong classifier classify prevent manner balance weak comprise classifier formal tune unfortunately discrete evaluation amenable error correct pair already interested incorrectly correctly classify formal replace classification make sense equally sr good training set equivalently find optimal weight ix think symmetric offset drop wish follow one define
apart compare actual value via carlo identify epochs iii initial put ignore fa bound involve involve give use b replacing involve far combine th sharp exact however constant constant problem importance identify drift vary specific example drift drift typically present tool computable space might apply successfully example state therein hierarchical paper cite failure hour assume conditionally I hyperparameter gibbs follow cite satisfie condition establish drift sm x vx sm vx obtain calculation iii prove jensen shall define assumption respect consequently n p proof repeatedly iii vx corollary rearrange write vx x jj directly term condition corollary nn n vx vx desire I drift induction k vx f vx x assume vx n ii ii n term v vx nn term put ii fx v x v acknowledgement comment acknowledge help comment associate paper partially education grant partially bound square mcmc estimator valid ergodic encountered computation unbounded variance bound geometrically polynomially ergodic chain condition corollary confidence distribution borel objective quantity high bound arise often solve monte average explicit reliable prescribed level begin ergodic chain admit sharp sense lead asymptotic central proof rely technique sequential bound geometrically polynomially appropriate quantitative transition mse geometrically polynomially ergodic establish drift utilize ergodicity use mse uniformly consider ergodicity polynomially ergodic reformulate bounding immediately confidence chebyshev trick inequality mse upper geometrically polynomially example hierarchical bayesian poisson normal toy mse derive geometrically polynomially chain section applicability proof defer markov various brief derive result ergodic explicit trick exponential conclude give turn concentration dependent large see example reference use result motivate metric expression ix iy set suited functional ergodic ergodic chain detail explicit bound apply ball condition detail chain ergodic verify rather dimension seem tractable tail ergodic however constant explicit may chain chain establish cl approach directly geometrically ergodic computable paper generic approach chain polynomially ergodic secondly computable bounding variance surprisingly estimate mcmc curvature geometric ergodicity wasserstein bivariate approach appear applicable different employ coarse assume curvature complementary rate convergence geometrically investigate quantitative quantitative computable together need importance translate allow burn irreducible construction exist bivariate way rule give qx chain moment epoch x specify sum bind always markov prove lead assumption ergodicity section explicitly computable quantity set lead estimator refer detail recall eq excess entail middle significant portion proof j version q I pair identity identity q attention mean sum invoke follow elegant excess obtain account proof section geometrically quantitative defer standard establish definition specifically drift exist model practical geometric geometric computing bound appear hold v pt theorem note th order theorem involve follow simple bound v f mcmc always case iii inequality might improve add burn computation discard initial part justification equilibrium technical reason effort inequality upper f polynomially ergodic chain drift deferred section follow drift counterpart assumption use establish polynomial ergodicity markov constant drift mcmc sampler walk langevin sampler example section expression pt pt pt pt pt pt establish analogous iv necessary involve depend difficult simple complementary inference small present simplify hierarchical example design actual quality literature poisson demonstrate realistic drift nevertheless obtain model van multivariate student bayesian effect model algorithm student regression body relate devote drift set kind paper l substantial establish quantitative cf geometric validate argument qualitative suited derive quantitative conclusion simulation experiment design compare prove actual normal reciprocal eq unknown thing improper q abuse symbol draw conditional start convenient letting rhs
form analysis miss entry know study I missing modify objective square cp second stay compute partial term respect conjugate update search toolbox opt accuracy randomly tensor whether underlying factor extract denote extract different scenario couple couple third mode one entry normal matrix example first factor r tensor r j adjust factor r entry experiment opt extract quantify match follow matrix denote mode vector norm similarly rewrite indicate weight original extract extract value change opt number iteration set two divide gradient tolerance generally stop due change function rarely opt stop reach maximum e run opt furthermore opt stop ccc c c alg success p opt opt opt opt e e e opt present result experiment factor normalize generating factor ratio experiment factor parameter run compute pair statistically scenario opt significantly outperform opt quite accurate recover underlie factor noise affect accuracy hard component near accuracy true datum first accuracy slightly change greater average around extract factor c c c c c c c c e opt c opt opt shift focus explore structure source formulate problem couple address couple tensor square opt factor extend couple extended incomplete well fitting algorithm opt alternating fit note current formulation couple factorization drawback r r formulate couple f scaling take consideration scalar address part modeling area future bayesian promise future loss incorporate interpretability laboratory direct development laboratory national security contract ac briefly couple construct rr assign plus entry member versa member form form factor matrix r normal construct g source improve restaurant recommendation system rating customer rate customer social facebook restaurant category well recommendation consist high rating tensor interested capture formulate tensor factorization tensor manner traditional approach optimization opt joint analysis tensor handle couple incomplete demonstrate access amount data advance internet medium device genomic medical diagnostic restaurant site access customer medical patient eeg monitor fmri functional imaging laboratory test restaurant category chinese couple high tensor discusse analyze heterogeneous simultaneously entity represent movie review tensor show rating similarly tensor collaborative filter example suppose dimension mode common model cp list cp factorization tensor show least bregman metric illustrate motivate couple tensor source fine grain capture set suppose customer customer store customer period customer live customer group group people live interest item imagine discriminate svd customer plot svd separate conversely imagine tensor discriminate cp albeit illustrate middle factor use group separate bottom plot recovery different miss limited amount single long enough datum recovery miss still use fail couple mode compute r r random km miss recover entry extract recover extract common mode letter subscript entry denote entry ni ij k kronecker result size k kronecker product property kronecker rao product tensor I hadamard define I tensor frobenius give nr matrix fusion collective multi field decade briefly datum couple source attract considerable community netflix competition movie rating order rating rating exploit tag movie movie collective simultaneously couple matrix rr factor general extend loss early study collective matrix scheme collective factorization back common variation later follow set factorization matrix work moreover simultaneous develop blind separation speech microarray data analysis tensor analyze multiple tensor third analyze nevertheless couple heterogeneous address different heterogeneous formulate first show al tensor couple mode
spline space approximate kernel solver classifier incorrect evaluation maintain multiplication look unlike dense change simple form initialize repeat follow large consist dimensional inner loop algorithm refer easily solver exploit practical embedding somewhat image histogram intersection kernel classifier pyramid histogram orient gradient website instance zero entry set dense image svm often take machine sample b spaced basis degree quadratic cubic spline offer smooth fit figure show degree accord spline fit uniformly spaced bin various additive expensive training additive linear svm expense additional training train vs label high response find cross validation model significantly outperform classifier closely match c svm spline spline spline fourier compute train spline additive outperform svm hour b spline spline b spline spline batch additive outperform additive embedding enable memory overhead see suitable store outperform significantly spline work svm high spline fast spline library publication train also new class classifier paper current kernel svms linear spline suit ease compute overhead connection spline intersection kernel spline intersection kernel svm enable solver slow order fast train parametric classification become attractive svms desirable view offer significant shift semi shift approximately train overhead naturally setting e kernel computer vision additive histogram intersection map lead compact feature estimation line explore feature learn additive propose construct spline estimate statistical ever additive introduce practical formulation extremely spline embedding efficiently embedding underlie learn linear embed embedding spline function spaced basis adjacent embedding svms develop train additive embedding kernel particular arise spline basis advantage control choice desirable moreover fit go prominent spline spline modeling space b basis difference adjacent spline formulation key whole discriminative training involve use hinge implicit reproduce hilbert rkhs additive shift invariant kernel feature base propose efficiently kernel propose model typical dimension feature enable dimensional derive one overall concatenation embedding additive dimension space spline difference adjacent spline let spline consist construct th difference repeat dimensional everywhere top result spline svms additional embedding approximate choice matrix invertible therefore k equivalent inverse triangular excellent spline spline ccc cubic various refer spline basis truncate polynomial provide interval x regularization overall additive embed practice small form classic derivative orthonormal orthogonal sequence three family make well purpose
integrate curve continuously differentiable tangent agree field curve interior approach fully infinite computationally intensive empirical component consist detailed field represent nonnegative window tensor field play role imaging calculate representative eigenvector correspond eigenvector use monte spatial point advantage quantify uncertainty effectively structure field strong alignment arise many different study densely one dense long rather connect relatively point window new consider stanford galaxy appear cluster along interest surface extend surface along open ridge locally parallel close arise align along reconstruction dominant orientation overcome construct dominant orientation directional ambiguity stanford principal generalization principal em choice smoothness number difficulty reconstruct region signal datum galaxy empirical density galaxy thin compare technique see ascent path path analyze useful insight method brownian motion enhance contour edge digital along structure around model segment integrate encourage area prior model bayes death mcmc formulation use exploratory focus dominant spatial modeling point ordering process homogeneous unknown cluster henceforth one object segment high dimensional space contain discuss appropriate identify spatial window overview propose appropriate outline sampling compare issue implementation describe segment orientation curve segment say orientation segment agree mixed poisson cox drive independently point generate cox point character tendency regularity along distribute distance often spline fit integral agree field orientation specify see characterize parametrization arc length poisson along construction introduce bias consider adjustment correction make negligible part seek field property finally background independent homogeneous onto generate directed acyclic dag dependency show acyclic henceforth denote reference arc list write arc arc th orientation possess may signal cluster anchor point isotropic bivariate normal spaced arc distance adjacent point dirichlet encourage either evenly spread along independently point allocate th proportional approximately add store variable resp allocate allocate across window total poisson parameter noise point sake simplicity assumption assume total poisson distribute estimate proportion point assumption proportional suited noise detection along number example prior common include suggest alternative instance indicate omit distribution make represent estimate regular grid point estimate orientation evaluate small distance orientation adjacent segment discretization course calculate advantageous field integrate datum high field bayes mean aspect datum alternative treat random identify derive theory see appropriate smoothness field field computationally calculation relate lead complexity huge difficulty ensure properly explore use easily suitable orientation produce orientation arise estimator likely field section vector mapping pattern apply euclidean field definite eigenvector eigenvalue indicate strength orientation assign eigenvector eigenvector unique say equal dimensional field commonly diffusion imaging use analyze scan water use water tensor orientation eigenvalue proportion water let transformation represent method set create principal eigenvector field interpolation see extended account calculation tensor matrix zero eigenvalue indeed intend one dominate calculate due point far round become zero remain error log tensor least eigenvalue uninformative suggest lack take potentially field smoothing kernel metric hence field principal eigenvector give integral drawback create rapidly orientation field analyze magnitude bias find smoothing b smoothing bias occur field unbiased allocate trivial j proposal death desirable add extra move utilize move amount reference keeping occur predefine hasting improve section propose noise propose beta move find deviation update field degenerate configuration allocate exclude state inspection show reach death process irreducible visit infinitely motivate low inspection estimating occur happen point short exponentially distribute region within burn assess spectral density diagnostic reject mean sample sufficiently last assess whether death death rate event birth death etc algorithmic stationarity estimate birth sum additional move implement pt mind complexity show record constant rate sampling reciprocal carlo remain mix decrease approximately proportional prior inspection datum continuous determine detailed balance per fix relationship actual hardware hardware describe birth rate adjust propose join unit time evaluate four neighbor window cluster server intel processor gb fully ram http www hour datum yet explore perform show brevity first simulate pattern facilitate stanford signal family allocated allocate higher allocate indicate allocate sample point enhance clarity symbol indicate indicate noise pair nonzero round c c c c percentile distance point birth death sample sample hyperparameter half well albeit curve fit probability property point closely correlate chance chance percentile point percentile origin bivariate normal variable percentile simulate partly explain slight beneficial less desirable evidence dense dirichlet density lie drawback multimodal proposal birth anchor good clustered background datum set usa find http www edu cat birth death unit sample state hyperparameter dispersion half table numerical distribution http edu cat window dispersion probability nonzero round cc pt percentile b indicate allocate indicate allocate find smoothing area density allocate sample enhance clarity symbol indicate different cluster often associate across one limitation assume number particular width influence signal unit figure central dispersion apparent width shorter wider effectively th percentile distance table overcome could cluster feature sample indicate agglomerative arise interestingly positively suggest additional arise part preserve lie depth set allocate indicate allocate estimate reduce enhance clarity estimate cluster consider birth death unit discarded take time near neighbor picture small scale phenomenon partly inter contrast thank smoothing step field succeed fitting indicate area location near edge show ideal fairly interval identify nearby could gap reconstruct data property posterior distribution window noise nonzero round noise point percentile new process monte instrumental arise nature investigate around line galaxy visible universe galaxy know align turn connect web overview different location place straight line aid currently prominent curve nature curve sufficiently smooth must angle smoothness desirable identify curve desirable computational integrate use algorithm em
construct coincide function elsewhere show eq construction continuous lipschitz constant version operator define htp technique adaptive loss function canonical radius combination ty threshold dual algorithm regularize variant proof differentiable derive proposition prediction let tune quantity unknown forecaster figure individual eq unknown forecaster two obtain without cf since otherwise adaptive regime losse high curvature corollary loss regret unknown forecaster adaptive original loss dependence obtain grow grow hence improvement property stem another benefit optimization regret gradient improvement correspond forecaster ask ball forecaster ask ball case regret good simplicity assume framework r tr sub prediction work access sub simultaneously perform regression forecaster aggregate simultaneously regret concavity two sub regret known every assumption average forecaster expert lemma fact nonnegative xy ty inequality xy cr u finally get conclude claim second follow bound scale knowledge automatic adapt quantity use update past adapt adaptation replace sequence r grid ensure carry trick however avoid use automatic term negligible author would thank comment suggestion national ec yu partly fellowship le la et les perform stochastic trick employ follow notation space orthonormal family dx space integrable ts independent precise xx eq remark entail comment since argument elementary inequality get cauchy schwarz follow random get choice technical page rely elementary fast hold constant eq conclude entail inequality fact xt get conclude bind regret keep eq supremum moreover regret regret latter remark conclude proof theorem past identically f expand via notation c denote infimum multiply inequality conclude follow loss negativity b construction adaptive therefore corollary particular loss gradient case function resp continuous draw consequence inequality gradient substitute combine result inequality rewrite solve suffice first bind elementary rearrange conclude recall round account q follow elementary cumulative forecaster satisfie next prove elementary exponentially forecaster base infimum supremum treat separately regret bind fy tu denote linear output forecaster satisfie step exist forecaster upper forecaster orient slightly ridge forecaster forecast forecaster access base forecast predict temperature forecaster know weighted forecaster satisfie q proof straightforwardly latter exp assumption argument proof integer canonical j rest dedicate bound last sum equal zero equality k combine remark inequality conclude variant reproduce convenience combinatorial cardinality k k k elementary tuple lattice path straightforwardly corollary sup paris individual forecaster sequential prediction one linear present dependency exhibit transition furthermore algorithm I knowledge adaptive arbitrary competitive good predictor extend task aggregation include analysis dna sequence filter country growth diameter predictor dimension provide algorithm ball linear follow environment choose forecaster instant environment reveal forecaster incur dimension relative step case sequel inner product du forecaster shall hold opponent bound may unknown forecaster linear linear ball refine express absolute transition bind partially type argument deterministic yet stochastic refine setting ty stochastic batch make statement order extend achieve inefficient unknown overcome term forecaster tuning prove confident cumulative efficient logarithmic regime contribution bind nearly optimal prior issue modification automatic tuning algorithm nearly section generic transform lipschitz technique generic prediction yield main improve derive curvature result grow naive way uniformly knowledge neither upper bound efficient adaptive known aggregate achieve uniformly discuss automatic tool minimax term intrinsic quantity bind definition ambiguity call f upper base prediction observation infimum supremum secondly partial question ask gap prove see term intrinsic quantity relate intrinsic quantity theorem term intrinsic bad regret choose continuity main relie type another apply sequence bound achieve ridge forecaster forecaster therefore third sequel argument refine bind e look discretization study grow instead naive first fact competitive dy yx remain aggregation forecaster predict b mp u forecaster tune elementary put together corollary quantity improve already dy c absolute infimum sequence lower bind small proof batch employ lower bind take inefficient dimension achieve minimax modify forecaster tune nearly achieve modification regression forecaster computational nearly regret technical loss inefficient version call require oppose original version automatic tuning suited loss arbitrary forecaster environment choose loss incur online radius dt function proposition general function gradient majority expert let adaptive sequence differentiable bound straightforwardly linearization argument regret
plus sec hence occur intuition like various ensure know inequality fulfil e explain propose practical via carlo following set run randomly draw independently bernoulli center perturbation unique trajectory continuous know behind angle regression homotopy condition ty tx ty result inequality tx ty immediate gaussian distribution recovered estimator top strategy increase solve equation globally condition initial refine simple compute solution first correction recover monte lasso false positive result lasso use well lasso available estimator deviation slightly figure top htb level derivative maximal see vs fidelity find decreasing summarize compute lasso l n find recover monte properly false component lasso penalty tradeoff value newton discard line trust region correct price number component vs fidelity tradeoff constraint see respect false positive figure propose simulation encourage evidence dependency practice ahead result confirm snr know large configuration lasso exactly true setting perform poorly component carlo experiment strategy snr much surprisingly word appear practical suitable snr snr choice could base occur varie question let except decide refer interested globally convergent paper finally remain question proceed support aic criterion recall result provide high trivial modification condition perform product lemma ec instead simple inequality e true component instance w obtain whenever actual x row inequality write hold compatible I previous th simply second take obtain z section prove sufficient optimality admit unique l hold decrease proposition solution property derivative moreover show immediate consequence concavity ty thus interval form continuity assume contradiction continuous sequence respectively l l uniqueness continuity imply sign due least interval uniqueness imply multiply ty tx tx tx tx obtain exist exist calculus contradiction bound bound fix also large subspace n imply imply tx increase nonempty denote infimum take tx continuity var admit trajectory generic generic moreover interval x tx combined tx ty tx sx decrease study deduce read tx tx since tx tx differentiable use author grateful thorough comment presentation argument france mail fr address issue generic jointly minimizer square functional parameter tune parameter choose enforce fidelity similar one plan ann support simulation show enjoy lasso signal outperform false dimensional unknown gaussian read denote unknown error study regression impossible study extensively context compressed paradigm design matrix number positive estimate selector refer simultaneous b control sparse introduce sample resp common incoherence coherence coherence appear basis significant recent context require whereas bind require coherence pattern uniformly op index signal condition possibly suboptimal address author aic use practitioner procedure avoid enumeration subset covariate intractable covariate motivate provide joint variance compatibility problem support provide explicit strategy present paper mainly aim understanding extend right magnitude estimator consist replace main difference summarize satisfy condition numerical viewpoint iteratively overcome estimate fidelity precisely enforce fidelity complex estimation regression natural constraint observation support probability explicit coherence assumption readily currently concentration property singular yet concentration value criterion difficult impossible oppose property make nonzero regression coefficient magnitude great require bind result suggest snr end paper strategy snr present give proof technical intermediate usual notation x transpose symmetric real matrix maximum resp order symmetric stand notation resp distribution line degree bernoulli distribution submatrix index orthogonal onto column norm j selector h diag fashion support main strategy discuss study particular tune numerically find support resp coefficient satisfy small slightly require may argue strategy experiment empirical signal experiment standard lasso ensure enjoy special continuity say ix tx lasso problem support non singular prove generic hold surely implicit var estimator implicitly exist sequel mention uniqueness show give increase prove precise interest existence scheme discuss satisfy bind component gaussian set order design column column haar unit sphere scalar x phenomenon sphere u concern require beta deduce sharp ahead require matrix imply sharp advantage sample sharp increase sliding imply usual beta see want specify may risk interest practice plain variance many lar instead order model aic bic etc vary support recover incorrectly detect component range quite vs noise compare constant various set r choice impose less allow end limiting allow make fair divide consequence condition prove oracle full e tx intersection discuss next study enjoy know ahead follow oracle b r might interesting tx formula th recall assume seek var ty p var orthogonality var var tx var p henceforth x tx particular subsection show tx high choice var p property tune great virtue deduce p yield r writing p satisfy r conclusion recovery pattern need rr hold proof hold high slight improvement iv probability appendix conclude strategy deduce proxy feature pattern sign
base delay unbiased jk delay note discovery indeed delay order moment delay provide delay discovery delay variance suffice variance variance additive henceforth variance distance estimate distance abstraction imply block discover node definition l eps pairwise la lb lb literature since la lb v detect graph practice relax equality tuple path path hide connect length middle h length linear end sum along respective algorithm merge employ additionally introduce modification addition discover random outline short idea behind reconstruction cycle test summarize intuitively avoid testing fail reconstruct limit avoid obtain tree reconstruction short edge length thus maximum bind diameter choose carefully cycle least point relax distance node attempt possibility consideration already nothing merge create cycle cycle create need firstly cycle guarantee merge list bad secondly cycle need create short whose attempt merge without create attempt merge create new cycle check distance original list processing attempt merge bad account bad towards spirit merge handle presence cycle distance length short uv uv uv merge topology short path distance second short length uv uv uv uv uv heuristic discovery shortest short I path uv b j merge create short cycle short join consistently add inconsistent uv attempt create new long new path else contract least hide uv uv node compare length assign miss assign connect already assign b w b path create exist split length uv create current node short short short tolerance compare length kl kl shortest available agree exist create cycle path split verification fig else add uv create cycle outline short distance addition short propose summarize use short distance shortest carry shortest retain distance specify second short distance short middle edge retain rule minor check multiple different length simplicity assume distance node algorithm l l dot line length path short middle cycle detect bad eps eps eps fail succeed wrong show test recall satisfied internal path outcome incorrect outside bad outcome bad use internal merge procedure detect wrong internal outcome result error give procedure edge length additionally equality constraint fig equality constraint instead cycle event occur length far analyze exact delay input instead propose require achieve distance delay variance delay delay imply discover topology distance topology node require low distance realize construction end prove low realization graph distance graph adjacency random belong decay graph reconstruction less otherwise output achieve distance node graph participant short enough distance lemma every great threshold impossible uniformly strong converse say short short path algorithm performance sub participant lower cover argument cover high l eps l reveal shortest greatly improve discovery graph accomplish end counter fig fraction distance yet identify reveal fundamental topology subset key tree species tree series occur specie low g correlation decay range estimate delay long delay delay sophisticated however reconstruction short distance discovery topology sub uniformly participant property obtain end path participant scenario measurement available topology sub participant participant efficient explore algorithm edge length explore locally well model employ explore change join interest guarantee minimal plan centrality analyze plan develop develop thank anonymous uci award fa cycle cycle length two cycle overlap cycle length cycle argument expect q edge overlap cycle obtain deal representative degree first condition enyi event result least modification minimum degree minimum proof line cycle length overlap cycle denote number enyi event three give node overlap cycle prove characterize good event addition distance reconstruct minimal graph concept bad length criterion middle generalize cycle accurate merge consideration edge short length short successfully part induction initially empty correct path yet add merged node add merged join point path create correct join point join part cycle join point imply middle cycle accurate merge correctness middle middle bad reconstruct contribute wrong amount reconstruct three correct number middle node middle edge event v b v lemma occur generalize cycle bad chernoff b v g r expect distance proof succeed accurately candidate join short path edge middle middle short cycle expect result delay sample take obtaining reconstruct short q know c require asymptotic obtain nk obtain journal use node participant rest information topology consider discovery participant exchange message exchange second short uniformly participant strong sub linear uniformly imply participant bind reconstruct original graph demonstrate tractable graph participant graph discover participant availability end path participant end characteristic network topology network many rely failure infer social knowledge useful infer characteristic information flow possibility traditionally topology however tool require node protocol request privacy security concern topology request scalable discover protocol switching path increasingly discovery network topology e without flexibility increase popularity detail topology end may discover place g population drug survey discover network fraction participant many topology need computationally provide fraction inference topology desirable low sample definition measurement achieve accuracy scale size achieve objective discovery topology participant issue phenomenon participant identifiability desirable topology guarantee performance r perhaps provide reasonable explanation network social address graph fraction provable kind participant useful discovery participant low distance discovery achievable address insight sparse provable end second information participant discover extremely participant reconstruction graph redundant consist delay short path participant refer test tree topology previously topology roughly participant end short node maintain demonstrate achieve participant thus discovery sample logarithmic end end sample needs state discover participant exploit locally random graph enable guarantee know topology control use test time exploit cycle obtain tree topology leave accurate topology use participant error participant specifically roughly reconstruction algorithm also general identifiability discovery participant reconstruct path participant contrast discovery end property applicable discover tree like social extensively propose instance mapping network prediction link consider consider infer latent propose survey discovery development topology discovery various topology discovery availability kind previously path internet network formulate discovery random query law node network protocol relate consider topology short several edge whether edge select query edge subgraph query return two vertex query know unlabele unweighted discovery extensive end end delay refer previously traffic thorough local test inspire previously topology accurate reconstruction tree recent temporal exist similarly notation probability measure say hold almost surely equivalently property generalize length edge count definition union short denote subgraphs subgraph topology enyi graph arguably denote edge occur regime exhibit contain component super critical regime regime discovery regime real world extremely participant discover topology discovery graph topology let node exchange message amongst fraction desirable reconstruct node decide random regime mean node discover message provide presence need goal topology experience delay delay sample metric depend consideration scenario delay pair end delay message message experience delay identically direction assume delay delay delay along participant delay topology message exploit efficient discovery end delay assume message participant short path path delay also scenario participant short alternative short fail
solution interior formal cutting solve initialize find interior respectively duality dual new interior optimize first statement violate dual constraint relaxed associated weight know np iteratively add maximize summation return constraint must discount terminate duality gap program add primal add theoretically interior primal computational finally iterative bregman guarantee solution relaxation note practice never need explicitly within product basis efficiently provide application effectiveness community purely art clique arguably simple evaluation sum inside clique clique clique element volume htp cc detect team team repeatedly clique detect interaction ideal scenario since indicate player concentrate equal matrix b les social social character les b clique identify clique overlap detect follow character co character relationship social regard figure social spectral result red cut three cut community network ht box plot clique volume clique approach volume clique clique meaningful clique see table verify ground novel correctly separate clique treat character small clique important clique clique necessarily satisfy bad clique volume slightly clique character ex n n social community cluster note think large clique score clique frequently clique seven contain six node medium broad field part clique clique volume clique identify pursuit comparable clique volume identify clique hundred ht science show identify behave persistent exactly cluster combine persistence clique clique meaningful bipartite clique cluster identify parent identify clique c network clique method clique volume clique parent identify paper clique heavy persistent c clique program clique track rank contain give rating rating top score ground truth top count whole path capable htp top curve select fourth persistent connect compressive adopt algebraic tool homogeneous formulate detection compress construct characterize clique heuristic level community model compressive result area create usefulness new framework clique network compressed sense algebraic basis group compressive interactive information management rank bi variate pair look compressive clique turn recovery time approach world acknowledge grant nf nf nsf google support basic program china program cb microsoft research university thank support package university compressive liu stanford modern acquisition produce amount though analyze largely statistical processing present framework connect different area compress sense perspective network consider clique tool homogeneous recovery conceptual solving keyword analysis compressive sense pursuit isometry property clique detection past decade research include scientific study link citation network connect citation relationship frequently modern application domain lead hoc exploring help scale drug control network principled analytical crucially researcher predict fall static static snapshot dynamic dataset index os enyi blockmodel blockmodel dynamic though analysis learn unlike usual measurement collect relational prevent directly art analyze bridge framework assume sparse compressive compressive system reconstructing due contribution adjacency node problem clique connect new space regard study sparse clique basis address exact algorithm recovery apply setting restrict rip paper new exact sparse choice root pursuit compress sense also practical usefulness content present polynomial clique example conclude paper general compressive nonparametric notation denote euclidean edge represent associate generality upper triangle triangle model vector evaluating infinite nonparametric without dictionary sequel element column index construct sparse thing simplicity pre section clique problem identify community arise management social application typically network pairwise clique frequency govern community clique formulate clique answer rigorously motivate situation sensor identity belong grey pass interaction observation due player mostly difficult involve team member infer information belong rank item set item partial ranking understand organization direct appearance activity typically network character exploit clique network basic interest group clique often govern community formulate relationship community social social majority study community base partition nod modularity base frequently practice overlap structure clique address clique model clique node compressive clique turn community structure clique explore network light multiple aspect frequency social bivariate function pairwise order intuitively interaction belong map low order subset fit compressive previous demonstrate basis purpose programming formulate name pursuit construct clique perhaps clique weight adjacency clique clique matrix transpose suitably space exploit transpose matrix transform dictionary observe discussion scope paper entry many strong realistic instead study universal seek signal pattern sequel clique clique could clique extract less rank need number column linearly expect signal observe complement extract follow characterize self include assume invertible cx come kkt equivalent lagrangian lagrange multiplier kkt give program two sufficient minimizer minimizer condition j j w inequality less since independent necessary free setting necessity span condition respect invertible sup lie condition sure condition must relevant basis irrelevant depend span mild necessary selector like subsection verify important clique let clique let example thing verify fact really clique clique may large recovery behind enough check define intuition try technical entry set condition exactly give fact contradict condition disjoint every set remain equality generality belong contradict member member remain say contradict proper condition case enough happen large choose large disjoint set say I ta show row correspond row construct long often clique reasonable network exist join together make community belong partition belong hold inequality sup norm note belong overlap size entry clique bound establish second inequality observe fix time belong intersection fix must fix belong otherwise far satisfie see implementation basis clique basis actually submatrix subset clique submatrix clique intersection respect suffice firstly diagonal dominant recovery impossible
calculate complete operation complete qr qr perform complexity enough size realistic away scalable brief discussion iteration qr factorization nonzero element number overhead iteration total parallel feasible pt localize range three peak apply basis resolution wavelet wavelet four vector sparsity stick loss test thresholde sparse pca specify theoretical stop compete recommend c c two margin large margin algorithm spike loading seem select coordinate well next take spike figure spike spike loss average spike threshold recommend first spike relatively separate compete separated method estimator imply estimate pick right subspace summary spike principal successive focus purpose low dimensional projection devote proof divide major step pt well approximate interest oracle sequence estimating sequence sequence various need actual sequence oracle forces oracle road extra quantity oracle construct rest organize oracle matrix replace pt formal guarantee full th factorization k principal satisfie identical step satisfy actual three section subspace approximate break part oracle lemma principal regard feature nj supplementary material key proof claim require well estimate since analogous eigenvalue nj supplementary material claim inequality oracle high satisfie end characterization sequence evolution role point denote show oracle j nk material claim require condition much high claim j evolution large lemma nj supplementary material ingredient subspace characterize foundation proposition error maintain decrease continue slow slow fashion interval inside proposition previous element high study sequence k consequently soon nm nm directly aspect sigma hold proposition take error sigma uniformly oracle give material complete rely actual sufficiently probability least supplementary theorem actual special theorem proof theorem lemma sigma c jensen event sigma l author like thank discussion claim section remark principal subspace eigenvector large propose model recover lead eigenvector consistently optimally many row usually thousand hundred issue analysis large space principal project onto span population variation capture dimensionality addition dimensional visualization covariance traditional plus recover vector generative factor loading vector factor logarithm spectra spectral component component number spectra collection reasonable asset stock people usually simultaneously scale hundred addition big signal processing generality high suppose rank covariance eigenvector therefore th spike literature spike make dimensional spike practical difficulty eigenvector challenge side sample eigenvector estimator sometimes even nearly different generality examine result recent start pca span dimensional explain approach start penalty become normality spike prove pca feature large variance eigenvalue lead eigenvector exactly loading nonzero location method al consistency fixed propose augment pca eigenvector show attain range lead separate span opposed find vector individually reason eigenvector identifiable eigenvalue dimension great new iterative estimate addition orthogonal basis model sparsity characterize lead subspace adaptively wide sparse appropriate moreover lead eigenvalue rest eigenvector attain optimal derive sense result large visualization avoid construct implement last principal examine normality demonstrate present produce table figure author website index denote dot submatrix norm say orthonormal might differ different occurrence number constant use observation eigenvector equal subspace object always primary circumstance interested visualize want consistently principal part convenience normality note onto projection function possible discrepancy onto range geometrically square lead orthogonal iteration matrix upper triangular schmidt dimension orthogonal matrix qr denote orthonormal eigenvector terminate iteration classical pca could problematic orthogonal dimensionality interpretation accumulate impossible span one sensible focus zero heuristic incorporate orthogonal effective feature screen orthogonal multiplication estimation summarize theoretical conduct normality datum threshold level orthonormal matrix multiplication k nj basic add user satisfie thresholde result column specify unchanged across indicate size apply range qr amount basis qr although estimator initialization involve without radius spike last establish wider simultaneously spike allow order spike constrain radius outline extension special allow recall impose growth large spike spike satisfie ratio j j part spike large magnitude interesting spike flexible spike grow infinity mild end coordinate ball radius rapid sparsity entry extend follow decay unified notion sparsity uniformity depend though require grow radius appear example bound away arbitrarily constant exist verify condition discretized eigenfunction eigenfunction isolate wavelet belong ball wavelet moreover determine eigenfunction grid away functional datum type define lead large magnitude compare coordinate coordinate actual equal coordinate let complement stand dependence notational convenience convergence cardinality show bound large cardinality constant parametric term interpretation discussion theorem turn ad spike allow depend ad eigenvalue spectrum principal say th satisfied satisfied large spike convergence establish convergence subspace relaxed generalize probability least state appropriately probability uniformly vanish constant vanish uniformly consistent quantity could elaborate little focus coordinate appear far accumulate later focus thus variance tradeoff nonparametric term vanish condition procedure rate adaptively sparse yield bound away iteration precisely stop could note nm follow direct setup property switch selection motivation iterative trade variance signal specifically
enough object recover representative analyze analysis rank write shorthand correspond case denote identify average main prove easily unless note average query choose collect comparison pairwise assumption query request correspond rank previously collect comparison fashion pick inefficient also version error persistent probably audio prove idea interpretation problem derive seminal learn crucial position dependency analysis dimension easy dimension make sort bad object initialize request object dot new label rank significant gap permutation assume passive pairwise ranking adaptive full quite inefficient ranking full recently note theory adaptively select need learn cause concern since consume pairwise example researcher collect pairwise comparison preference object understand due expense require collect underlie informative pairwise query perspective component primarily passive query admit space comparison especially subject highly rank familiar need million possible well worse inaccurate accurate intuitively intrinsic use binary response object embed distance learn comparison correctly object embed study main contribution specific standard unfolding familiar relationship distance space structural constraint ranking use generate arguably ranking assumption rise interpretation view rank ranking request whether consider connect hyperplane define close close thus equivalence query pair hyperplane possible intersection recall ranking cell indicate refer cell cardinality set ranking assumption assume cell hyperplane I recursion partition cell recursion addition fashion prove positive q suffice ranking low algorithm require comparison bit query provide full bit ranking example dimensional object order binary achieving impossible higher induce hyperplane bad require query conclude bad situation instead reveal randomly inefficient answer ranking cell consist cell query pairwise random replacement integer query rank hyperplane bind probability unless query need ask infer probably correct rank label call cell pick random query informative fact hyperplane point lie check fall p free recall reference equivalently response represent hyperplane observation see equivalent primal interpretation dual interpretation point label linearly scenario assume comparison exist hyperplane still separate point pass point primal separate hyperplane correspond define associate pairwise sequential initial object nontrivial partial ranking probable denote object equally probable partition cell partition cell cell partition equal cell comparison choose j conditionally event separate pass hyperplane representation cell see hyperplane object bound constant see correspond probable object divide total cell immediately query query eq algorithm situation response query probably correct label incorrect response equal fashion algorithm exception query encounter several equivalent vote accurate respect ranking adopt popular comparison convenience time report incorrect pairwise closeness response random group people realization decide vote identify request deduce exactly recover condition one need request comparison possibly incorrect persistent human ranking distinguished guarantee good hope recover rank object merely probably correct recovery object approximate henceforth persistent persistent ingredient design voting encounter suppose query set identify object rank contain define rank rank contradiction furthermore rank uniformly initialize order rank uniform ranking explanation sequential encountered call closely sufficiently large accurately determine threshold draw replacement call decide voting otherwise list determine next enjoy favorable approximately rank initialize k jk l l output object consider algorithm end object rank query pass guarantee mention object pass one three chance mind constructed rank query average intermediate stage full initial randomization voting pass uniform assumption object prove sequentially random large one ranking least e pm robust pass first object first pass ordering randomness voting algorithm figure recover least ranking initialization repetition correctness correct ranking rank quantify estimate ranking consistent probably correct additional uniformly correctly know comparison combine lemma figure n initialization sufficient object passed arbitrarily recommend believe early greatly affect present free represent hypercube experiment repeat new simulation reference response identification ranking plot exceed twice agree deviation query noisy response solution dimension std error algorithm symmetric similarity represent similarity audio row possibility repeat persistent suggest relationship approximate non multidimensional embed pairwise label sequential fraction similarity rough guide report average suggest hope agree many make perform binary implement query lemma n dd simulation permutation object embed cell rank hyperplane point denote cell keep one unbounded obviously similar construct upper induce hyperplane object hyperplane hyperplane partition intersection partition intersect general way cell hyperplane partition hyperplane ease pairwise comparison st object event conditionally follow binomial query relevant sufficiently none sort implement query th ranking random randomness assign th rank probable ranking placing straightforward follow argument equal generality may cell hyperplane relate observe word uniformity constant trial vote chernoff query see I j random py calculation place may aid see converse possible ranking probable choose partly theorem statement
express regularize incomplete computer algebra trial process know success system sec voting interval yield agree derivation principle straightforward mathematically least rigorously justify bootstrappe implicit distribution error regime bootstrappe say seem mention indeed well replacement confidence coin beta bias result estimate problem ref percentage percentage book se compute confidence method sample near red rest white sample take replacement red yield confidence wrong fraction probably certainly instead nonzero describe call sample medical trial case might consider interval never several obviously wrong result exact describe detailed want final eqs exact value order discrete case stand whenever limit coin tail specific tail head specific contain way motivate main trial later infinity coin coin trial eq trial coin coin example n mean observe head coin coin head confidence computer large continuous probability head
absolute relative resp da da da f da extend frequent mining q absolute association definition rule resp w w w c relative close solution mining rule property def point bind dimension sample approximately outline detail dimension sec science infinite member range range cardinality space cardinality subset large arbitrary range interval define subset vc range vc dimension sense subset relative range subset eq construct see resp cardinality resp assume draw replacement constant depend show constant currently know thm interesting range space size sufficient approximate solution market transaction subset transaction transaction empty da range empty vc transaction follow corollary let dataset support set exactly definition di rp ds vc transaction subset exist di transaction di di contradiction transaction correspond space maximum transaction transaction build easy transaction transaction compute bind range efficiently upper maximum integer transaction length one dataset item index anti anti large anti transaction c chain determine reason anti transaction subset transaction contain also contain contain easy transaction dataset case vc dimension dataset def transaction clearly contain transaction transaction appear would transaction short transaction appear long must anti transaction transaction member contain labeling get transaction contradiction therefore false strict vc formalize vc different transaction length transaction transaction transaction range transaction since dy transaction argue vc dimension two exactly anti build maximum anti compute maximum polynomial slow take solve match hence scan memory htb tie break arbitrarily easy bound contain constraint transaction anti fashion transaction keep integer transaction memory transaction avoid already argue transaction transaction deal correctness compute maximum contain transaction length less maintain tie break transaction integer different length complete thesis size transaction read invariant transaction begin loop iteration transaction examine loop invariant condition fail begin transaction equal would neither invariant transaction contain th iteration transaction transaction great transaction hence end transaction least transaction indeed line end th different transaction express invariant instead th transaction transaction contain transaction mean see begin transaction strictly transaction p follow sx dx algorithms absolute execute mining estimate depend integer thm frequent least great property def def property size absolute need compute need approximation integer dc time easily element dx sx k I def property theoretical concern lemma build cover deal constant correspond thm know association definition anti w association rule pf otherwise anti monotonicity property therefore def property def thesis size close absolute extensive experimental approximation analytical size theoretical motivate market previous work artificial come repository moderately effect size frequency dataset build transaction possible thm create follow generator ten transaction use fp growth association rule compute resp resp reasonable characteristic ar currently value find work practice select range frequency threshold top association rule size measure rule sample deal extract frequency real real figure plot take association rule collection single resp quantity run rule always collection indeed theoretical guarantee collection collection interest collection return least relatively false collection output remain extract pos ar acceptable positive distribution real frequency dataset within frequency highly probable pattern real high positive low threshold depend algorithm real exact false positive scan tp goal error obtain look close correspond pos index picture bound analysis error size derive seem suggest room improvement report compute artificial dataset transaction million transaction draw absolute absolute close problem tp evaluate extract note possibly association frequency two appear transaction frequency similar conclusion motivate intuition market mining entire mining transaction frequency running make useful frequent experiment artificial sufficient speed grow perform create scalable dataset become mine grow dataset generate associate report bound transaction show actual relative e vc associate dataset exactly transaction length behave equally fig fix dependent evident suggest present always vertical logarithmic encounter size pos derive extract approximations frequent association rule linearly vc transaction tight family theoretical statistical develop solution computer well guarantee moreover ar random aggregate frequent quality exploit adapt resource achieve parallelism quasi correctly account guarantee technique small dataset believe apply mining traditionally use important discovery vc dimension dataset rank suggest conjecture computer science university edu discovery association fundamental computational primitive application market database heavy find great identify frequent association define confidence solution multiple dataset expensive explore application sampling sample tight relation size satisfactory solution analyze frequent discovery item frequent exponential distinct item begin represent chernoff difficulty simple application therefore sophisticated work loose bind novel application dimension tool roughly collection complexity sect formal major vc dimension indicator obstacle apply vc problem range theory presence transaction contribution characterization vc range tight vc quantity transaction large vc time require greedy fashion analyze frequent rule tight extract mining top ar bound absolute relative sec quality mining table compare technique see vc consistently dependent minimum dataset extensive experimental evaluation advantage work first extraction random sample believe exploit data k sect formally define goal analysis main derivation bind vc association rule sect mining sect extensive conclusion find sect mining extraction dataset start problem generate rule imply frequent association rule improve ar reader survey time heavily depend use present empirical validate random build contain frequent candidate frequent candidate frequent pass entire suggest ensure union drawback sample linearly static chernoff possible analysis suggest work try sample technique theory hybrid chernoff derive loose useful corpus extract select add transaction self fast evaluate suitable satisfied major guarantee another approach give frequency use sample mining guarantee recent first analytical size transaction also present approximate present well apply problem top extract
desirable machine storage rather bring additional complexity natural reason distribute decentralize typical user click storage process avoid bottleneck powerful server relatively distribute platform amazon ec oppose server constrain hardware improve size past decade several reason question cluster broad pass optimization delay distribute consider batch version distribute article technique parallel framework close centralized knowledge implement scale report implementation fair compare exception leave something large measure run interface speed fast achieve throughput gb interface throughput achieve cluster example time course per throughput difficulty parallel efficient parallel occur platform transfer support generality force deal cluster unlike programming effort essence call one key across node computation robustness rapid synchronization overhead good environment core evaluating provide intuition work discussion open become abstraction ill machine researcher iterative algorithm abstraction operation start number name impose proceed phase phase average entry typical main architecture simple architecture section spend time attempt optimize parallelization bfgs gradient accumulate locally average exist moreover implement little address compatible span server cluster job process connect connect span create ip address ready service desirable reliability fail fail trick reliable enough practice computation hour idea computation gain abstraction similar primitive hybrid batch approach pure desirable drawback overcome attractive optimize objective rough precision pass sequential algorithm however make drawback attempt batch newton quasi newton bfgs optimal reach rather require aggregation every attempt drawback start one pass adaptive modify good rather locally accumulate square maintain square confident e assign node feature indeed call routine diagonal bfgs global summing gradient locally initial rapid guarantee quasi point communication small hybrid strategy repeat learn similarly pass different online section get moderately test fast reach strategy share carry iteration break phase vector size size application order communication much second naturally nature share open invariance see jj jj j across weighted start bfgs typically avoid execute job job finish framework handle span topology create span net trick hundred length highlight benefit gain experiment online page key matching page visit click ad get improvement user result click ad example result click negative click good visit represent user ad visit end element illustrate conjunction imagine place bit hash string conjunction unbalanced subsample negative set day dataset find human public dataset effective strategy day induce learn never result sequence regularization test hash feature zero explicitly impose implementation introduction small site recognition curve drop subsample substantial thus optimal roc curve precision log likelihood report iteration node record spend spend spend finish communication minimum interest speed median outlier time slow execution successfully cccc max without execution execution study measure need one run repeat turn speed slow oppose aspect pass median time function experiment bottleneck main degree slow also time version display datum example pass minute describe speed throughput result distribute boost run per report core factor processing report throughput range appear slow investigate speed hybrid interested fast objective fast learn online pass optimize plot gap define I pass strategy bfgs hybrid consist bfgs l bfgs appear site bfgs yield explicit representation create overhead algorithms job job scheduling transfer datum parse implementation table confirm speed extremely remain plus compare gradient tune sgd site online stochastic accumulate mini batch minibatch gradient carry yield mini gradient site mini batch pass update overhead update hour less finish pass much superior run communication instance conclude update possible mini reach similar much batch batch size communication overhead expect time reach pass small theory site inferior prohibitive tune evaluate characteristic communication computation aim hope design choice scalability consideration strategie clean simplifying restrict weight extend scheme additional detail convergent bfgs bfgs provide gain substantial certain regime comprise example uniformly node example objective couple continuous neighborhood around bfgs understand pass hybrid analyze pass perform pass node approximately jensen yield eq complexity arbitrary minimizer function combine inequality fourth remainder discussion denote switch contraction bfgs pass data bfgs hybrid amount overall pass ensure computation cost quite observe bfgs batch similar quasi alternative ever relatively hard local pass online averaging notation approach discuss beyond special one hybrid virtue phase overall approach offer first level level argument clean pass hybrid oppose pure pass regime extreme noiseless desire impact certainly strictly bfgs site typically moderate accuracy evident rate overall competitive computational identical weight per pattern cost modern cost variable total nonzero pass way relevant format pay across
general hereafter limit kernel interval corollary towards first introduce complementary additional whenever stand ii family function satisfy usual hereafter display consider distance assume function differentiable exist condition satisfied function class van norm whenever radius cover set condition easy display set lipschitz index take take diameter condition satisfy smooth let nonempty possess integer derivative set van page inner evaluate quantity u dy dy taylor expansion around everywhere smooth consider clear interior moreover make subsequently clear maximize function easily eq give function f increase notice xt eq follow eq similarly whenever proof straightforwardly principle continuous consequently speed first contraction principle eq low suffice nh l nh lx lx establish lx constant away lx lc nh nh lx lc proceeding proof lx lc decomposition statement l l lc lc lc lc lc lc lc lc lc obvious nh nh lc lc lc r lc nh r nh lc nh nh lx lx observe first real x consider statement obtain lx lx lx lx lx lx lx lx lx lx lx lx account shape lx lx lx lx lx lx lx r lx r lx family observe x whenever similarly therefore whenever e l whenever therefore value take follow consideration definition consider function whenever tt whenever therefore establishe achieve processes york york normality nonparametric de la dans des semi es paris functional series international conference trend characteristic process theory new york inference practical aspect nonparametric nonparametric ergodic deviation principle university york cm remark universit universit paris france abstract devote functional index vc chernoff state deviation index functional deviation vc class g great interest motivate great time nonparametric continuous explanatory finite play important role account reference therein due availability phenomenon modeling year view worth increase number recognition studying model continuously present work al al recent therein introduce lebesgue measure banach distance function kernel go go define notice estimate stand whenever deviation derive asymptotic case xu xt xt f xu speed good integrate simple whenever uniform rate display notation assume whenever differentiable uniform explicit function whenever display whenever differentiable end set complementary large
message update conjugacy message along inference recently apply channel interference decode bp speak exactly bp pass graph bp check code lot even tree reweighte recently develop reweighted geometry decode use projection find mf mf convergent message rule compatible cycle compatible constraint code constraint especially probabilistic bp mf combination drawback unify combine approach equation bp mf minimize kullback region derivation pass bp mf main technical theorem message pass equation bp mf correspond stationary state couple bp mf correspond pmf separation bp mf pass point represent whole factorization pmf interference problematic research community posteriori probability feed receiver coincide propose channel thorough justification message factor factorization pmf stationary constrain correspondence belief arbitrary point marginal always bethe observation message namely factorization pmf certain organize fix devote energy approximation recall bp em approximation use briefly extend avoid calculus message bp lemma pmf bp real convergent implementation message equation graph pmf special combination bp mf joint decode advanced architecture numerical find application communication direction capital cardinality write convention set e capital discrete realization pmf convention representative realization run realization realization pmf define realization convention denote capital letter stand th row x iy I stand proper pmf variable ia aa set ia ap pmf generality factor positive positivity factor connect depict subset region associate number number approximate normalize variational normalization energy kullback leibler base rr belief give two valid set count show region count q bethe free energy bp bethe bethe free energy impose marginalization normalization lagrangian marginalization belief bp point positive belief versa solve schedule message constant drop belief indicate normalization ratio message update accord ignore belief change rescale message irrelevant rescale belief obtain solve lagrangian elementary publish rescaled solution suppose n ax ia rescale solution see tree message run forward backward interpretation approximation free factorization constraint good approximation error plug expression region correspond belief normalize normalization mf exist convergent belief simply use iterative belief converge note order I ax transform pass mf case lebesgue message pass interpretation special mf belief rewrite minimize message except message eq q pmf ia ia factorization bp mf furthermore define I aa counting region define compared count number bethe guarantee valid counting approximation aa normalization marginalization need belief marginalization ax I ia bp part backward belief ax available mf ax ia satisfy normalization constraint equation free proceed describe free guarantee converge show compute simple message enough class complex architecture together simulation bp gain convenient work hard mf approximation message mf apply bp intractable cf pilot symbol respectively represent bit respectively represent code bit n lx ni remove cyclic receiver symbol vector pilot symbol multiplicative represent nz time channel set condition imply factor htbp mf split mf utilize advantage mf refer figure note fulfil bp factor graph convolutional initialize eq ix bp part bp bp compute update message eq get parameter consideration message bp mf figure pass mf pass bp part whereas message pass mf part ambiguity ambiguity reflect family replace decrease compare single complexity mf approximation discussion ambiguity make evident message simplify exploit conjugate mf gaussian factor equivalent apply bp factor I admit closed form term consequence message term message bp combine approximation comparable message complex alternative figure notice perfect consecutive approximately bandwidth channel twice bp inference like g comparison mf find mf yield complexity tradeoff instability bp mf combine describe receiver pilot number evenly pilot scheme symbol convolutional channel channel coherence bandwidth realization replace exist message additional gamma conjugate mf estimation message pass equation mf approximation correspondence solution system point equation split mf bp node result pass point belief stationary update certain show mf bp demonstrate efficiency computationally intractable propose bp computationally demand message extension bp contain generalize lagrange marginalization objective fr differentiable fr definition book continuous bp mf self localization another region message pass reweighted correction mf reaction author thank ia set equation message positive belief combine fulfil pdf integral region verify mf apply discuss q minimize formally significant simplification make marginalization normalization marginalization q shall compute stationary imply ix ix ib point stationary rewrite normalization fulfil ia ax ia ax j jx ia stationary point belief reversible rewrite fulfil analysis remain exclude exclude vice versa prove contribute see ia finish run backward algorithm base income mf bethe free minimizing accord allow plug
classic classification matrix video dictionary mp formalize section especially learn tool recent review device size critical effectiveness publish model determine bayesian g impractical aforementioned lie answer question know selection information aic minimum description principle cost search minimize sense regard practical state description phenomenon simpler usually good length principle give data sample describe bit define describe practice define ml choosing model tool mm familiar explore wavelet denoise corrupt additive white datum exploit iid variance coding encode define art aforementione work follow way dictionary learning atom adaptation low critical deal successful sparse pose particular item systematic fit naturally result category robust thus information learn check meaning treat rigorously natural naturally add dependency feature art bring fundamental understanding bring world detail introduce different part encode describe actual datum mn error active pz I refer matrix say achieve notation extend quantization solution either use greedy pursuit mp convex commonly body certain mp case property denoise wavelet domain parameter choose optimal respect orthonormal avoid cost guarantee alternate optimization dictionary patch patch decompose overlap patch image noisy patch variant user algorithm keep patch patch belong plus small despite method later stage patch partial problem assign however still increase burden introduction practical well angle selection subject follow call class nest p want objective criterion selection mean possible class encode traditional broken guide possible obvious learn patch parameter embed fashion well thus free advantage practical slow one critical clarity presentation encode principle call refined author date extensive reference subject compete coding good capture description avoid description select short description translate problem ideal shannon common lx assume constant quantization encode include cost depend come summarize fundamental establishe model encode depend geometry initial encode separately use expression develop bic significantly main difference early block shannon fully know classic theory encoding produce need produce example describe instance compare pick parametric likelihood ml well encode code base require something laplacian dictionary consequence art maintain encode separately use universal part next encoding scheme describe model approximation noise model regard description main incorporate prior information thereby include certain transformation markovian encoding model quantization component respectively order realistic precision drop simplify example write quantization several mind encode separately rest end discuss single signal form three independent need cost extend sparse discretized laplacian wavelet encoding depict use sparse coefficient encode model number encode ideal smooth parameter result encoding scheme code describe bit universal scheme actual sign coefficient encode magnitude continuous quantization determine cross validation handle stationarity variability universal universal mixture standard mix density informative shape expression zero density ideal shannon quantization coefficient reduce coefficient practice quantization natural indicate advantage attempt optimize keep mention solely actually clean follow nn reliably sufficient component due call show figure purpose verify derivative influence type estimator see theoretic experimental distribution unknown encoding call universal employ come component gamma eq guarantee informative also reliably say parameter free numerical formula estimate coefficient atom th atom west markov atom pixel markov random assume unbiased f obtain case two zero previously universal model statistic typical see learn dictionary observe row encode respective kt plug encoding scheme sequence encode family coefficient th recalling give markovian adjacent dependent example use estimation depend value image process markovian neighboring west one encode occur depict take encoding encoding nest describe call well however code approximation constraint alternative pursuit another relaxation function brevity describe relaxation sparse coding publish elsewhere minimize mp selection empty value give dictionary current coefficient coefficient high add contribution enough produce part implement candidate increment variant turn slow compression gain per sample stop iterate previous assess validity variant stop mp residual coefficient evolution iterate marked black circle note describe pursuit p pl make datum base make easy class dictionary speed forward describe start empty approximate depth subsection algorithm learn frequently atom initial dictionary dct lp l p give set traditional alternate cost need regularize estimation current iterate accumulate iterate process along accord mp correspondingly fitting term none differentiable triangular efficiently use backtrack variant fista focus experiment code actually datum bit gray selection bit gray decompose patch overcomplete dct obtain ability compare apply dictionary database measure backward variant add backward dictionary compression cost convergent forward yield require result implementation ii ghz three reach similar backward cost significant partial fast yield task clean observe corrupt known contain overlap patch denoise patch backward selection global start clean atom second stage admit distortion distortion case describe prescribe c consistent derive dictionary obtain j develop j portion clean think clean problem j markovian dependency occurrence previously encode neighbor denoise summarize detail figure case improvement relevance limitation rd comparable carefully tune scale image aggregation patch rd variant well tend rd visually rd case include cccc rd rd denoising rd clean noisy learn dictionary recover estimation residual portion add back final sample task assign texture patch patch encode dictionary center pixel short patch rate consistent picture figure inconsistent formulation adapt patch respective texture expect decision cumulative patch patch basis compare dictionary patch success comparable see explicitly markovian improve dictionary automatically different patch wise correspond texture success rate classification map average rate extension mn equivalent certain able recover noisy arbitrarily corrupt framework camera surveillance
prototype cluster give convergence address perform baseline hundred scheme adaptive wasserstein start class datum second show fashion e distance artificial show outperform experiment quality paper deal distance histogram histogram complex description phenomenon spatial environmental wasserstein compare wasserstein histogram skewness histogram histogram propose base wasserstein cluster contribution histogram cluster kind variability descriptor whole account descriptor obtain partition measure index function wasserstein distance real collect represent characteristic image usually field description phenomenon flow bank account well statistic aim collect respect deal cluster number histogram introduce context symbolic contiguous histogram symbolic domain discovery relate pattern intelligence cluster factorial describe cell weight proposal dc suitable dc proximity call prototype element cluster prototype dc general cluster look good dc dissimilarity phase dc criterion dissimilarity schema around line factorial comparison distance dissimilarity vision color texture worth histogram pixel intensity wasserstein probability wasserstein permit internal distance spherical possibility identify orientation term alignment cluster step include tune weight associate distance histogram schema easily extend wasserstein present adaptive wasserstein allow compare euclidean account support wasserstein histogram two variability histogram decomposition adaptive approach decompose globally cluster cluster tool ratio base cluster sum inter cluster square histogram compute wasserstein organize wasserstein histogram dynamic non criterion base adaptive wasserstein distance histogram application usefulness propose cluster histogram end conclusion histogram discover phenomena similarity choice relate capability wasserstein histogram representation assume represent histogram contiguous bin different partition contiguous interval frequency let histogram ccccc var I n histogram function dissimilarity introduce histogram distance distance distance wasserstein permit result distribution wasserstein derive wasserstein distance natural euclidean metric quantile respective mean correspond histogram word square wasserstein wasserstein two latter difference wasserstein decompose location shape quantile represent problem calculation need computation quantile histogram avoid computational drawback identification wasserstein define measure histogram quantile means respective quantile histogram within decomposition use property define empirical standard object specific formulation wasserstein equip wasserstein joint histogram wasserstein multivariate wasserstein give weight variable distance equip general induce several associated partition set decomposable weight weight squared wasserstein wasserstein first propose well fitting individual algorithm cluster proximity function individual homogeneity minimize wasserstein distance histogram dynamic proposal wasserstein distance weight define describe histogram histogram description prototype histogram dynamic cluster partition fitting cluster locally minimize update accord cluster follow weighting allow component assign class proximity convergence criterion reach criterion minimize wasserstein histogram prototype criterion assign wasserstein distance prototype eqn general assign cluster dependent wasserstein prototype center eqn base stationary eqs minimize mean represent histogram belong resp result intra inter heterogeneity partition algorithm index intra tool non euclidean distance interval use evaluation histogram square global prototype cluster criterion pp k accord bss distance wasserstein prove additive term within square bss eq descriptor describe histogram descriptor adapt wasserstein data eqn general prototype histogram quantile average variable prototype accord prototype histogram description prototype description sum minimize criterion algorithm recall adaptive square q evaluate index define hold wasserstein index follow equal decompose inter wasserstein distance consider clustering unsupervised clustering evaluation quality specific quality limit synthetic crucial lack public describe histogram histogram compose repository test data generator monte experiment allow quality index agreement partition one china index performance describe performance dynamic square wasserstein distance control variability datum dispersion component histogram dispersion well dispersion histogram wasserstein distance decomposition end set monte carlo histogram choose skewness set assign set belong obtain four parameter generate random number parametric histogram criterion datum measure distance validity adopt generator experiment compare agreement whereas negative agreement chance correct classified end statistic cr accuracy variability generate histogram variability variable globally variability normal describe st cccc st visualization show baseline parameter compute cr deviation cr std clustering adaptive outperform classic performance cr base histogram cluster globally fix baseline sampling distribution skewness cccc parameter mean st skewness order visualization generation baseline cr deviation htbp cr std adaptive outperform dynamic see cr structures dispersion list begin dynamic cluster distance pressure wind people china recorded purpose month month contain histogram propose describe statistic relate global square variability histogram mean percent percent pressure mb pressure mb wind total mm show report description correspond standardized moment spatial coordinate relevant except cluster active c st st mean c st skew skew st st skew skew skew st fashion choice method cluster cluster index within case execute initialization three index fig reach suggest cluster
study univariate logistic model sis seem table contingency estimating way contingency range exact goodness contingency satisfy give problem basis mcmc state via contingency state already easy program intensive drawback bottleneck markov way contingency table show difficulty basis allow entry trade run markov independently take assumption lastly clear general converge importance implement sampling contingency table proceed contingency final terminate cell sample identically iid proposal thus sis expensive prohibitive basis table sis guarantee long sis present new problem compute lp sis reject sis contingency due interval table satisfy sum moreover way contingency sample sis distribution rely sis contingency give excellent interpretation sis fact lead target subsection table satisfy marginal via sis procedure notion short sis rejection limit I rejection conduct logistic model study procedure rejection even table cover sis detail matrix show contingency input table give sis rejection sis rate logistic contingency contingency integer integer generality view contingency n set eq write simply mathematical sufficient usefulness enumeration theorems rao extend element th rewrite simply count let set condition trial draw sis sample count compute integer notice always constraint sequentially apply sis programming ip solve integer q already sis equation lp however importance sample sis sis procedure compute solve integer sample integer interval return distribution mean big probability number equation inequality x exist nonnegative cone call feasible encode rational sis rejection polynomial fix namely generate q rational showed size rational function herein index size rational generating bit rational short rational generating suppose assume term input table sis state fix term short z fix rational generating input size short rational generating form lemma generate ij z j apply unbounded polynomial short generating respectively compute form sis rejection sis pick generating via generate find return pick side input solution thus nonnegative thus proof polynomial size step since count lattice sample implement seem function sis arbitrary rejection integer programming integral assess rate univariate bivariate choose sis seem rejection even generator poisson generator item integer output table cell pick else assign else assign randomly pick pick integer table random
learnable first statement classical argument estimator lemma prove restriction distribution game hard game relationship batch online focus depth study supervise game complexity bad surprisingly adversarial vc learn indeed irrespective regret within rademacher logarithmic factor algorithm blind game blind learn supervise adversary supremum satisfies pass associate game whenever lipschitz rademacher scenario distribution translate restriction mixed strategy interestingly upper bad hybrid function ty tp armed composition say rademacher hybrid scenario lipschitz rademacher complexity analogue contraction class lipschitz eq rademacher choose necessary corollary follow absolute irrespective bad rademacher matching bind attain choose variable irrespective matching rademacher bound define absolute adversary allow coin purpose prove enough adversary pick whenever restriction history tx restriction bound case second bind hold restriction allow rademacher restriction supervise learn protocol sometimes adversary yet observe subsequently reveal protocol allow yet equivalent modify improper allow choose side cumulative cost inside decade arguably optimistic bad smoothed see realistic measure smoothed explain despite exponential section analogously concern former batch sequence adaptively argue neither reasonably world smooth bad sequence well I simple learnable supervise scenario yet online case supervise dimension dimension threshold infinite mistake choose adversarial world opponent adaptively bad case smoothed scope greatly smoothed infinite tractable conceptually smoothed smoothed smoothed sequence technique smoothed yet hypercube example choose random smoothing pac consider corrupted error formally perturb measurable example noise correspond generally consider move smoothed adversary generic online trivial upper guarantee enjoy smoothed adversary adversary player observe move perturbation proceed nothing restriction study adversarial parameter smooth somewhat surprising learnable exponentially noise adversary learnable show bad notion restrictive supervise function binary learnable online trivial particular complexity appear formally eq uniformly adversarial follow setting whose move corrupt bin well smoothed fall bin take discretized threshold threshold difference generalize threshold correspond argue adversary bin supremum supremum cost bin sign range result exhibit martingale difference fix hoeffding observe discretization coincide element interval ensure fact threshold sake element fall bin mind choose throughout game bin q use need infinite indicate half learnable perturbation half classification smoothed note upper value game inefficient recover formulation problem smooth half space discretization job directly bin ball loss non discretized acknowledgement nsf grant dms proof proof couple understand move inside arrive repeat step expression q way prove fix tp sequence functional expectation q b fix infimum strategy past eq q bayes response supremum infimum strategy depend minimax strategy move upper martingale employ technique introduce sequence construct copy independent condition true hold plug hand obtain informally indicate conclude think path determine outline worth mapping draw independently sense define tangent supremum last equality verify understanding correspond supremum successive distribution irrespective contribution successive distribution condition value path irrespective clearly contribution argument alternatively expand claim verify simultaneously term generally perform center expand q pass upper tt p increase equality pass coin define constraint pass supremum tree path tree apply path rest tf tf linearity last lemma slight pair equation arbitrary tf tf linearity pass step however equation range rademacher associate quantity select pass arrive eliminate outside increase without follow appropriately proof proceed sequentially property start towards mapping supremum fact depth note rademacher bind need justification reverse argue supremum supremum supremum provide removed supremum long appear conclude notice play copy distribution rademacher draw whether via selector produce sequence restriction protocol define bound proposition expectation expand stochastic allow conclude proof adversary rademacher unlike sequence define conditional tree lower concern path similar lemma classical statistical scenario adversarial pick learner neither possibly due game adversary move capture stochastic building approach range deduce adversary supervise hybrid way consider smoothed half space learnable smoothed adversary decision turn infinite learnable continue line array assumption notion theory complicate nested tool unify bad learn scenario two present make progress towards instead measure external nature restrict play notion place nature assumption adversary yield associate name player name far traditionally refer decided keep whenever problem sequential adapt game language think sum adversary learner minimax heart statistical half restriction put adversary allow aware similar sequential minimax central estimator minimax function rough capture vast array consider associate minimax supervise loss infimum goal predict function combinatorial vc uniform risk minimization unsupervise learn erm quantity zero formulation long make generate opposite manner ahead game reveal frequently minimax write randomize randomized adversary pick bad sequence course protocol adversary choose move proceed next move player instead write quantity way sequence adversary round stochastic capture assumption online scenario move various smooth present minimax many question adversary organize minimax rademacher see case main careful devoted rademacher behavior adversary batch online complexity irrespective pick learning dimension learnable small introduce extensively use close metric distribution learner capture adversary restriction scope adversary restriction adversary restriction adversary player write define restriction name restriction strategy deterministic point support x tx constraint allow budget pick bad case corrupt equivalently restriction adversary choose distribution restriction shift gx pick force restriction eq adversary restriction restriction might point equivalently write q crucially strategie mapping q value main object payoff mapping tree extension bound q statement particular center relative shift adversarial move restriction denote banach play allow increment condition increment depend walk understand rademacher well rademacher complexity equal sequential sequential rademacher upper apparent reader minimax formulation mathematical necessarily yet difficulty become familiar next classical tp writing path equality fix path rademacher make expand inequality supremum interesting case hybrid adversarial refer proof dependent rademacher distribution sequential sequential tree depth result state maximal upper function mapping control integral function lipschitz constant z cover composition play useful scenario yet satisfy pick include game move previous move variance generally adversary specifically game adversary sequence round constraint play adversary view adversary conditional way game constraint compact necessary corollary satisfy necessary hold range set tree minimax q tf armed extend result say adversary allow away decision player exploit learner paper function consider constrain
reduce criterion netflix burden perform ignore remain rank approximate enforce constraint metric step within manifold rank lead art general address minimization riemannian update riemannian riemannian straightforward gradient tangent update geodesic direction intensity hand equip usual product straight note procedure intrinsic independent easy calculus symbol need much easy map tangent yield update give idea sphere endowed consist follow operation avoid calculate geodesic riemannian critical subsection remain converge size manifold manifold rotation real subsection continuously differentiable use instead exponential subsection slightly modify riemannian take account effect tend mild convergence prove manifold e disk half plane trajectory compact riemannian uniformly step exist compact build usual remain parameter compact measurable hz hz hence imply converge like fact use sum trajectory variation denote non bound variation e n n e sum quasi converge absolute converge converge taylor expansion ep f k right martingale convergence value map riemannian bounded continuously differentiable size satisfy condition compact gradient rely twice small exp riemannian martingale converge variation exp right unchanged second previous convergence long remain converge hadamard manifold hadamard manifold connect manifold curvature effect size yield relax even case hadamard map invertible usual case opposite towards curvature denote function manifold converge mild assumption imply upper appearing finding may require suppose converge build go trajectory easily expansion half riemannian geodesic bound hessian square eigenvalue tw fw w generate fw us taylor triangle fw fw v v v prove necessarily h quasi martingale mean thus fw continuous ball radius stay inside fix compact trajectory belong center compact moreover weak imply theorem derive used proof present illustrate theorem interpretation third throughout precede track eigenvector several application want principal e principal reason suppose vector element gr subspace ambient identify manifold column cost minimal basis dominant rotation state manifold gr study event consider stochastic riemannian application states geodesic vector input sequence gradient manifold zero compact prove point prove invariant subspace prove dominant subspace average invariant covariance application particular decomposition argument follow ambient particular already general present result directly stems illustrate viewpoint geodesic step computational especially know disk tangent conventional product diagonal angle riemannian distance move disk reach illustrated arc equip fr riemannian unique hadamard manifold grow interest filtering manifold propose closely optimization parameter randomly pick squared tensor riemannian redundancy equal riemannian require operation redundancy randomize reduction computational burden besides used filter measurement track slowly see low pass filter mean current new figure verify radius open geodesic contain hz positive quantity definite asymptotic similarly span subspace stochastic kalman theory gain become recommend gain gain make estimator initial sufficiently fast avoid concerned insensitive final could generally use compare behavior average gradient descent hz latter estimation decrease second slowly converge iteration experiment discrepancy rapid error require initial comparable norm gaussian input identity via semi rank matrix asymptotically top plot versus iteration plot solid estimation hz line coincide choose line gain method full rank technique parameter difficulty maintain definite thus attack algorithm project semi definite cone proved perform equal illustrate however algorithm little stochastic advantage convexity average full propose apparent operation become moreover rank lead thus low address emphasis refer variant extensively art variant test netflix riemannian multi distribute replace exchange various limitation gain popularity procedure neighbor average node reach intermediate value little distribute play consensus well know consensus problem consensus space wants agree motion phase adapt g consensus receive procedure address decentralized measurement distribute computation assume possess estimate allow exchange node accord pick randomly represent availability node reach value implement interesting cone riemannian view symmetric information tangent distance agree kullback leibler symmetric denote kullback leibler geodesic amount information separate understand statistics rao bind accuracy state draw look sample unit increment whereas identify variance discrepancy admit strong invariance mean attract ever image year follow procedure tackle step pick randomly neighboring node towards geodesic half geodesic select cost manifold explicit expression advantage viewpoint nice property merely geodesic proposition mean new accordingly unchanged yield orientation axis versus merely theorem assumption endow complete find geodesic ball geodesic ball step towards lie current remain convexity belong compact moreover radius away appendix cone definite riemannian behavior always illustrate figure run show fast initial outperform usual versus diameter superiority simulation robust together property average riemannian graphic superiority node heterogeneous bottom graphic ht versus run riemannian converge top node heterogeneous manifold reasonable convergence numerous control manifold enforce constraint intrinsic concern lead substantial source cast successive propose riemannian parametric natural asymptotically intrinsic intrinsic rao future explore reach rao bind detail future aforementioned completion prove critical lead mathematically identifiability possibly consensus stochastic hope extend algorithm consensus fast author thank subject l discussion geometry realization parametric joint log parametric law conventional space definite chart assumption w w term w expectation law basic rao prove statistical efficiency prove space endow gradient blind bss performance gain recently intrinsic rao replace distance fisher usual update could replace intrinsic paper achieve fisher reach beyond geometry let g riemannian carry metric length unique path minimal geodesic geodesic position cut roughly stop circle cut geodesic ball radius riemannian tangent kk along geodesic paper develop extend stochastic gradient point numerous potential novel test numerically provide great practical minima optimization demonstrate idea toy briefly mention traditional trajectory gradient correction correction angle tend reasonably hope
training test base index smooth marginal may benefit weight let smoothed sx appendix criterion convert inductive hypothesis establish guarantee smoothed inductive furthermore might nr analog smoothing need cubic root guarantee investigate advantage perform unweighted smoothed empirically weight choose work uniform sampling emphasize benefit even might weight attempt reconstruct noisy random signal ensure error perform prof throughout trivial entry recovery cite define since combine hold case step true since inequality fix write ab eq continue denote smoothed empirical generality matrix apply theorem bind probability empirical marginal cn define expectation split obtain si yield result weak requirement bound product expectation prove combine proof theorem define define argument two department brain cognitive sciences technology title trace norm might fail distribution index present empirical know empirical collaborative filter small reveal match well theoretically understand case nearly row require completion motivated issue variant entry show superior guarantee clear trace theoretical norm recent report e index product distribution seem realistic case netflix user rigorously trace correction netflix indicate helpful rigorously empirical weight advantageous arbitrary rely instead present consider arbitrary target matrix reveal replacement like prediction expect respect trace norm marginal distribution trace particular pi although certainly estimator sx sx empirical observe although inductive draw future state fix theorem set use trace rademacher complexity pair value modify trace sampling index empirical rademacher sign possible random use bind yield provide unbounded product marginal restriction fix sx elsewhere guarantee expect complexity duality spectral I mean matrix combine r calculate row assume marginal potentially unbounded truncate class spectral sl pz extremely sl p sx px rl px sx result marginal lower precisely satisfie result row marginal marginal unweighted trace none perhaps trace give arbitrary excess give trivial weak guarantee discuss lipschitz loss require amount require entry boundedness lipschitz fix rademacher product uniform excess error cube give cube setting whether improve theorems factor construct degenerate use product non uniform special cube good hold arbitrary error unbounded construction arbitrary unbounded regime error demonstrating let rl py sn sx rp ax summarize guarantee tight c unbounded loss uniform degenerate example generalize need enforce sort uniformity computation lie large small row improve guarantee smoothing provide guarantee suggest smooth large advantage situation check smoothing weight beneficial denote smoothed hold loss fix sx x p apply essentially identical definition si apply result eq think significant order square trace low rank column magnitude much place require dataset netflix netflix rating user netflix rating sampling scheme user deal test create rating aside set rating movie aside rating rating zero via reason truncate trace optimize regularization choose cross mean k smoothing smoothing even differently sample
family probability inferential lemma say universal universal family inferential equation per conservative hold pa connect concept density universal pp universal g equation pa aa equation say odd terminology change odd use original surrogate g recover hypothesis g apply measure aa may either section type minimax bad among since average solution density thereby finite jeffreys originally invariance whereas parameter describe reality default serve inference scenario imply support observation review asymptotic population density universal optimality subsection minimize minimax normalizing act convergence consistency since asymptotically indirect evidence incorporate define q notational parallelism bad hold I w accordance population observation consider logarithm commonly finite cancer transform abundance conditional absolute protein simultaneously estimate maximum obtain appear fig sample minimax approximation could achieve comparison simultaneous I amount support nuisance comparison argue support one bayes support quantify odd odd hypothesis pure previously replace maximized parameter constitute composite hypothesis thus strength evidence union fx fx perform nuisance parameter extent intuitive principle viewpoint l primary generalize pure predictive make applicable hypothesis measure difference empirical bayes proportion compatibility even example odd calibration compatibility information uniquely measure protein standard odd measurement protein come simultaneous inference nature discrepancy support increasingly increase dimensional application overfitte support penalty factor acknowledgment research partially innovation innovation university david biology department road thm david r department frequentist hypothesis accurately ideal support observation second minimax normalize compatibility normal sample expression datum closely odd available protein random parameter reliably indirect evidence description likelihood science observe p belief ideal salient easily term value hypothesis quantify odd odd available frequency although baye assign alternative unless hypothesis factor improper conventional divide test generate proper expense specification sample concept measure applicability routine scientific present herein another odd rely relate probability hypothesis biology genome protein determine nan hypothesis hypothesis operate distribution nonetheless retain model across include necessarily datum paper reliable often parameter measurement available often reliable estimation argue due microarray less maintain reliably record statistic meet explain framework one odd thus report support enable reader roughly determine either use value minimax address support abundance protein analogous multiple protein finally conclude density parameter hypothesis call member nan density reflect randomness specify value employ eliminate nuisance parameter interest identify thus nuisance eliminate distribution strength statistical theory effect real uncertainty nan respectively pp indicate hypothesis sufficiently
fair al publicly crowd use amazon color color ask crowd indicate ground color comparison distance color distance check validity collect response comparison task response similarity worker answer subsample would number response per crowd amazon quality indicate since phase around algorithm perform majority scenario assignment assign priori collect limit task worker latent draw provide minimax result minimax factor term budget core concern answer cost query budget achieve target good task assignment inference crowd parametrize minimax range range assignment ask query total probability assign query achievable oracle prove minimax case low rate worker low rl mistake task flip fair coin half convexity jensen necessary crowd assume query actual assign choice hold bad assignment minimax regardless worker term number query achieve necessary adaptive query algorithm average worst worst analyze however result prove worker quality budget establishe minimax pay factor necessary simple voting approach voting provide achieve voting least I achieve voting need ask per voting costly term budget well operation require prefer practice process parallel switching assignment hope adaptation help careful requirement much gain allocation identity worker reliable worker manner crowdsource amazon identify particular worker persistent identifiable open exploit worker adaptively batch task next worker base thus far hope adaptively collect less perhaps surprisingly gain adaptive worker error good task allocation true total exist assignment scheme query budget error worker scenario achieve average less case compare corollary significant achieve factor scheme limitation adaptation strongly fact worker exist design corollary worker improve constant fact family worker algorithm succeed worker exist achieve probability show adaptive algorithm vanish increase simple adaptive show go algorithm worker task group repeat reach allow mistake run allow query finish prove allocation scheme worker ask worker among response run query worker order terminate condition concentration result prove vanishing grow scheme prove non instance adaptation significantly worker fail crowd apply produce error half grow section explain vector reveal answer fill zero prove constant numerical cf due result scale know effective perturb mt conditional expectation mt reliability exactly perturbation mt spectral signal correctly extract use crowdsourcing introduce prove scale result analyze spectral gap spectral conditional denote prove lead randomly initialize normalize converge sign v power iteration identical pass task worker iteration message upper scale analysis technique singular crowdsource graphical weight bipartite connect edge posterior ia ia abuse denote probable graphical bp maximize approximate probable closely standard line provide support bp operate set message message worker message message bp task correspond worker message worker standard framework iteratively message follow randomly initialize ip fp p aa k aa x ib ib rule indicate proportional algorithm produce end predefine I ip bp ib ib I either tell always give wrong answer rl bp update iterative I ap update cf belief propagation despite surprising perform optimally task prior discuss implication research always suggest stop transition discuss provide iteration iterative voting algorithm notice meaningful half inequality half main emphasize behave try error iteration increase algorithm generally section scale always cf suggest iterate help cf gain maximize exist worker bad intuition variety algorithm identify ask whose answer also gold standard unit assess quality gold unit seed inform gain gold utilize lot gain embed gold estimator still include strategy utilize seed seed gold pilot gold pilot question pay question benefit pilot gold collective crowd worker pilot describe gain pilot still pilot crowdsource crowd might worker crowd collective quality payment use mix say worker subject per immediately maximal implication pricing scheme crowdsource crowdsource platform collective crowdsource platform quality ratio least good crowd possible paper factor difficulty worker response worker question task reliability bias towards answer represent reliability formula worker binary model sign large answer likely regardless correct answer response latent study crowdsource early ij another ij ib ib model paper task share worker worker distribution proof symmetry result estimate average task probability randomly task assume density task worker edge worker distance root inference worker pass task run worker message grow subgraph evolution error make actual next vanish regular bipartite bind error condition variable whenever variable negative distribution decision know density code theory distributional probabilistic equality density operate whose randomness randomness graph realization variable variable condition regular locally regular tree distribution message initialize worker variance initial worker iteration ij symmetry density evolution equation definition also variable rl p represent worker distribute worker conditionally independent initialize message incoming message response worker quality incoming message response variable chosen numerically compute computationally message take heuristic novel sub recursion upper prove sharp decay exponentially message behave evolution accord p p p simplify p substitute evolution variable r assumption bc v k q second k k var k x chebyshev hand prove sub weak message fundamentally regime good regime converge grow q get tight gaussian following hold e k r km precisely definition distributional independence k follow satisfie k k get km r write formula evolution mean substituting follow proof inequality regime recursion k k bc c worker node configuration half edge match worker let denote worker nr similarly way half half probability r total assumption bound p apply fact majority voting majority agree formula neighborhood random without rescale random variable node assumption p monotonically increase assume value odd use closely approximate error low expand substitute bind node distribution task e follow q prove even know reliability worker th worker collect reliability decision next assignment stop criterion meet label maximum make worker assign task follow p rl worker assign stop worker assign oracle know computed worker reliability budget achieve allocation necessary since total definition pt hold focus task reliable worker estimate worker worker let reliable worker answer therefore jensen worker reliable jensen maximize result notice change ensure variance step variable binomial gaussian calculation limitation general crowdsource model mistake accord worker make worker reliability task equally challenge inference crowdsource model bias heterogeneous answer approximation generalize algorithmic interesting solution might modification belief propagation probability small worker approach scenario worker counter achieve term budget around well majority formally crowdsource study theory mechanic rgb rgb crowdsource numerous piece worker solve character recognition nearly devise confidence answer typically assign answer majority vote general crowdsource total achieve target reliability worker infer worker inspire outperform majority voting comparison know dynamically assign task worker might hope surprisingly minimum manner scenario scenario rely worker exploit build worker design crowdsource system human annotation optical character recommendation crowdsource amazon market batch worker benefit crowdsource highly task even among worker collect strategy reliability answer worker pool reliable exploit fashion crowdsource amazon large anonymous trust particular correct correct answer may never truly make hard resort worker irrespective answer aggregate majority reliability answer crowdsource worker neither persistent identifiable may identify particularly reliable worker nonetheless worker answer draw weight however worker aspect unlike batch plausible ask pilot question worker decide ask question answer decide ask understand variation define probabilistic capture question collect simultaneously may adaptively worker answer provide task error achieve error establish task optimal base characterize collective reliability crowd achieve target reliability necessary replicate interest assignment ask achieve rate somewhat manner adaptively help setup crowdsource integer categorization example state correspond label child solution worker fashion crowdsource two task allocation query sequentially accord choose subset constraint choice batch worker batch worker reliability parametrize rl character clear next adaptively answer collect thus far assignment typically query meet inference estimation answer say task answer collect answer adaptive batch batch process parallel however switch adaptive investigate aware worker consistent real crowdsource worker go persistent solve new particularly game sequence trial pool identify multiplicative crowdsource hope particular capture worker difficulty hence performance across task discuss generalization worker distribute example answer mean different literature model random label crowdsource parameter proper normalization crowd role capture collective crowd worker indicate crowd case achievable population distinguish sample quite meet batch requirement prior execute reliability determine time task many iteration reliability discuss simple way overcome perfect crowd correct justify truth task crowd agree consensus without crowd definition ground worker quality proportional accuracy want crowdsource minimal deviation make crowdsource identifiable worker worker batch crowdsource efficiently worker identity worker imagine crowdsource platform identifiable worker choose explore exploit one significantly simultaneously estimate worker select worker good worker phase al voting identify good worker exploration exploitation however test crowdsource use pre track worker market popular crowdsource amazon world crowdsource impossible track popular crowdsource market worker crowd reason assume worker provide algorithmic worker show worker technique furthermore worker track develop worker start account attack closely formally address crowdsource payment infer accuracy worker worker pay worker task amazon optimally crowdsource contribution technique inference introduce operate message priori overcome novel establish recursively prove class message certain answer query regular answer achieve order task precisely approach low rate adaptive minimax factor necessary rate bad worker propagation allocation scheme worker amounts design bipartite node indicate include batch crowdsource platform batch complete task work batch bipartite many worker resource g task worker typically task automatically edge achieve low bind graph degree thing significant worker budget collect decrease grow immediately affect performance rate however worker degree generate regular want graph slightly propose simple random task half edge half might edge node graph number double hold result analyze performance sharp intuition behind graph spectral follow vector weight adjacency excellent spectral gap enable low allocation connect assign index worker node node edge correspond worker worker introduce novel inspire approximation connection operate value message worker message reliable worker message initialize sensitive positive initialize message need extra degenerate accord task neighborhood worker task update worker give sign believe message represent hence answer worker iterative I jx max belief approximate bp detail crowdsource density evolution bp crowdsource numerically evolution cf density develop density iterative analyze broad message upper achieve iterative configuration decay dependence effective gaussian tail inference r follow bind run iteration crowd collective
implementation ease state along generalize range restrict e tensor square problem method tb input b bb output histogram histogram recovery briefly discuss extension multiple histogram convert sparse dimensional orthogonal transform version wavelet apply transform e omit refer interested reader wavelet coefficient non zero coefficient turn practice cluster significantly histogram similar matching pursuit inverse transform represent histogram correspond coefficient unseen cardinality section present histogram batch load level comprehensive engineering believe conceptual choice address question extension algorithm modify query maintain appropriately histogram step incur consideration similar solve small non wavelet propose modify say accumulate modification remain unchanged histogram maintain online characteristic however old information adaptation might assign recent discuss learn formulation involve empirical assign alternate assign old modification algorithm detail algorithm present current histogram histogram training dimensionality histogram attribute evaluation involve one histogram percentage average relative number use experiment detail present histogram report relate dynamic census repository dataset essentially mixture pt census various attribute synthetic varie clearly range linear dimensional synthetic projection census c synthetic gaussians random synthetic ii five gaussian around attribute census age census database synthetic census spherical dataset mixture gaussians random census age work attribute choose attribute status education describe range query generation dependent query center total uniform model data range hyper generate around volume query train test various detail dimensional implement experiment well use implement matlab average solve solver solve scale pt query incur synthetic converge around error synthetic type incur incur error test age attribute incur error incur histogram vary learn increase train first compare incur attribute naturally incur increase query however able around query incur primary round fit query dependent fig query converge boundary discover align peak true frequency mis peak contrast accurately boundary incur type ii query converge accurate specifically incur incur incur inaccurate bin boundary much wide boundary interestingly query histogram histogram method align boundary true fig census uniform incur less able learn ht cccc c dataset model incur error query query dependent incur census query incur less vary synthetic query respectively figure incur incur error incur error incur however query query model significantly well perform next query vary compare three three method incur incur incur error census query draw vary incur study vary compare synthetic well theorem heavily query trend query summarize converge incur incur synthetic consistently vary demonstrate advantage two small incur multi census ht synthetic vary query incur incur error incur incur census vary query incur error incur incur incur test histogram incur incur plot incur skew able reasonable histogram incur follow figure report finish census training census incur significantly cccc pt frequency number work attribute census estimate histogram census training query work attribute capture accurately dual core ghz gb ram mostly finish finish stream incur version dataset database update query train dimensional compare vary dataset generate datum dependent histogram outperform reduce rapidly query query incur incur next three vary outperform small incur around experiment census dataset recall census project census attribute database work concentrated try empty consequently incur incur error still approximate underlie incur estimate density peak contribute incur report incur census vary incur relative scalability increase conduct experiment synthetic gaussian show incur vary incur incur incur implementation terminate day entropy problem round need iteratively number run method minute figure training error incur number significantly incur repeat census consider education attribute trend accurate inaccurate incur incur performance presence datum wavelet keep unchanged effective converge robust synthetic generate batch batch around similar th converge update database summarize enjoy number number remove end fit high error contrast well run solver form large even though require well high scale fairly high inconsistent extend update simplicity critical query histogram use work histogram histogram reader tune first accurate split bin density provable restrict grid high case grid however grain impose feedback small estimation call learn entropy state art tuning recent involve distinct dimensional histogram effort seek execution wavelet tool haar popular wavelet histogram piecewise signal haar wavelet extensively histogram estimation wavelet probabilistic method decide wavelet provide guarantee tuning contrast introduce appropriate introduce theoretic tuning cast learn well histogram approach limitation still good histogram self histogram width many empty region haar cast adapt pursuit omp transform easily extend dimensional update effectiveness method scenario provide variety result especially critical world database able reasonably census future difference histogram expressive individually tb standard fu follow f formally assign furthermore I intersection last cauchy schwarz record q attribute select use inequality selecting hence incur solution histogram recall regularize function hence fu along axis use follow q last follow property theorem corollary theorem axiom tune histogram theoretic feedback histogram small memory minimize study histogram highlight width histogram histogram histogram haar reduce histogram sparse multiple scalable multi scalability histogram feedback natural incorporate handle advantage art histogram modern database query typically histogram approach limitation first uniformly histogram might space construct scan sample histogram nontrivial update inaccurate histogram build limitation idea collect execution refine query together query expression feedback collect minimal execution plan characteristic histogram build histogram feedback arise year select histogram boundary fix exist theoretical dimension current self use feedback feedback step valuable information quality scalability another limitation update update feedback theoretic histogram informally goal histogram standard principle advantage leverage translate efficient inherently inconsistent strategy recent old next begin study dimensional width reasonably approach well practice analysis incur know histogram fail contribution recovery problem informally wavelet histogram cast vector provide adapt width generalization histogram extend haar wavelet transformation show query maintain presence extension incorporate database histogram include propose dataset prior term accuracy cardinality database outline histogram denote column algorithm generalize categorical domain record histogram consist estimate partition estimate use range notation represent query histogram bold matrix letter term b u kk represent range histogram histogram arbitrary histogram next width histogram optimal relax histogram search search record search induce correspond search convex observation avoid function lead relaxed specify mb ir mb distribution section instead empirical solution optimally standard optimal multiplicative problem l I width constant note show property fact generally nr hence histogram consider confirm intuition histogram query however factor lipschitz lipschitz function
value become constant evidence eqs total eqs live live arise return possibility evidence accumulate estimate live versus red series spike live dominate live volume live likelihood decay volume continue decrease represent nest want continue nest ratio illustrate live mean take extra effort uncertainty therefore interesting effort comparison live stop fractional sample present uncertainty histogram agree well bayesian volume sum yet quantitative whether agreement equivalence two yet apparent depend likelihood nest sampling moment currently foundation theoretic continue result problem establish estimator compute statistical maintain core nest draw inside current surface determining mean effort acknowledgment thank valuable discussion suggest error us national uncertainty nest valuable determining volume I evidence estimator new make evidence constraint valuable challenge explore compute evidence dimension complicate possibly multi modal shape carlo popular mcmc sample great range yield evidence intensive recently nest bayesian speak away volume layer evidence volume statistically focus evidence yield challenge nest surface nest pick new discuss surface likelihood include multi modal importance moderately hamiltonian evolve new keep differ pick call nest always step proceed computational nice statistical purpose evidence nest numerical choose applicable nest nest establish detail likelihood space normalise prior volume span allow range incorporate flat bayesian high monotonic function rewrite increase word examine step heart method idea straightforward difficult associated volume treat statistically slightly statistically small volume uniformly word distribution large prior define volume relevant region prior distribution begin live live nest live point estimate associate draw replace extract live point prior iterate sampling estimate technique nest mainly iii associated posterior much difference gain leibl see integral straightforward approximately live dominant statistical fairly uncertainty distribute evidence rigorously variance volume carry estimator uncertainty like small volume eq correlate independent density q volume sampling volume q eqs term since statistically volume obviously evidence convenient begin component include include one index average write product collect rewrite care distinguish appear twice product notice back moment partial evidence combine usual new estimator uncertainty current notation yield two expression interesting contribution enough expense make expression nest nest remain evidence live uniformly live point live live average live eq moment uncertainty account involve term evaluate put together live total give step live negligible end formalism assess since sufficient also flat prior coordinate align principal axis q prior cube origin box test examine gain last box dimension conclusion live estimator fractional analytic volume mean simulation predict average yield agree indicate uncertainty small analytic accurately volume yield
h idea recall selection complexity vs answer heterogeneity ex conjecture axiom comparison search target manner ask select similar object list present selection repeat point terminate strongly world network focus database popular case demand heterogeneous heterogeneous demand novel heterogeneous demand comparison mechanism bound intuitively demand constant capture topology interesting connection comparison classic search problem search user ask object list new object list typical unable query exploratory life cite human subject example pose automate method feature fashion human person select mind unable query mind able express identify among close web priori post present let search comparison amount determine object user possible model choice output goal object object query recently attention content embed edge minimize cost variety database unlikely frequency follow world demand early homogeneous novel problem context comparison bind mechanism also low content adaptive content meet heterogeneous demand intuitively appeal relate content important distribution target capture capture organize formally state search comparison adaptive devoted proof extensively early embed intrinsic net deterministic support metric embed satisfy certain sphere pack restrict connection constant mechanism restrict demand advantage object metric space rather require similarity object object rank performance cost call capture triangle play role demand work search heterogeneity restrict analysis additional distinction existence explicit question oracle datum phase incur contrast scheme adaptive drawback search lie incorporate history arguably first membership ask belong interactive handwritten access human moreover cost designing dataset prohibitive contrast behavior ability limit mechanism seminal condition embed explore heterogeneous demand investigate propose structure expand definition notation assume embed represent mapping feature object object abuse keep mind object partition equivalence respect empty close close formally imply basically aim capture human target picture similarity associate pair equally human stress place target analogy user mind present relationship character hold oracle order order order pair call demand vary different target eq demand play quantity search distribution introduce notion define define connection content access object accord find need oracle average occur frequently require identify entropy target bound search also topology capture radius probability distribution minimum nc n note contrary determine cube arrange uniformly cube arrange dimensional entropy l order demand entropy local give expect formally object embed although constrain directly distance access comparison section serve start greedy content propose object object process return something happen greedy content search process terminate analogy actually never reveal oracle stress priori present oracle content try response oracle response include object content one history object current content search allow policy case reveal locate history reach select number particular target policy randomize expectation comparison define demand select minimize note small consider assume direct property pair distinct object exist object lead object content goal start oracle oracle q greedy message repeat greedy terminate property eventually reach currently moreover close comparison object oracle neighbor close typically call determined proximity arrange rectangular grid gap distance every rectangular indeed locality satisfy edge property goal possible select greedy message edge random subset expect greedy heterogeneous target select accord demand embed edge demand greedy essentially obtain technical appearing problem restriction select second exactly edge go object connect joint edge eq minimize cost cm content sample draw suppose problem search start policy generate source see one vary across neighbor local content move object target message move effectively movement reader proof rigorous relationship optimize greedy np found section reduce version interestingly instance distance remain np hard even consider suggest problem solve motivated content restrict particular cost term entropy content whose object suppose define property
analyze set contribution learner sensitive supervise control sensitivity ensure learner bad assumption fail invoke feature assumption manifold density metric diffusion analyze degree unlabele control strength strong use sensitive kernel provide number adapt degree choice learner fail preliminary confirm alpha risk hold fail reference therein knowledge paper allow choose strength assumption outline section error used adapt xy p yx px py state sensitivity follow distance modification curve speed make correspond smooth respect smooth distance recall derivative exist degree condition real projection boundary boundary small condition inference additionally access access let bandwidth unlabele support boundary mx follow use sensitive characterize performance sensitive proof negligible unlabele enough sensitive mse joint condition supervise depend inf supervise estimator couple term integrate intuition lemma need take advantage keep familiar smoothness cube add series two number implying design specifically low easily essence boundary establish square sensitive estimator know sensitive parameter attain well outperform index parameter control strength semi hence take advantage unlabele semi extra preliminary alpha supervise fail report possible density sensitive relax besides elsewhere state repeat appendix indicator unlabele mx write measure curvature volume follow h give point combine component part grant fa dms sensitive distance q outside must proposition characterize plug behave assume px x x triangle mx x xy must similarly proposition mx sx mx mx x write apply follow condition dimensional euclidean plane argument proof q constant compact compact connected quantity let nd q lemma euclidean even result base vector affinity supremum measurable semi mi mi dirac also measure define product note measure denote logical arbitrary define clearly e easy satisfied easy see complete ef rf
sde dt x motion ergodic tx chain reject probability pointwise family hasting hilbert provide metropolis hasting markov k k main value sde equation ball probability maximize radius ball minimizer generate concentrate minimizer possess almost fine scale reflect operator brownian motion brownian quadratic draw invariant measure almost sure finite sure property reflect metropolis limit sure asymptotically paper limit ode globally quantitative rate algorithm motivate create noisy flow dimension random variable evaluation come complexity suffice walk metropolis rwm increment condition condition require analyze restriction tend independently indeed diffusion paper demonstrate langevin mala condition step prove closely introduce nevertheless analysis chain evolve theorem temperature increment gradient minima standard heuristic neighbourhood global minima stress asymptotic effect schedule hilbert simulate annealing present challenge distribution mutually singular brownian contain statement theorem proof quadratic result often term limit conclude section notational concern real function canonical inner trace eigenfunction assume orthonormal like rx x rr rd operator say orthonormal outer z operator denote norm l l satisfy satisfying notation sequence constant notation randomness present notation fact hilbert brownian hilbert may self adjoint class gaussian cx expansion x notice variable denote diagonal rx r equivalently r frequently functional arise exponent fix distinguished exponent change formula notation form orthonormal c continuous gaussian stationary increment real brownian define brownian motion equivalently brownian equation express functional connection bound act must full underlie decay eigenvalue domain define appropriately formalize element dual comprising may identify derivative weak localize assumption operator domain exponent satisfy derivative satisfy imply x functional behave sense make dx dy second remainder expansion clean exposition highlight central derivative localize assumption cost considerable technical argument present version formulation precise show weakly lemma reversible measure temperature hasting acceptance use mean acceptance position via formula reversible chain variable metropolis hasting x repeatedly proposal p give conclusion horizon piecewise linear markov article refer weak markov sequence value differential motion conceptual clarity consequence diffusion sx martingale drift chain read x k notice martingale array increment rescale drift limit sequence rescale brownian sequence piecewise converge weakly diffusion globally lipschitz x process weakly motion priori priori state rescale process tend generator next exploit arise limit particular convergence together explicit idea appear literature article context markov drift martingale decomposition define converge weakly differential dt equation x du brownian equal unique value continuous process du quantity approximate rescale piecewise k continuous argument weakly cauchy employ weakly du weakly converge weakly verify process go condition exist p condition priori last conclude proof prove sde read weakly mapping end proof order lemma suffice verify satisfied quantitative drift x linearly principle prove principle let rescaled define equation weak covariance priori metropolis approximate property see introduce satisfies quadratic wiener variation like almost sure piecewise solve limit ode whose globally stable quantity quantify equilibrium expansion every definition quantity n possess possesse possess quadratic speak condition terminology behave quadratic number vector possesse surely possess vx let prove suffice surely hold borel every readily temperature discretization propose dynamic show vx finite time temperature mcmc variation let hold converge vx accepted position x happen indeed use borel exist define gain insight behaviour ode metropolis continuous piecewise interpolation process probability trajectory non equation assumption let start converge already indicate showing go prove lemma proceed accept move bernoulli variable acceptance converge compare show number accept use trajectory quadratic behave accept ingredient low acceptance piecewise vx represent accept function differential consequently theorem suffice accept note u k u vx vx k go supremum content conclude minimization functional note h competitive jx ds ds second functional unique euler lagrange minimizer sign different exist display blue minimizer minimizer whose ability evaluate gaussian emphasize lagrange develop lagrange solution dot maximum minimize recall finding precisely demonstrate densely define regard viewpoint although course measure together identify see basic give bridge law brownian bridge x brownian bridge differentiable algorithm implement define support embed locally set develop paper relevant version temperature time discretization move move accept space start global minimizer functional dimensional set distribution lebesgue proportional show log scale finite htb rescale full behave dotted concern concern detail dominate study require ability parameter behave control systematically construct dimensional metropolis proposal take infinite rwm rwm multiplicative component identical rwm infinite matter apply approximate hilbert choice diffusion sde decrease target contrast produce diffusion limit function mathematically last may limit dimensional hilbert diffusion sde limit variety two define hx zero prove give control quantity x corollary jensen prove suppose indeed x hx x p x integer lead martingale x let cauchy schwarz independent source randomness consequently x follow suffice x x give describe x sx fx consequently fx fx x cauchy schwarz fx sd fx proceed give give ball radius xt sake completeness equation differential xt consequently xt xu du xt proof algorithm divide find finite markov lipschitz lower bind finite stay xt stochastic prove finite universal thus imply prove mcmc proposal x least ball rr bind give bind accepted b acceptance k bind k hold every sx since trajectory mcmc stay bound bernoulli I k behaviour exploit x k prove instead control lemma author thank constructive grateful david frank helpful discussion concern behaviour discussion theorem grant dms grateful financial department statistics university thank department conditions mathematics institute uk department university cv uk hilbert gaussian
semantic pixel sift benchmark well maximum markov exact appearance pixel paper experimental similarly move beyond demonstrate incorporate information scene take scene patch tag model strong scene label image global capability model among image share pixel label bank detector demonstrate consistent crf part object capture label elegant hierarchical classic benchmark principle scene inter relationship scene ps undirecte model part connect specifie specify human pose specify plane part configuration follow specify unary potential part specify form potential express interpretation part vision firstly star object video context pose appearance enhance robustness localization appearance part mechanism np tree graph interpretation normalize function energy permit principled estimation structure significant limitation general may miss precise require rather strictly present g five part pixel define basic individual patch specify semantic car label maximum labeling element nonparametric comprise jointly also semantic pixel level description appearance part shape plausible significant passing long relation among support structure scene plausible upper typical g five dataset specific unary binary individual plausible explicit form plausible spirit sparse appear similarly relaxed full need indeed mixture section underlie part potential specify definition unary term capture appearance shape term mm model appearance space map filter bank filter follow optimize filter appearance particular specify foreground background histogram model use foreground histogram specify appearance measure fitness foreground vice denominator measure fitness class histogram numerator denominator matlab right shape mm capability shape pixel member centroid coordinate height part function frame otherwise store call part induce membership finally shape shape road class show object modeling labeling object location gaussian centroid centroid location covariance mahalanobis term ht angle pairwise connect part coefficient map pixel support centroid sophisticated plausible capture structure part evaluate parameterized distance ij angle von unimodal angle relate von direction concentration parameter variance jointly relate configuration essentially cast unary potential part seek define configuration respective connect general finally part potential mle contain current intuitive explanation accord role sample foreground background vice versa although clarity inside outside converge initialization assign ratio appearance mh annealing process posteriori sample add pl restrictive schedule sample challenge result unless resolve principled schedule manual basic desire acceptance shorthand consider move schedule principled manner manually tune begin estimate assume ratio needs manually acceptance acceptance impossible experimental sift flow sift gold manual percentage label gold standard split note split sift flow drop th different global n ij color intend legend mm mm quantify gain part quantitative comparison mrf case assume method basic mrf basic element make good structure reader assumption know quantitative among display three allow analysis aspect basic overall clearly outperform independent mrf per labeling mle dataset improvement building strong global increase exhibit modeling bring visual character sift intra dominant sift global bring car person rich category sift mrf description insufficient accommodate nearby phenomenon case quantitative graph shift crf crf nearly class label crf extension hierarchy field great labeling directly interaction rather object paper label labeling accuracy numerous careful literature assume important comparative furthermore unary potential simple color yield power respective perform crf immediately evident compare class mention one type road cat body tree water road show pixel object error group tend stable appearance shape uninformative object informative claim key allow emphasize global shape look comparative accuracy explain model follow infer location appearance class allow rich potential seminal firstly segmentation incorporate significant secondly plausible partition seem seek mode whereas know part plausible carefully mixture configuration mechanism return spirit highly team player set pixel sparse make scene labeling label rich remain tie pixel label common experimental restrictive benchmark sift flow solve assume need extension method jump plausible full logic two potential global pixel dependency principled big respective class incorporate grateful provide grant mind nf finding author plus minus plus minus plus ex plus ex minus plus ex ex minus ex plus plus ex em plus minus ex scene community visual importance scene vision vision limit global problem semantic labeling propose approach pixel structure overcome limitation part directly pixel support permit toward pixel modeling report first
characterize employ fine covariate fashion difficulty cell small hard big dominate partitioning allow prove logarithmic bound static mind improve remove encounter bandit previous sense reveal somewhat surprising price pay partial information pay obtain setup good round one arm observe arm information fall family oppose minimum detailed difference similarity two type side online side discretization employ worth cumulative paper define weak compare literature reward propose certain policy bound depend type receive suggest arm notable stream work covariate convenient notational elimination setup oppose ucb policy adaptive attain regret policy well reward arm improvement ucb policy static armed bandit see pt assume optimal analysis traditionally denote indicating observation strictly measured drive random bound high operate round decision time subset arm decide eliminate hypothesis significantly remain arm keep repeat make obviously quantitie policy arm prescribe horizon horizon perfect horizon horizon choose always lead horizon iy note exploration arm exactly time reward denote average arm convention essentially high deviation arm bandit pseudo presentation also complete time regret policy armed bandit horizon random random reward se exhibit otherwise contribute introduce quantity go accumulate suboptimal arm quantity accumulate accumulate arm induce eliminate happen choice define eliminate round ki arm contribute decompose union k k complement event side decompose follow side c hoeffding every yield right eliminate round arm I k bound I xx put display yield regret equation follow slight variation allow run respectively eliminate arm see page unclear elimination matter arm side dependent show unknown spirit small aforementioned ucb logarithmic previously gap gap nevertheless recover near bound much strong gap since hold reward necessarily independently none superiority arm expectation case bound dedicate nonparametric bandit denote yield throughout lebesgue respect reward expectation take place sequentially machine arm observation complete oracle arm large break pointwise oracle measure arbitrarily subsection describe natural regularity condition possible achieve sublinear policy nonparametric impose smoothness condition pointwise function every consequence smoothness old control weak closely employ terminology setup naive policy bandit machine describe partition initialize arm partition collection define decompose independent denote bin policy accord pseudo bt nm nb tn expect policy difficult parameter kk study suboptimal bound ucb policy running policy thus theorem indexing constant measurable integer appear margin proof collection nm j associate yield large smoothness bin bin indice eq well behave otherwise ill bin h old inclusion hold fact contribute w number weakly bin indeed yield behave part step bin condition sum define ix j f ji x ix ji ix c f f j j together j r nm ji j j generality gap order way ki ji ji j low generality indexing ji f j satisfy smoothness parameter together fact c j inequality yield j hold prescribe point specify bin g reader potentially factor appear upper generally expression light significant limitation surface arm bin regret least bin bound away arm could reduce overall bin logarithmic definition bin definition operate refined root tree child node bins form partition subtree construct partition leaves sense construct time set bin initialize initial covariate bin live replace child bb round intuition policy run birth arm average end round high smoothness eliminate satisfy suboptimal bin keep arm uniformly tn policy consider expect time q minimax sense low bound imply improve except round multiplicative say dominate imply exist bound arm operate static discard covariate theorem recover regret imply keep track constant might differ section newly create new initialize reward obtain successive remain round integer define parent q recursively bin incur define leave active policy arm remain leave bin adapt convention go treat incur live differently quantity decompose regret accumulate live accumulate live rely event b ix point great result hold hand also remain probability define occur play bin control note b interval treat arm event arm probability put together nb display together margin right side dominate leave bin proceed pt proof dominate lemma difference sequence moreover denote average yield integer martingale q argument p grant nsf grant dms dms armed bandit noisy reward realization depend observable armed bandit setting dynamically reward describe reward capture maximize introduce adaptively elimination suitably localize static construct partition use bandit optimal regret seminal extensively field computer science economics multi armed population arm point receive reward property devise reward population high regret population sample homogeneous identically policy regret small regret see
apparent two edge point point ji em em em em identical point pointing pt recursively remove point line obtain em em em em point line since remove point line matter remove pointing form em notice refer path notice since point notion minor inconsistent look path mix hold path introduce use base definition path would remove obtain point removal must turn arc inductive either adjacent line adjacent remove conclusion else adjacent connect path subset path separate induce replace criterion extension separation introduce separation graph line connect graph compositional compositional axiom connect connect contradiction connect short connect contradiction connect give contradiction close connect contradiction connect shortest contradict addition inner contradict connect suppose connect short symmetry enough non node contradict short contradiction connect type call path suppose contradict focus pair nod composition separation previously chain four markov graph discuss classify ii iii dual amp property iv property consist entirely arc multivariate induce separation typically interpretation graph loop however modification mc dag conditioning graph independence graph generate dag deal graph definition follow arc case em em I preserve say subgraphs illustrate straight simple cyclic establish graph identical sequence remove point full previous starting unless line another preserve cycle point arc still therefore reverse induction identical clearly illustrate markov equivalent connect path connect path give preserve edge connect path conversely suppose connect non instead thing remain preserve path point member preserve argument direction preserve edge instead connect ensure edge become absence essential induce induce independence model cox paper without bi edge direction preserve path internal let node take connect intersect connect follow let direction simplify graph acyclic mixed check primitive induce maximal identical pair adjacent reader establish maximal graph contradiction primitive induce node short primitive induce trivially node unless direction preserve turn identical contradict primitive induce markov model far also compositional result undirected graph independence model property separation lemma theorem pairwise maximal establish lemma compositional disjoint subset compositional six compositional fact six property marginal independence notation give sufficient condition connect graph graph suppose connect connect mutually exclusive b pointing pointing edge obvious connect deal two pointing connect connect connect node pointing path preserve kl connect pointing edge set compositional r compositional global markov pairwise compositional axiom composition far observe establish moreover property follow result induction trivial first I subgraph independence model c compositional pairwise adjacent property g l l mi cg connect node ij property inductive consider connect path contradict point symmetry pointing preserve path imply connect contradiction node lemma contradiction conclude symmetry compositional composition union obtain get base part inductive ci first cl l process lemma contradiction connect cp lp lp cyclic adjacent shortest connect contradiction path case symmetry c independence get follow independence composition axiom pairwise markov r markov six compositional axiom mention six axiom show composition compositional axiom intersection axiom imply axiom imply axiom imply compositional axiom imply axiom pairwise markov imply intersection violate undirected five axiom equivalence global statement associate compositional semi axiom markov I proof intersection conclude state independence model define compositional property w global property grateful comment version corollary theory graph mix separation compositional graph case acyclic graph graph pairwise independence compositional widely recent relation node random range simple mixed similarity motivate attempt mixed graph criterion cover graphical independence independence form exception mixed ensure reasoning indeed behave motivation define mc summary graph relation conditioning acyclic dag study graph graph obtain mc dag summary case pairwise latter edge compositional model ensure independence represent miss support intuition concept pairwise context independently several abstract satisfying axiom graph use separation condition equivalent prove compositional mention conjecture theory compositional mix mixed associate separation mix compositional section introduce class graph concept concept demand independence markov property compositional graph section notation triple consist relation edge distinct write say refer order every hence em edge neither loop edge induced edge edge walk repeat graph uniquely throughout node sequence describe path edge apparent belong say path j jk j general path path path may intersect path
informative dependence parameter orientation consider et al subject guarantee two inner fix use affinity closeness subspace maximal root perfectly factor ssc recover point random ambient fully detection ambient differently ssc achieve previously understand depend obeys assume point subspace exponentially result hold property bind course explain general shall discuss restrictive well difficult dimension regime concern many reflect comprehensive effect slightly version detection probability fraction hand probability point ratio early dimension grow linearly ambient note statement factor relatively assume unit apply sample detection ssc decompose solve expansion threshold make belong subspace dimension short expect make rigorous show outli iff remain outlier value together n ne nc numerical subspace uniformly de ne n de subspace outli detection scheme reliably number root ambient orientation point similar hold shall succeed notation subspace exclude hull introduce concept q point euclidean show dual direction dual arrange correspond denote radius euclidean incoherence subspace point property local solution column subspace incoherence subspace affinity incoherence imply cluster become spread distribution point direction difficult skewed toward blue special point huge subspace situation would successful sparse recognize set lasso subspace orient spread ssc geometric perspective concrete subspace model definition affinity subspace angle k orthogonality alternatively column cosine angle large affinity point subspace subject hence ignore square subspace assume simplicity subspace perfect occur ne notion affinity match sure close define affinity cluster less allow subspace intersect ssc still provably cluster discuss subspace small number subspace inherently reflect intuitively small increase probability modify hold probability ne similar manner presence claim make suppose outli threshold correctly constant semi detect q point outli threshold multiply ratio dimension increase get hold probability small need able inherently small prove believe conjecture support conjecture couple theoretical advance definition dim direct disjoint geodesic subspace dimension property long independent formally q detection singular rank sub appearance principal angle hand particularly paper introduce deterministic restrictive obeys r checking np slightly tractable assume subspace since column unit side strictly look entirely clear achievable exclude ease side subspace intersection dimension subspace intersect long fraction angle explain ssc disjoint simplicity subspace subspace see impose fully cn would put restriction comparison subspace ambient improvement come insight apparent ssc succeed inner product another zhang study effectiveness recover sort minimization simultaneous subspace minimize convex semi zhang subspace dimensional hand typical analogous outlier number combine therefore say hold general result zhang perfect recovery even noisy manuscript address outlier suggest outlier exceed possible gap correct subspace follow similar ssc il feature detection belong far choose neighbor term equal value cluster build ssc clustering take correctly otherwise eigenvalue ii check detection property hold detection vanish wish intersect end generate dimension uniformly among subspace denote inside another uniformly set feature detection uniformly three instance property ssc vanishing intersection show success ssc subspace upon point subspace hard affinity per decrease trade great experiment every point express point subspace basis angle linearly affinity affinity evaluate ssc criterion value affinity sufficiently large figure display proxy interesting cluster ssc generality achieve perfect subspace increase subspace random subspace different subspace normalize laplacian evident property subspace case gap always effect spectral ambient dimension point per sphere noise normalize noisy I value level correspond laplacian evident figure regime noiseless correspond evident decrease present subspace correctly advantage ssc ability broad circumstance subspace eigen gap discuss quickly review subspace start rank affinity interestingly also affinity perform perfect affinity approximately block diagonal empirically presence assumption violate affinity put subspace none violate ssc approach eigen broad circumstance demonstrate sample dimension ambient affinity ssc gap plot ssc subspace robust possibly add case methodology noiseless setup noiseless dd l turn subspace choose random outlier equal point plot correspond see gap appear solution correspond outli argue detection small work detect outli correctly consider able outli detection prove threshold perfect detection worth concentrate prove would ratio different value heavily concept exposition sphere norm eq cauchy schwarz deal appear page concentration follow lemma modification geometric face polytope face positive banach banach eq body concern volume compact I n n primal dual hold primal primal equal first variant support c identity solution identity optimal primal dual feasible subspace set check satisfied definition small ball consequence symmetric notice condition thereby conclude incoherence state choose assume numerical parameter modification subspace union establish independently furthermore dual distribute unit uniquely uniformly random justify orthogonal dual point problem know random sphere transformation prove apply union sphere unit positive eq assume begin map upper plug low step furthermore direction sample sphere know area spherical union pair theorem relate norm body variant x conclude relationship bb polytope theorem step use equality inverse one another eq apply identity establish uniformly random derive unit sphere column well sphere radius family norm shall plug volume sphere proof mean lemma eq pt lemma bind expect combine union part b part proof es thank discussion paper plan grateful comment stanford fellowship award fa grant collection assume near union many dimension develop name provably ssc ambient prove ssc data intersect develop ssc succeed corrupted outlier insight numerical demonstrate effectiveness fundamental step analysis reduction approximate low pca collection plane furthermore know approximate list unsupervise build representation input make predict input unsupervised union manifold furthermore well approximated handwritten handwritten character recognition simple transformation rotation shift character reasonable model insensitive change characterize invariance transformation digit approximate et al subspace problem visual corner million web visual development appearance motion segmentation multiple move approximately need move object hence subspace computer vision include image face propose researcher work comprehensive review reader reference therein disease kind specific factor test test construct row factor level contain cluster group suffer cause disease subspace together factor subspace cluster hope clear lie dimensional subspace class may surprising study science begin framework develop
nonnegative factorization inner polynomial separability meet segmentation additionally separability condition nmf unique separability hence work sense separability fairly practical context learn various retrieval david nonnegative r rest organize exact sf section prove sf nonnegative throughout factorization inner factorization must full first impose context scale column factorization interpret column introduction column topic nonnegative explanation document express column lemma hull matrix disjoint instead combination entirely disjoint nmf task retrieval even set extract useful would whose rank solve maximum rational semi value variable al rational approximation use fact column let matrix factorization iff basis factorization conversely factorization let basis row correspond matrix row let check condition become make polynomial full factorization factor matrix case algebraic reduction complicate goal cast nonnegative polynomially obstacle entry minimal span function transformation span transformation satisfy choice possible partition algorithm question constraint inequality constraint subsection minimal structural informally maximal corresponding column element order iff nonnegative matrix dimension minimal minimal respect respect nonnegative dimension proper choose basis use nonnegative repeat condition satisfie transformation recover column recover subset eq equality identical replace respectively remain technical polynomially many choice efficiently even two transformation column order definition minimal choice polynomially demonstrate restrict partitioning establish regard partition virtue arise chain partition set collection hyperplane chain row w iv I contain I j tw imply say important partition reduce partition hyperplane specify specify partition generate domain hyperplane polynomial time constant vc dimension hyperplane imply distinct partition fact result give upper bound efficiently structure checking result intend partition partition column disjoint separable partition hyperplane lie subspace give improvement remove condition algorithm encode hyperplane choice slight hyperplane separation define separation hyperplane mapping hyperplane hyperplane hyperplane independent extension hyperplane already contain perturb non eventually hyperplane contain remain agree define hyperplane nested apply obtain hyperplane correspond implicitly column contain large recursive add column contain choice initialize active run hyperplane partition hyperplane correctness algorithm follow partition find semi theorem nonnegative factorization cast nonnegative existence question algebraic nonnegative compute rational l nm r rational nm lemma partition partition decomposition row transformation span row span respectively long nonnegative transformation similarly recover nonnegative lastly factorization write constraint choice formulate constraint define matrix inner factorization call return al elimination bss bss bit find rational approximation quantify determine give polynomial exist set inequality particular quantify give polynomial also give number bit solution special time binary search find approximation note result sf structural special positive nonnegative rank give factorization solve time hypothesis hardness reduction unable reduction range goal e recently prove exponential fact exactly state definition distinct bit solve reduce intermediate intermediate point simplex point simplex reduction intermediate simplex versa preserve value immediate result e universe intermediate figure intermediate simplex reduction represent recall input inside dot except simplex contain triangle figure instance angle determine edge length denote coordinate place intermediate simplex get triangle thin triangle intersection triangle intersection triangle vertice triangle segment respect rotation symmetry situation illustrate triangle later specify line boundary two line extension thin difference intersection segment segment high line also inside since always distance large inside intersection edge intermediate simplex triangle possible hull angle sum symmetry shall case interested contain direction angle move intersection view vertex coincide case show figure observe take generate generality every triangle particular triangle outside get know intersection hyperplane intersection check vertex figure triangle contain also problem define construct number use dimension sum ensure box conv x origin simplex plane choose close constraint enforce large recall ab sure value constraint e gradually relax extra proof observe almost effect dimensional box simplex specify intermediate let possible still point reduction sum establish completeness reduction completeness straight triangle choose include choose equal point line contain line hull clearly contain next point convex hull triangle correspond three coordinate choose recover coordinate lastly hull point vertex combination partition triple set let contain contain hull two dimensional cone like plane plane act intersection origin affine know convex triangle know number abuse use section make number ab ce strict sake contradiction since place weight equal want use three dimension contribution convex weight imply enough simplex sum lemma intermediate must contain solution choose know know lemma one k value give evidence nonnegative consider allow give provably non trivial condition widely cite identify condition factorization database give assume one separability right usually separability factorization separable entry intuitive context column specify occur nonnegative coordinate thousand topic appear happen simplicity still normalize preserve factorization writing factorization norm eq unique nonzero separability nonnegative hull nonnegative separable inner dimension row convex row modify non coordinate coordinate nonnegative column let construction separable say column zero everywhere else operation end condition condition end output apply lemma loss separability appear row plain say row copy hull remain row iff equal vector index row convex hull conversely hull row separability hence convex hull determine exactly thus row nonnegative separable equal inner separability assume input add separable alternatively notice separability satisfy column column row entry satisfy condition unknown instead small hull remain unit condition separability separability generative model column distribution pick identifie seem suitably generic pick column vector tend satisfy robust property separable robust polynomial time nonnegative factorization row separability index whose nonzero entry column claim since convex row find robust upon ignore row convex hull linear claim robust robust claim show hull leave row row least close hull conversely least robust robust row cluster row w rather approximate unlike factorization assumption nonnegative rank well truncated decomposition e solve approximation note frequently outer product normalization main equality intuition behind decompose responsible little ensure nonnegative remove onto less singular vector singular triangle norm v ta w enumeration programming enumeration vector lie w find appear svd choice unit suitable multiplicative solution convex separate column denote close enough v r substituting claim argument still w solution value claim contribute second bind proof complete program candidate right find square w w lemma term bound choose nx rx xy randomize may ensure svd let inverse near q row column multiplication entry enumeration entry cause carefully deal keep row approximate candidate namely entry orthonormal schmidt orthonormal solve
apply edge setup path level efficient integration focus undirecte contain loop edge unweighted triple value generalize direct unweighted belong measure make notation side path naturally direct set eq restrict graph conclusion range topological computation short matrix choice topological measure quantity derive consideration length path vertex unweighted neighbor g function topological interest graph unweighted graph topological express function network density unweighted realization low take element follow equation write follow topological computation path integration equivalent computationally expensive update adjacency strategy need invoke algorithm level rank entry weight path successively add topological short return topological three stage firstly compute rank interest true tie randomization extract rank run rank order eq suffice pair recursively collect short finally integrated metric difficulty algorithm proceed addition edge exist short search around vertex conduct check neighbor secondly check short south black south anchor south south background rectangle every execute cell execute cell execute cell row style style row column style node anchor south matrix south west west anchor south anchor south black anchor south south anchor south anchor south text node style font execute execute empty style row column column style row column row style row row row row style anchor south anchor anchor west anchor south anchor south circle anchor circle anchor south black circle south edge time edge graph component conduct respect accordingly degree neighbor represent red panel respect blue first degree corresponding modification path inclusion short storing efficiency efficiency case code well efficiency search efficiency would coded short path thus combination interest code recursive shortest graph integrate topological metric respect generalize direct order pair vertex check one topological availability describe paper become fellowship uk centre health foundation trust college foundation valuable pt remark example recursive path application integration institute centre sciences research centre health south college institute send centre centre college se uk email send uk theory generally constitute weight topological density proportion graph topological short interest integration usually density short recursive short edge updating replace procedure first search code adjacency iterative among seminal systematic calculation various topological characteristic biology several
look give detailed online protocol datum base input forecaster time round reveal input forecaster choose linear environment forecaster dy protocol forecaster hold choose adversarial environment later regression random bold letter u u q endowed borel denote kullback integer equal thresholded natural regret forecaster advance trace gram analysis game forecaster bind study variant variant exponential weighting introduce stochastic tailor individual heavy heavy tailed point early heavy tailed sampling approximately one quite temperature scale associate tp satisfy jx case dt follow pac loss constant consequence theorem loss aforementione recall exp comment remark get inequality restrict infimum note therefore truncation prediction square previous define natural pac perspective crucially particular shape thus outline change inequality technical tool convenience reader last take translate namely symmetry term rewrite inequality corollary yield inequality approximate combination sparse appendix regret actually term small former empirical bound proposition assume forecaster gram matrix requirement prove sparsity bind initialization get purpose differ threshold temperature allow time forecaster idea forecast past regression set unbounded square ingredient perform truncation automatic previous several corollary derive access quantity forecaster batch analyse tuning tuning scale proof corollary forecaster previous proposition pac satisfy argument change spirit little comment replace prediction zero hand convex duality kullback cf recall sum replace let note concave next dedicated upper bound particular definition fact loss exp jensen take divide bound happen b concavity version precisely set property ty jensen concave elementary calculation summing put together pac inequality yield complement distinguish distinguish case x b ty b convexity line except instead restrict infimum cf get adaptation possible long gram forecaster trick repeatedly algorithm regret multiplicative round regime instance call round date begin regime particular past tune require begin game fully automatic possible viewpoint automatic extra viewpoint without repeatedly like vary might question price involve pac appear since change proof visit tuned instance regime summing result see upper uniformly forecaster proof stochastic design design prove risk logarithmic variance question sequel online dictionary regression forecaster copy whose estimate regression set pair I I surely measurable satisfies oracle risk begin game treat use estimator sequentially tune therefore depend knowledge tune bound extended technical base forecast ty enable base main deterministic jensen corollary jx satisfy expectation apply jensen proof amplitude later question next oracle lemma l almost subgaussian comment stress term avoid least achieve quantity respectively correspond classical design type aggregation rate automatic online bound unknown adapt sequentially bind risk bound batch ty tail recover still question heavy tailed output tune automatic purpose third variation note several assumption fx type assumption subgaussian subgaussian conditionally subgaussian factor avoid together q key particular subsection noise subgaussian upper slightly tight corollary twice oracle eq x get conditioning conclude bind oracle inequality notation straightforward jensen seem since case choose reasonable make free another simple analyse tuning assumption r q ball whole advance ask drive answer positive still quantity risk correspond bound infimum risk however contrary thank drive open deal prior inverse practice author ask answer batch forecaster almost small mean fx like consequence sequence tx tx ty tf tx deterministic theorem main setting jensen appendix design almost subgaussian variance assumption hold hold assumption remove idea appendix algorithm require factor la european provide state corollary need comment begin regime past past ambiguity regime convenience bind empirical gram fact apply period infimum get xy yx sum substitute inequality note conclude view check extended remark elementary holds non corollary pair definition figure remain expectation therefore concavity infimum side note right conclude definition constant individual risk appear front equal would previous vanish assumption appear avoid multiplicative factor prediction interval change see sf corollary loose fulfil assumption replace without elementary bind assumption thus sketch main argument sequel expand square tf might comment design case divide exactly state distinct convexity square definition jensen take expectation conclude exactly instead several inequality kullback leibler notation measurable set expectation bayesian reproduce case translate probability linearity integral sum side bound indeed expand equality prove u concave q conclude next corollary theorem respectively center real constant eq random constant subgaussian q entail least remain precise get line rearrange conclude proof definition associate show convex xx base type maximal xx last jensen inequality convex exponential moment conclude jensen moment sup sup paris france ambient round sparsity regret weight drive truncation version algorithm solve question open bound adaptive gaussian sparsity individual sparsity extensively stochastic decade dna netflix amazon hyperspectral high dimensional cross country message impossible statistically feasible situation focus many practical work decade bound express oracle fact guarantee consequence statistical possibility scenario deterministic namely regression sequence newly deterministic prove online sophisticated automatic preliminary parameter apply deterministic nature thank imply unknown noise logarithmic open introduce main motivate online viewpoint main respect statistical literature machine arbitrary deterministic forecaster output assess goal almost forecaster sequence form small sublinear sake omit set version ridge forecaster
limit statistic node kde computation begin reference compute far moment reference use node moment node compute dm leaf r rr r show include rp r determine query rr dr core dual routine computing kde g l r tree recursion structure kde query current tolerance reference approximation continue fine terminate consider avoid exhaustive leaf achieve accuracy computational cost query change reference regard locally give query entire subtree reflect change immediate child query fine g g rv computation query pair call exactly refine hence query pass node reference value sum note add onto query sum refine bind g finally post level pass child scalar point post routine g r low upper maintain properly call maintain properly main call point q incorporate passed contribution un visit call query consideration change two correctness prove call leaf subproblem hypothesis call maintain low child r argument query incorporate pass change correctly among leaf call maintain low upper hypothesis contain bound query account either exhaustive exhaustive incorporate node belong recursion global simplicity limit available exhaustive denote centroid leaf node whose contribution centroid set sum r b r ir snapshot low sum subsequently triangle four available method p naive naive c x x naive naive e e position color dimensional dataset angle dataset last scalability kde guarantee criterion table symbol tolerance common validation conceptually visualize store conceptually dimension iterate scalar give scalar bottom storing ensures meet time global include take recent automatic tuning tree method automatically parameter fast base fast combine kde gauss gauss gauss expansion computational implement implement inherently query moment similarly node moment allocate array imply mapping digit number position array base p position linear index dd index position basically convert representation multi expansion basically form single reference x p pm p basically compute multi see implement r c p implement far operator doubly accumulate implement r compute h univariate function ad e k df evaluate expansion implement structure outer loop iterate far query dot far computed figure evaluate far reference term p pf w r c q implement translation doubly loop doubly translate accumulate moment term translate local implement j pf h c p implement accumulation basic doubly nested loop accumulate inner position implement equation r l p implement translation reader local accumulation doubly moment apply implement j p evaluate outer loop moment among reference local accumulate query cp p pz q theorem thm kde achieve optimality computation kernel summation algorithm transform derive additional algorithm utilize hierarchical demonstrate truly gauss second within dual compute user kernel density bind wide procedure density kde point choice spherical density th th reference assumption true mild estimate kde need find initially dataset validation bandwidth query kullback subscript point maximize bandwidth sense cross square yield score gaussian kernel base force make reference practitioner apply commonly evaluate sum efficiently kernel expensive sum many cross kernel algorithm develop expense precision consider criterion measure error absolute value I error bind bound hard initially quantity many focused bound relative criterion achieve specify tolerance build upon relative barrier evaluate grid flat briefly strength gauss method rigorous bind bandwidth bind widely contexts sum grow dimensionality cause bottleneck center another exponential box empty gauss similar utilize flat advance point group proximity whose propose expansion translation expansion algorithm multiple offer user fully absolute still inefficient section b neighboring sum use discusse kde extend handle structure specify raw compute coordinate dimension discuss type rule recommend neighbor implementation recommend I h reduce fouri dataset zero grid count matrix two key ingredient fourier transform convolution count matrix corner grid point inside c c l k l calculation point boundary interpolation grid provide moreover quantify incur kde discrete algorithmic summation algorithm tree consider query point recursive structure reference variant centroid fast kernel summation cross optimal bandwidth optimally evaluate demonstrate dual bandwidth large improvement dual first summation gauss transform derive extend demonstrate truly hierarchical gauss tree gauss transform integrate expansion approximation compute algorithm world dataset bandwidth selection guarantee relative offer fast range cross validation build tree fast transform add thorough comparison general computing sum extension dual provide kernel query computation point construct index component enable tree structure computation single flat clustering collection collection tree hierarchical location recursive r b b x x procedure hyper n u center split two coordinate continue point threshold compute bounding involve kde computing summation formalize query tree traversal demonstrate reference tree compare reference box partition record bound highlight recursion upper distance reference tree r compute summarize contribution node query query query traversal thereby exhaustive computation univariate crucial mechanism gauss integer time continuously field expansion query four term r c involve array query array expansion kernel separate reference centroid far field region part sum approximate moment node term dp sum query centroid reference query point location expansion sum representative query denote disjoint function remain reference point location moment accumulate store within query coefficient generate third q involve take product array coefficient dimension far field window function whose reference main summation pre compute evaluate point dot vector require far operation compute show local accumulation operation moment see centroid translation call local translation operator far lemma truncate far centroid term fr p taylor centroid c proof must moment accumulate c p show operation bound choose order section wrong correction evaluate truncate evaluate truncate truncate error reference evaluate far reference reference r r n c expand expansion function x intuitively evaluate field reference form bandwidth centroid node inside hypercube truncate local centroid query accounting contribution reference l taylor r respectively c achieve multivariate bind lastly evaluate truncate form truncate require node local expansion convert already centroid eq expansion centroid fr l p r n q p geometric propose use node constraint large possibly derive incorrect derivation determine need far answer bind nevertheless lemma question maximum term achieve within tree partition show field expansion error allocate far field reference algorithm ratio length twice absolute error side determine expansion form reference query node twice determine error evaluate determine form reference p necessary
obtain proof bind set note write derive simply impose prove c ignore fail begin ir follow l low c identically choose variable conditional lyapunov cumulative big explain calculation calculation point variable obvious subtree uniformly neighbor local dimensional tt child eigenvalue expression variable inside expectation whenever recurrence hard child expression denote measure restrict try since addition expand power decay interesting condition hence consequently ss hard tell ss nontrivial notice solution expression expand expand connection remove write regular expand define write law second introduce mind recall expand expression inside take inside simplify finally first notice eigenvalue immediate c family already discuss explicitly incoherence ss proof class succeed reconstruct show fail reconstruct dimensional everything use expansion power one expansion priori first notice depend fix independent zero eigenvalue reconstruct enough note differ require behave numerical function expression along understand behavior concentration incoherence turn tell incoherence violate enough question success minima due symmetry pl path constant high axis graph solution yield reconstruction exhibit make curve path eps plot separate include show tend value high finite plot positive region curve reconstruction also unless fail vanish identically scale allow converge require make fail scenario graph surprisingly graph correlation big neighboring notice fact consequently simple use prove large neighboring great non neighboring node algorithm operation range prove g node great expression calculation concavity easy furthermore condition assume case get inside plus pt lemma definition conjecture fact structure ise field concrete low often fail field range correlation phenomenon appear apart mechanic ise seminal find numerous computer distribution dependent interest introduce stress follow study structural qualitative encode identifiable unbounded resource class operation sample general emphasize definition alternative reconstruct correct consider definition result qualitatively spectrum general understand trade devote tradeoff structural ise fact independent instead conditionally dependent particularly large connected discussing get toy illustration learning study figure family vertex interact directly interact numerous conditionally since fix choose ise distinguish unbounded covariance statistic infer assume graph distinguish covariance match x ix remain involve subset number spirit achievable even marginal confirm need ise distribution formulae word although let match eqs correction ambiguity arise weak indirect graph phenomenon correlation connection reason edge correlation graph bound degree system distinguish graph former vertex trivial configuration counter away regularize regularize regression rest paper bounding pseudo method defer general pattern fail toy demonstrate trade via maximum interaction strong answer gibb predict behavior uniqueness gibbs far roughly independent vice versa family far apart dependent polynomial exist arguably hard uniqueness phase dependent compare analyze mention provably fail raise question structural provable dependent question overview finally paper hard learn logistic fail regular specific family indeed mostly analytically tractable try likelihood ascent gradient log likelihood require ise carlo reason first output approximation zero overcome suitably regularize never value use mcmc fundamental limitation yield polynomial sampling apply dependent case develop computational provably maximum field degree present limitation author exploit view consist regression ise model vertex appropriate problem variable logistic structure extend early notice short present neural system two explore challenge forward average structural decay generalize analysis adaptive family mathematical strength simple thresholding thresholding correlation compute choice numerical bounded degree fail result confirm idea complexity correlation apart decay exponentially learn happen family range characterize advanced behavior algorithm strength challenging degree exploit conditional independence encode consider fix whose want reconstruct rx neighborhood assume produce condition remain value produce conditional subset vertex motivate select denote minima maxima contribute follow consequence analysis omit degree exist constant first implie reconstruct impractical restrict decrease rapidly graph distance mention assume small enough constant exist take consist likelihood fluctuation select often regularize likelihood v r x p j regularize logistic calculate numerical degree exponentially hold success write converse make say probability neighbor isolate sequence successful particularly encodes let vanish order asymptotic converse whose deferred section degree regression fail regularize logistic regression indeed fact describe fail enough fail reasonable dimensional fail high reasonable call necessary successfully similar lasso necessary model ise model intuition behind quite solution quadratic center add regularization analogous lasso plus error control dominate contribution lead incoherence expectation ising check graph contribution consist gap mechanic ise grid weak ising explore relevance result carry use logistic among algorithm balance ise gibbs change temperature take day processor tree weakly couple regime case p report success subgraph obtain independently probability average scale satisfactory clearly illustrate phenomenon page despite threshold irrespective indistinguishable plot versus figure random probability sufficient reconstruct analytically compare proceed convenient notation make form whose lie submatrix index whose try reconstruct hereafter shorthand subgradient e variable distribute accord convenient hessian limit denote matrix argument clear quantity evaluate clear subscript introduce copy recover exactly throughout expectation connect least hoeffding ji ei possibility impose right hoeffde expectation derive bind avoid walk self avoid walk imply theorem prove degree vertex add extra node converge exist fails obviously choose use leaf node show complete failure first establish small ss ss min min c reasonable sense unbounded choose sequence star node edge central increase star condition ss violate rely convergence prove regular degree exist positive exist proof lemma enough c q v generality eq exposition even differ exposition incoherence bound eq show stationarity r n ss notice q recall absolute value index relation know hoeffding yield conclude ss min min n c ss theorem expression ss I subgradient take ss iw straightforward computation bound bound detail result mention lemma regular root associate ise leave unique prove expectation ball whose neighborhood rt pp node expectation measure expectation g rt consist
real self configuration still difficult get big solution prove rich enough especially stream environment science cm cm configuration http www gm de contribution self machine train data experience come exist achieve tuning detection pattern self tuning advance construction boolean forest increase current barrier discuss systematic partly solution potentially new human adapt rapidly situation environment generalization environment work case game mining game ideal self system world complex like mainly formulate immediate goal playing well learn rather program expert opponent discover reaction rl solve feature remarkable td show alternative evolution follow many contribution start case rl games reinforcement toy wrong one interesting tuple recently game remarkable g uci special robust forest larger careful detect aim numerous especially area stream mine recent development parallel feature bring optimal use feature systematic model development rely test confidence finding high input look configuration self w teacher look n tuple sec configuration mining dm discuss desirable flexible dm feature dm finding research question remarkable example game learn behaviour play master td reinforcement complete solution dramatically simple piece piece winner pick right detect leave opponent formulation trivial successful td learn raw human index white black hypothesis tuple similar trick relate dimensional indexing input situation indexing perceptron system architecture original perceptron unit form random input strength problem perceptron tuple perceptron signal concept vast amount capability near future machine considerable progress decade forest robust support number build implicitly knowledge complicated usually build c tune model preprocesse able want system mining mining streaming underlie configuration calibration daily routine adjustment configuration although well quickly get become curse contain boolean integer modeling often consume adjusting run budget red line training among competition clearly high tune column high winner rf build help krige optimization run es bfgs dm result novel ultimately pattern find numerous infinitely way input fourier transform transform gp genetic pca assume linearity assume linearity dim form random one belong general principle good could successfully htbp gauss alone rf evidence work require module
ask person world nuclear anomaly change detection frequency alarm anomaly method point detection change detection anomaly frequency detect job job keyword early thus algorithm four collect twitter wide spread job quick propagation news conference response promise detection approach conduct framework planning scale handle stream content boost microsoft core topic interest rapid text video dynamically new anomaly sequentially aggregate anomaly relationship network demonstrate twitter mention approach detect ill anomaly discount facebook daily life since text video datum text contain may rarely may friend time receive every mention sense mention like number interested user topic like discuss friend conventional approach concern ambiguity cause may preprocesse apply information form unique little available social first nothing many friend friend twitter friend topic something new show link user user occur anomaly use reflect behaviour anomaly obtain apply technique detect topic effectiveness four set twitter good show propose link anomaly topic frequency mention detection tracking extensively task one topic none topic text mining factorial research concern seminal firing inspire change momentum topic social content content utilize citation citation analyze lie focus social content document flow service user mention assign anomaly post feed analysis stream contains mention formally mention training compute predictive assume beta integrate beta integral numerator beta predictive far follow total number derive predictive likelihood set handle appear anomalous instead chinese crp crp addition keep probability accordingly compute anomaly post past trend anomaly obtain discretization time include aggregated anomaly point statistical time series monitor new datum employ layer length autoregressive ar scoring length know optimal code employ point detection procedure outline follow anomaly aggregate sequentially learn smoothed sequentially appendix change density alarm score exceed threshold histogram probability exceed positive histogram n th update histogram threshold alarm collect service collaborative service belong detect recognize people set job organize list list participant change occurrence frequency topic manually topic describe sparsity seem bad method experiment detection anomaly anomaly fire rate alarm drawback relate must advance always think detect datum participant job person student anomaly detection frequency
cpu program know consequence requirement gpu device computational capacity theoretically certain usually apply perform realistic subset compute statistic independently distribute sufficiently computable typically compute exceed value value think extremely pass test say test fail whole test perform small large present thorough test quality produce early variant suggest particular design carlo mind generator algorithm distinct generator request bit bit popularity element sequence express parallelism exploit algorithm hx hx last produce yet sequence produce careful specific generator sequence recurrence direct quality distribution library framework library default generator library generators generators generators generators still pseudo integer right shift exclusive shift architecture typically operation division generator retain paper generator period period software package generator advantage period memory requirement instead subject criterion design give good linearity generator comment proper generator suitable parallel family comment combination step perform linearity generator shift class generator simple odd recommend odd integer generator generator shift ensures omit periodic period addition mod linear operation operation allow pass fail increase factor increase extend nontrivial consideration parallelism represent recurrence consequently count generator circular operator circular buffer reveal structure circular buffer denote access circular buffer element replace element calculate parallel examine flow within buffer produce sa sa sa sa ib observe maximum inherent provide versus define generator parallelism class generators generator consider approach create sequence number independently block architecture technique logical subsequence mp parallelism thus space generator period advantage make dynamically allocate generator whose exist gpu implementation gpu device present table generator state period generator small requirement generator great period short next throughput device generator generator generator fast old architecture order generator fast generator design current initially event speed platform dependent compare generator library benchmark fail benchmark fail test generator design fail interestingly fail two test generator none none none none none test consideration generator fail test combine generator generator pass generator test approximately probable relate consecutive seed value block within avoid unclear take parameter set value explore overhead increase memory generator consequently reduce generator conclusion show perform comparable speed require space device frank com generators graphic generator rapidly pseudo gpu statistical demonstrating generation graphic processing monte interest carlo mcmc mcmc numerous physical demand large random quality acceleration carlo gpu component bottleneck gpu accelerate gpu statistical
potential unlike generate return unnormalized computation likelihood many conduct inference stream nevertheless hard parallelization attractive part exploit advantage parallel hierarchical markov chain monte variant importance datum parallel issue demonstrate non mcmc direct argue need like chain autocorrelation generate independent absence parallel chain acceptance ds rejection common demonstrate improvement resource applicability failure separate prior another moderately sized et high conjugacy method speak share ds direct focus alone approximation ds proposal draw proposal ideally improvement improvement need concerned estimation advantage maintain conjugacy common generate target entirely parallel placing core permit likelihood start mcmc conclude discuss issue implement limitation like ds unnormalized let mode proposal obviously inequality hold negligible little uniformly yu marginal simulate marginal du write equation repeatedly et proposal construct continuous bernstein polynomial poor extremely large tackle strategy sample variable denote cdf draw proposal proposal draw density proportional segment partition probability fall multinomial proportional draw standard first random sign expression sign find unnormalized log compute proposal draw draw repeat draw place proposal draw draw multinomial variate draw single draw choose naturally efficient principle spirit dominating implement metropolis hasting multiply nothing trial concept tuning happen mode know importantly analysis collect investigate take advantage et notable al evaluation resource execute linear markovian dependent draw technology generate require hand parallel dimensionality motivate latent prior mean robustness outlier tail one joint distribution give contour around uncorrelated tail extremely poorly indeed note diagnostic chain chain attract modal uncorrelated chain tail chain posterior mode proposal draw area high pick correct shape acceptance collect draw collect mcmc mala constant hessian iteration start draw predict jump ever gap move tail consider covariate intercept coefficient gaussian prior weakly simulated simulated intercept value case population parameter summarize replication sample multivariate normal mode hessian multiply roughly average proposal collect accept reject phase take posterior mode start transform line take acc take execute accept reject collect sample total require algorithm conduct mac cpu core ghz ram core allocate h proposal proposal acc assess examine estimation draw generate core responsible collect parallel however one use chain independently arbitrary rejection burn reach also worth implement reject begin comparison adjust langevin drift choose commonly hard mcmc general exploit use posterior start million iteration several diagnostic density decide burn minute acceptance rate collect adaptation proposal also sparse hessian block grow turn ability computation researcher popular newton acceptance method sample pseudo propose dominate substantial effort likelihood dataset mcmc marginal accept express expect draw rearrange treat proposal draw shift geometric acceptance count draw remain integral proposal draw also convention put estimate follow accuracy eq q multivariate compare conduct covariate intercept linearly spaced collect density number draw factor precision density case exclude proposal proposal draw present along harmonic percentage take draw mode summarize table likelihood remarkably robust factor appear approximation negligible improvement note comparable harmonic compare method one fall density importance run mcmc collect illustration advantage scale sd mean sd sd time sd sd discuss implement entire mcmc continue insight gain experience provide suggestion mention investigation search inference example standard package find mode estimate difficult tool however appropriate immediately mode barrier adopt ill problem density unimodal one newton trust region instance algorithms search method subject nearly flat method stable trust short find optimizer posterior neither trust gradient find approximation newton ad ad write compute ad library small take otherwise estimate mode covariance software hessian work expensive heterogeneous unit hessian diagonal dense right margin substantial storing compressed format zero estimate computer computational exploit sparse offer implementation library exploit matlab store consideration multimodal posterior mode discuss unimodal instead normal global well mode unchanged long global proposal guarantee optimization token statistical statistical language literature facilitate efficient multiple mcmc guarantee distribution high happen general algorithm offer distribution selection proposal multivariate center mode hessian think proposal scale gradually experience rate still collect draw remain time try
low preserve geometry project far utilize due standard next top implementation develop speed scale consist uniformly replacement row independently sample without form compute spectral reconstruction submatrix return indeed require factor mf assume projection step generate compute parallel multiplication parallel execute generalize perform repeat computation ensemble improve approximation straightforwardly leverage parallelism modern error parallel time project onto rather project onto onto choose random submatrix base mf submatrix generalize parallel average highlight empirical offer great benefit unite accurate expensive completion robust factorization admit poor repeat costly sec attractive scalability thorough estimation section give rise mf formulation guarantee present intuition proof entry outlier advance informally information entry correlate limit extent singular correlated coherence miss notion let standard define coherence contain coherence coherence coherence eq definition call sec coherence well property capture entry entry mc mc solve bind mc guarantee thm noiseless incoherence coherent prototype notably preserve degradation require e whenever maintain coherent entry suffice nuclear step base mf moderately coherent moderately allow base coherence present lem introduce draw upon randomized spread responsible accurate fidelity reconstruction projection guarantee projection rise master coherence thm error error step master estimation guarantee subproblem new establish mc high present present well consequence mc sec present coherence mc mc coherent eq replacement satisfy probability positive assumption reformulate parameter error uniformly distribute noisy incoherence problem f f provide rate parameter coherence mc make precise next algorithm proof build master return either row cl coherent proportion quantity always simulation follow result utilize incoherent fix large subproblem coherence submatrix demonstrate noisy consequence thm solver operate much mc algorithm solve follow coherence master incoherence coherent uniformly fix parameter base choose achieve f f fraction reveal column whenever understand conclusion base algorithm thm satisfy scale scale key control success noisy noiseless setting missing corollary guarantee solver base convex master theorem thm incoherence moreover q fix rate rescale suffice tc place mild grow comparable quickly exact noiseless base uniform mc exponential q entry accurate imply precise zero thm base sec defer sec master guarantee I small high base subproblem submatrix use control subproblem demonstrate thm estimation noisy thm probability suppose mc solve master theorem base mc constant observe algorithm solve suffice decrease entry sample whenever understand base apply f high exhibit scale exhibit plus non matrix incoherent matrix sec variety simulate world accelerate algorithm mc algorithm base report ghz core gb memory default suggest error square comparison execute subproblem since subproblem execute plus compare two base carry focus decomposition similar scheme create matrix entry uniformly support generate take average ten noisy eps eps figure eps vary percentage outlier gap perform figure figure slightly match performance observe optimal frobenius match omit explore speed outlier row mc nearly identical result present mc curve overlap fact highlight minimal cost mc eps eps recommender collaborative interpret rating infer rating publicly collaborative filter netflix available ensure remain split notably outperform time present well slightly outperform problem mf inherently easy estimate netflix rmse rmse cm cm background model practical activity surveillance background object background change illumination evaluate video foreground variation include illumination frame video pixel value reduce video rmse run time eps eps f eps eps frame sample sample scalability factorization leverage compute architecture divide trivially suit environment low provably maintain super suggest first complexity guarantee master alternative theoretical practical benefit open ground incoherence replacement thm require randomized proof thm rely heavily accord distribution slack probability relate incorrectly statement matrix multiply transpose case matrix tolerance failure satisfy trial th column whenever th lem probabilitie binary diagonal rescaling replacement lemma lem given value lem lem immediately yield sample thm lem lem typical prop involve sample slack term allow guarantee projection sufficiently incoherent probability desire incoherence eq immediately lemma claim assume consist first row write fourth full define old norm norm coherence claim old norm definition coherence claim column projection incoherence b since prop yield minimize minimize must incoherence notice thm q noiseless admit note recovery appeal factorize decomposition approximation low randomly rank submatrix iff treatment hence lead vector lem rhs rhs bounds statement independence prop prove statement proof gaussian eq fr coherence suffice bind hold lem probability union projection result result block submatrix lem begin event event coherent master thm whenever holds desired reveal q uniformly cardinality variable eq q hence ip master coherence master thm aa moreover eq identical yield begin prove let eq coherent coherence master theorem probability least hold hence definition number satisfy bernstein n n assumption fact n ip identical master thm coherence cl q spirit even noiseless reasoning replacement appeal svd complement f sufficient condition suppose write f hence orthonormal old assumption high four lemma establish entry provide state uniformly multiple eq least infinity q assumption entry replacement admit identical replacement note bernstein inequality replacement lem construct consider replacement set consist location sample replacement replacement show location event location sample replacement batch replacement batch existence sample replacement conditionally replacement satisfy second condition imply first second projection final exploit coherence conclude lem lem fail lem fail replacement hoeffding sampling desire projection row argument conclusion column row replacement proof two key relate multiplication thm column second probability lem failure matrix lem master whenever suffice lem event probability establish fp mc f l nc n master guarantee event event entry variable distribution hoeffde yield bind follow identical manner master engineering award stanford edu stanford statistics stanford cs berkeley electrical engineering computer berkeley berkeley california department electrical engineering science berkeley author
penalty incorporate singular specify bic joint pattern association explore association gene gene correlation figure show gene implementation gene association additional score gene nonzero entry closely panel sign distinguish suggest sample driving appear gene square target load absolute loading load colored red loading color proportional loading display component component neural panel indicate second capture association apparent correlation alone association capture joint biological gene display gene interaction construct large loading gene link predict expression prediction module database current target inexact gene predict interaction far contribute joint gene joint expression frequently report tumor cell facilitate cell link survival loading study number set consist multiple type relatively general integrated provide powerful activity type account individual versa tumor provide tumor interaction point associate estimate outlier exploratory suffer see interesting computing account structure property bootstrappe regard however factor must carefully focus general integrated finance improve within market study acknowledgment wish thank constructive especially associate concern existence uniqueness decomposition rank algorithm discussion property simulate set grant nsf datum several genomic tumor paper joint integrate rank approximation capture variation structure type noise quantify variation type dimensionality exploration individual popular canonical gene expression tumor association provide tumor type software analyze measure set increasingly include diverse set distinct mode computational protein abundance spectra atomic sciences temperature traffic link page motivation particular application biological study number diverse amount available expand collection publicly database molecular cell biology genome collect scale project cancer genome type establish separately measure individual analysis critical association relationship type unique statistical association source inference research collaborative national cancer institute national genome institute characterize molecular multidimensional genomic integration information genomic comprehensive understanding focus tumor sample brain tumor systematic tumor classify neural expression copy addition clinical difference across copy gene recognize important aspect biology biology analysis gene biological biological suggest function primarily target gene list relation research partly responsible known tumor gene investigate individually important relation interaction gene expression joint variation gene expression lead tumor publicly use gene expression share shared vice versa may biological many individual signal joint exploratory capture joint type identify potentially type account allow illustrate size algebraic identify sample gene structure complex share subset identify gene structure variability joint relevant biology blue correspond similar pattern green object row matrix combine often baseline helpful subtracting differ variability total frobenius contribute rank structure matrix represent joint unify matrix independent model impose furthermore row joint structure responsible responsible individual constrain orthogonality structure orthogonality joint uniquely supplementary material remarkable orthogonality constraint rank rank important accurately quantify amount supplementary describe selection minimize error account joint structure minimize little present red yield contain loading structure load loading score individual factorize summarize explain loading score row summarize pattern type loading sample score structure reveal expression principal structure individual two gene present effect visually apparent color initial consider appearance surprising apparent joint plot interesting apparent suggest biological remarkable variation joint standardize joint structure permutation conclude four distinguished structure gene represent type play great role biology think unsupervise integrated analysis investigate component survival direct identify distinguished association association alone identify global mode association type perform popular examine set first canonical loading weight vector maximize geometrically interpret pair canonical loading subsequent find enforce direction set cca overfitte pls similarly cca covariance pls data examine variation vice drastically pls call pls seek variation linearly vice versa pls find explore common standardized loading extension pls develop mf pca group functional spirit mf decomposition group variability type group mf among main level variation return introduce job find joint weak association principal component response pca variation figure analysis pls pls score joint response panel association indicate pls individual structure structure cca score panel f individual tendency color within
situation become human poor temporal clearly indicate largely build ahead become considerable improvement moderate time incorporate open include become costly branching ahead become consume ahead ahead cognitive aspect sense mind sometimes understand lack branching become reward circumstance advantageous event formalism next subproblem rl subproblem solve factored support robust dynamic description state module choice module first module evolution optimize complexity serve factor markovian large rl problem polynomially optimize macro ii markovian detect measure state macro apply macro use follow ms temporal reward table ahead look ahead estimation immediate reward discount macro behavioral look ahead since look ahead change memory work need solid promising ahead factored scale property extraction le motivate hierarchical spatio appeal individual offer selective globally optimize method give rise I behavioral evolutionary pre learn td optimize rule td macro state decision surface factor look ahead predict behavior control learn inverse dynamic improve control solve agent model lack knowledge agent enter especially factor machine description mdps laplace approximation different tractable problem reinforcement extraction close concept transition control mdps spatio temporal modular state description markovian achieve markovian factored look ahead argue factor introduce ahead factored look ahead markovian description scale cognitive working memory evolutionary try separate aspect give property individual ahead markovian macro state td description collaborative planning considerably thank help european financial project grant agreement www home page reinforcement inefficient markovian environment policy investigate propose architecture utilize combinatorial test behavioral run deterministic factored finite learn illustrate property deterministic ms view decision insufficient old yet still markovian markov process mdps solid whether description whether state description relevant rl obtain raise whether compression world unless mechanic inform laplace activity necessity markovian state optimization mdp formalism know within turn time description markovian world markovian change agent restrict consideration approximate finite state factor practical factored mdp tractable relatively description question certain checking property increasing occur consider expect discount temporal markovian direct policy markovian state ahead might intelligence demonstrate good state quickly overcome poor value estimation state improve represent learn agent save evolution use may lead markovian behavior actual world start behavioral information td td develop sensor surface improve agent overcome estimation markovian utilize ahead method deterministic ahead poor estimation follow review ms pac illustrate concept fig ms pac proper read available enable characterization prove favorable factored rl aside problem elaborate conclude decade reinforcement large include review subject g reference therein agent action rely mdp formalism markov action state assign policy value value discount discount denote large value ps practical utilize rl know comprise improvement may short temporal td td matrix regard rise state error r ts td ts learn rl g description non markovian large size markovian property extent rely representation maintain episode search description optimization action combination act simultaneously enable representation discrete fitness fitness particular goal correspond fitness maintain family parametric solution subsequent mark n adaptively high specifie item determine actual sample fitness sort way update kullback leibler divergence decrease selective resemble genetic online treat factor tie supervise control rl create convergent reference therein use correspond individual case factor process factor probability require exponentially exponential often simple sampling polynomially exponentially large space stay convergent tie factor concept action module I complex state whereas activate module state two enable module macro combinatorial flexibility factor description behavioral upon optimization dynamical rl planning estimation example new dynamic sum upon exploration rl factor rl planning pac game later restrict transition easily describe ms pac walk dot since machine approximate risk factor ms cost sake ahead planning look game ms therein ahead rl learn face method task feature available logic algorithmic component reinforcement possess action extraction property use pac illustration player pac pac particular dot worth dot try pac life initially extra life reach corner pac power dot turn blue short period try pac pac worth player would four dot worth study achievable original ii accomplish entropy combination rise frequent building powerful ms life dot follow consume group simultaneously behavioral pac list value behavioral e td code description nd digit digit code apply ms action behavioral effective average optimization action macro behavioral upon average description behavioral pre selective evolution pre rule improve performance thought macro formation macro sequence mean abstraction basically behavioral pac agent turn power dot select rule low away rule achieve performance give close look ms dot chance may simply selection policy rise proper macro behavioral pattern module optimization episode optimize far macro macro compute td histogram predictor decision surface td new alternatively ahead procedure state information introduce within option state via rise behavior increase collect ms pac satisfy greedy replacement new however markov show improve change optimize poor average slow overall drop even optimization estimate column td fig exclude frequent td behavioral fig reward pac dots peak broad dot third fourth peak pac four would ideal ms pac avoid td collect one build purpose collect forecast define rl starting collect extraction solve rl execute categorical therein factor come back search factored look ahead argue decision make decision surface list encode surface lose world human recognition utilize category categorical example base change decision surface version useful easy component environment furthermore component surface even artificial pac slight actual type big sensitivity change component high dimensional easy categorical categorical appear mostly low example color example decision surface ms long learn ms game dot coordinate ms dot behavior ms pac randomness deterministic observation pac behavior sometimes deterministic factored branching potential world different speed easy observe branch ahead la commonly e ahead collect approximate closely relatively close end exhibit ahead model branching reduce heuristic branch ahead ms pac ms iii ms pac stop turn graph simple tree backtrack possible branching factor
optimize scenario real statistically background discovery physics wrong level guarantee nature obey something think yet even way free explore parameter value consume major systematic error uncertainty background separate signal new like b train wrong would regard datum contain tb search try beyond look deviation physics search whether idea forward cdf use music scan deviation algorithm search physics supervise anomaly oppose detect deviation standard produce use analyze physics method inherently multivariate free dimensional histogram among measurement statistic solve semi anomaly design deviation label experiment semi detection solve independent classify observation anomaly fall region background tail region anomaly solve exploit physic occur anomalous deviation estimate collective deviation background emphasize datum maximize parameter recognition anomaly classify anomaly use probability discriminant anomaly proportion anomaly anomaly could far section anomalous anomaly statistical nan statistic enable discriminate statistical background anomalous statistic bootstrapping replacement background fit new corresponding compute collective anomaly simple illustration model show generate gaussian density set simple anomalous model contaminate anomaly find anomaly mix proportion model gray illustration histogram excess histogram estimate black line anomaly gray number covariance model background maximize computational carry proceed step q estimate th equation step improve anomaly anomaly model anomalous log background em free simply skip fix complete equation base idea implement overcome issue choose detection detector realistic benefit event event produce association decay bottom look slightly show new physics consideration single simulated cdf detector neural facilitate datum principal detect signal background unlabele contain reality weak limited artificial toy detect contribute percent magnitude background statistic cross validation evaluation log maximize contour background fix model algorithm converge anomalous dimensional solid contour gaussian low would enough close attention figure receiver roc regardless high mass lie background low anomaly train mlp compare roc anomaly show signal mass anomaly similar start likely anomaly successfully identify experiment efficiently train severe consequence anomaly limitation tb semi identify priori knowledge train network supervise scan focus think standard significant anomaly background reconstruct observable reconstruct physical anomaly physics interpretation likely anomaly detector determine introduce region anomaly repeat anomaly stage anomaly explain adjust study physics computational semi clearly could tool limitation curse dimensionality seem perform relatively possibility parsimonious gaussian handle
descriptor descriptor descriptor rapid molecular property atomic associate descriptor store library molecular descriptor summation atomic technology provide rapid descriptor traditional descriptor topological autocorrelation descriptor surface integral atomic path topological path topological autocorrelation descriptor without dimensional minimized structure density energy correspond molecular van nuclear potential feature detail elsewhere version protein score score score represent characteristic construct difference similarity linear notion dot product covariance dot achieve dot hilbert possibly nonlinear entry kernel size require computational effort work call kernel produce kernel square regression square response cite work column equal maximize assess round center zero absolute round execute code adapt maximize retain cv pls calculate validate cv permit pls hyperparameter combination calibration loo obtain maximize force matlab routine regression supplement descriptor exploit gaussian loo cv place finish supplement descriptor result prediction place moreover second place third round descriptor derive descriptor row descriptor one model chosen loo cv across finish post look dataset tight high descriptor despite prediction assume parameter loo cv calibration loo calibration prediction improvement pls autocorrelation address generate pls consistent feature generate descriptor set present descriptor find close place performance response retain upon calibration performance call descriptor descriptor loo translate noticed prediction coefficient ignore moment descriptor outperform high descriptor round exponential perform input descriptor performance across calibration prediction winner back feature possibility pls find advantage use question descriptor bind contribute descriptor descriptor contribution work p fellowship reference square pls journal statistical reproduce journal affinity global journal computer molecular reliable class bind affinity explanation complex protein column energy use van descriptor atom j thompson modeling system computer method quantum r semi molecular discovery virtual throughput institute york sciences sciences computer ny paper obtain blind bind part comparative prediction least pls outperform pls incorporation capability keyword least square comparative prediction algorithm cv square pls rkhs atom
framework easily stationary identifiable stimulus assessment assume bayesian assessment specification assess broad alternative without make direction advance highlight attractive predictive bayesian assess fit testing normality since multivariate normality assumption build two assess tree alternative normal dirichlet offer compute easy univariate base computing density may inefficient estimation alternative sub detection normality simulation claim dirichlet normal smooth dirichlet normal know variation possess adaptive rate normal line normal beta collection process normal normal correlate alternative scalar parameter parameter underlie parameter determine potential component carefully map parameter avoid identifiability despite slightly different mixture normal amenable computation sequential imputation posterior bayes computation factor computation adapt liu sequential imputation parameter propose type frequentist test univariate moderate size test classical test address refer bayes go go technical section sample size simulation consistency dirichlet mixture model independent draw variate cholesky p pp triangular testing improper along natural parametric provide develop stress maintain balance nan nonparametric specify alternative nan alternative normal wishart modification scalar beta normalize constant write distribution dimensional proof ensure alternative model discuss degenerate mass breaking write independently consequently give independently nx v stochastically volume see fine evenly shape curve center likely parameter alternative nan control base shift volume argue otherwise may weakly degenerate chapter alternative vanish concentrate remain alternative away make irrespective vanish nature limit choose bring useful flexibility value stick representation make normal close component shape allow curve possess sharp scenario satisfy limit remains carry monotone reasonable use specification reported determine little overall shape possess sharp broad shape notice intermediate dashed alternative spread away nan intermediate issue study nan magnitude alternative always way normal right haar light choose weight alternative insufficient justify need obtain proper either predictive property notation fold I surely absolutely respective lebesgue collection space df df family call invariant random law absolutely respect measure absolutely haar p p consist distinct observation calculation associate gx gx f h carlo draw unbiased root depend conditional density choice approximation later draw give ss choice justify conditional density partial sequential calculation therefore simplify gx nh n x due write suitably adapt implementation order randomness tractable approximate make approximation embed density expect tail way likelihood importance simple section sample oppose scalar inverse density wishart component tuning efficiency code website likelihood choose recommend conditional recommend gibbs rao alternatively smooth technique suggestion min st rd max summaries replication bayes normal refer monte sampler fairly west posterior base smoothing package bandwidth computationally expensive choice result improvement refer run posterior gibbs smoothing run importance substantially latter help extent competitive illustrate pressure minimum showing normality minimum three synthetic respectively bivariate bivariate student freedom bayes factor drop little normality outlier detect heavy tail dataset elliptical scatter sharp decrease minimum normality leave right consist take epoch multivariate variance residual mahalanobis deviation fail reject figure minimum bayes indicate strong large factor wide respect nan unique alone univariate produce mean spike extreme hence non pick approach bayes fit normality classical subject calculation simulate nan normal specification run compare test test derive similarly minimum factor partition minimum calculation normal student simulate result power non normal mixture produce outperform surprising tree alternative process frequentist property bayes nan whenever normal kullback prior substantial literature indicate mixture normal distribution broad support mild continuity expect nonparametric formal challenging nonparametric densely around prove simplified mixture simulate standard evaluate bayes sequence display path factor appear converge appear alone scenario substantial du nu u accord eigenvalue complete form change find characterize
consequence stop lemma extend technique obtain bind greedy rsc hold sparsity terminate lemma noting hold greedy substitute backward stop characterize terminate stop stop support backward step stop support stop q rsc stop notice satisfied value useful graph depend stop stop nd parameter far satisfy false condition corollary rsc note conditional logistic covariate bound analyze rsc logistic order remainder exist exist constant argument function rt rt w verify stop parameter well neighborhood recover union completes impose require node contrast regularize counterpart greedy observation result pairwise model variable take multiclass greedy backward greedy would add group greedy rsc far lack experimental several structure outline experiment b near star assume binary type generate learn structure match range sample batch suggest experiment via validation n coupling logistic suggest greedy less star mixed sign success versus parameter demonstrate terminate fail entail consequence upper stop support q optimize rhs stop sub range carefully arrive contradiction eq hence algorithm backward fail go entail stop stop parameter eq contradiction reach forward backward kk along similar hold terminate iterate variable completeness reach rsc consequently entail suffice forward optimizer j latter contradiction provide k claim conclude begin forward lemma optimizing conclude reach begin carefully k rt rsc eq fact proof structure graphical study property special case discrete model neighborhood estimation general sufficient size recover edge high greedy strong convexity numerical end undirected variety domain statistical among concerned markov mrf identically encode independence subset thus broad mrfs estimation greedy estimate successful exhaustive quickly estimate neighborhood set conditional need solve expensive large another approach score scoring candidate typically indeed discrete mrfs search large intractable thus heuristic local procedure come focus consistent possible even dimensional relevance sparse zero show practical strong cf paragraph induce nature program log likelihood recent model focus stagewise greedy add possibly parameter yet strong guarantee estimate finite greedy variant appear counterpart observation graph node pairwise specify e largely ise denote sample vector form infer sample estimator neighbor equivalent characterize define goal condition give neighborhood greedy stagewise condition combine neighborhood add occur respective node could use selection succeed recover neighborhood rule exact describe general next apply general next section outside graphical suppose estimating assign cost ease shorthand factor
c pure warm table number iteration value utility intermediate iterate algorithm contrast global different completion namely soft iterate simulation use code due matlab numerically expensive operation burden singular decomposition likewise implementation soft operation al accelerate involve solve lead burden carry approximate et proximal stronger accelerate three heuristic rank project estimate accelerate algorithm soft iteratively replace element involve post like performance vary computation iteration simulation dominant descent ability solve fix soft without like truncation assumption sample remove uniformly tr soft stop either iteration stop relative duality ht plot behavior greatly affect soft suffer moderate outperform iteration behavior optimization training kept entry descent tr soft initialize fix initialization include suggest entry stop relative variation stop duality fall convergence later exceed similarly stage tr rank trust region ht scale vary value generate random initialization iteration take stop number exceed stop absolute variation relative error plot converge take ht two regularization comment completion summarize complexity suit moderate required iterate scalability well descent tr benchmark suit low need compute move strictly go warm regularization coefficient minimize response multivariate optimization low rank motivate follow although smooth directional euclidean complexity term like cost full low note duality regression define x w trace numerical dominate numerical generate unit deviation white noise add cost regularization validate error rmse ratio snr fix minimum apart fit claimed gap similarly trust algorithm stop fall ht c snr main present minimization rank analyze convergence criterion duality problem numerically thank simultaneously ensure decrease cost function term p space riemannian technique geometric devise order proper trust region convergence contribution predictor step geodesic descent approach predict performance superior warm idea low completion encourage result n r u b x sub trace norm stationary uv v w global optimum virtue strict derivative constant update result function obtain maximize introduce x trace norm q sup lagrangian dual prove cm address low equip particular riemannian trust guarantee solution maintain parameter naive warm manifold completion regression know nuclear attract year relaxation intractable decrease unbounded consequence whole minimizer proxy minimizer rank dimensional propose second control svd orthonormal matrix span contrast allow unique rank long riemannian devise trust generate converge local rank minimum reach select decrease convergence minimum rank also thereby transform global minima procedure path numerical surprisingly literature primarily idea semidefinite framework riemannian develop improvement discriminate thank choose combine one appear particular context completion spirit set global tight trace norm priori use operation demand potentially singular one potential single efficient warm sake illustration comparison application matrix iterative numerical favorable represent manifold definite stress redundancy implication key success illustration optimization euclidean manifold manifold proper point whereas scale positive invariance necessary optimality solution optimality condition differential trace throughout either write lagrangian space section local optimum op operator I singular fact local identify global threshold criterion check global closeness optimization duality gap duality solution dual dual singular dual trace sup x f expression duality gap operator op operator conjugate easy propose problem first optimization early trust idea behind region quadratic solve iterate depend decrease objective function reject detail rewrite u notational convenience important second minima isolated rotation v pp symmetry total count mp equal rank remove symmetry identify belong equivalence class manifold conceptually optimization minima isolated computation manifold tangent tangent representation equivalence product eq z z subspace vertical skew size horizontal space pick positive horizontal tangent characterization b projection horizontal solution solve lyapunov manifold special convenient riemannian horizontal lift due equivalence class horizontal expression element directional lead representation likewise horizontal lift well riemannian positive cone derive riemannian connection eq euclidean directional derivative horizontal lift trust manifold guarantee quadratic rate propose trust region eq trust region radius solve subproblem lead direction quadratic model iterate mapping mapping refer map case mapping metric known virtue mp op omp mp mp op mp show linear operation algorithm start alternate fix manifold table ensures decrease belong stationary rank update ensure dominant leave right f fact project onto subspace stop point step obtain rank value orthogonal projection write orthonormal constant justification value give good stop rank guarantee fact fix unconstrained descent iterate rank theoretical however always contrast propose tight iterate problem separately increment make trust addition priori solution interpretability iterate global different motivate path n initialize warm regularization argument paragraph regression extend idea u I factorization perform warm scheme show minimal computation sufficient add multiple square x optimality condition look geometry rank obtain dual consequently smoothness smoothness decrease solution first order path compute exact solution step ht I geodesic extend need u v I refer logarithmic numerically expression mapping numerically approximate follow horizontal projection accomplish operator v p u eq approximation approximate eq predict step backtrack motivation observation line numerical denote descent tr low completion regression penalization recovery full ghz intel core machine ram illustrate case optimization completion behavior search p p mp diag diagonal suffer impose generic complete entry addition problem denote frobenius matrix multiplication convex high assumption
seminal paper approximated quantity appear optimum parameter inference exact implication oppose approximate implication justify optimum goodness reduce family generate optimum serve compactly lemma log likelihood number reasoning assign prior schwarz proposition bayes optimum large poor criterion aic aic bic evidence measure compete tell economic give goodness selection evidence maximize true nuisance substitute use theory estimation economic finance reference therein histogram fit theoretic popular economic particular physics theoretic analysis student model ac ny kolmogorov solve regularity condition subdifferential calculus duality infinite duality mathematical mathematics york ed bic york free test fit fitting nj second ed york behavior mechanics mobile university york des paris york university communication criterion correction equilibrium reasoning transaction university non likelihood ratio edu recently optimum kullback leibler subject approximate obtain model unconditional oppose method axiom nonparametric optimum information already measure kullback information theoretic reference therein gain popularity theoretical foundation provide product obtain closely bic bic pick compete measure tell poor I take compact believe unbounded compactly ball radius etc denote generally lebesgue moment define represent inference choose uniform hereafter carry set drop subscript minimize theoretic point justification special like finite ii regularity discussion regularity integral compact always substitute duality algebra understand development later unique dimensional affine tx strictly maximization solution concave solution newton optimum verify substitute equality optimum family guess conjecture true maximum tx maximum show imply coincide maximum maximum valid truth put wrong one exploit estimator family bic unconditional conditional solve population counterpart kullback asymptotically freedom optimum estimator unnecessary exponential arbitrarily
set convenient inner consideration abstract form establish ensure perturbation set perturbation work depend nonempty relatively complete bound perturbation give must complete hold form eq denote scenario random vector constraint discrepancy value branching convergence building work let decision extract scenario pair represent stage branching structure formulate optimality number decision jointly recall decision beneficial represent expand statistical attractive nonparametric process relatively particular space update mean stage function coordinate notation kernel semidefinite similarity also dataset noisy optimal stage optimal infinity cover decision select radial bandwidth tend coordinate policy replicate decision optimal act numerical regularizer confidence make prior learn literature deterministic problem nominal extract decision parameter affect regularity update exist policy exhibit usually possible procedure output know process density predictive distribution mean decision maximize feasibility program feasibility figure induce option correct feasibility heuristic maker near policy denote correspond ultimately ability explain scenario output lead problem true simulation scenario scenario finite estimator policy moment normally inverse guarantee true basis could eliminate rapidly investigate methodology three factor variation scenario approximate relatively solve accurately play role predict decision experiment evaluate criterion spirit application product distribution resource represent observable contribute reveal decision quantity decision allocate production composition demand numerical conduct simulate pure programming decision solve horizon decision fix benchmark processor run compute set absolute optimal two adapt production available demand reveal feasibility procedure describe variant report obtain must one radial bandwidth variant cumulative radial complete tune behavior feasibility heuristic adapt coordinate create reach quantity consume proportion namely heuristic generate order permutation shrink horizon procedure implement shrink horizon compare branch quantization assigning minimize probability integrate cell exponential growth scenario branching accuracy branching grow exponentially net percent recall structural property guarantee branching cm branching second test set scenario new horizon scenario tree branching stage choose optimize online scenario tree branching horizon use simulation processor result feasibility gp test collect tree branching well make treat overhead fast possible explanation tree thus much solve heuristic speed table benchmark program gp attractive form policy decision simulation eliminate completely call program achieve feasibility mostly solution problem scenario tree concrete branching structure turn study present consider study risk parameter budget generate could stage speak otherwise solve branch already decision turn branch branching scenario deterministic induced scenario branching scenario negligible know advance resource branch day realistic value motivate branching purely randomly sense optimistic bias turn scenario tree fast enough several tree thus essence branching base well branching structure lead produce tree assume node create randomly large number create eq iterate recursion small recursive mostly level tree expectation effect ex scenario create number child branch branching increment step otherwise branch tree approximation compact feasibility summarize table overall simulate scenario matlab processor randomized benchmark policy neutral cccc tree cpu somewhat surprising multiplying scenario compare set randomized decision discuss tree ensure good good obtain good one probability low program decision maker improve risk besides expectation possible empirical scenario paper approach infer policy rule stochastic present lead nonparametric investigate scenario tree generation develop use useful scenario algorithm exist current acknowledgment present network dynamical office scientific associate financial carry department engineering university li grateful suggestion presentation parameter test pt stochastic research university usa department engineering li combine run sample independent solve program technique choose scheme scenario branch parallel could ultimately retain test excellent stochastic scenario powerful rapid growth scenario grow limit size make possible optimize first look ahead compute problem propose assess tree sample pose span planning horizon relative sampling updating span time remain previously process find sequence decision value accord procedure produce unbiased unfortunately simulation demand stage restrict technique relatively update combine tree return statistical tree decision supervise repeat exercise produce policy function tested determine primarily approximate reinforcement community use actor computer low dimensional constraint approximation constrain feasible minimize deviation policy scenario optimize scenario could view near regression however context recognize direction makes quickly perform approximation create use scenario safe guarantee build fundamental implement particularly reach pass node scenario tree formulate optimization associate scenario associate optimization path root identity among fx k subject optimal first stage regret implement machine mapping generalize value feasibility set per simplicity scalar parametric value linear build stage index advance sample new distribution
extreme identically value max generalize extreme fr member generalize margin estimate generality assume margin fr transform step multivariate identically replicate maxima degenerate exist sequence extreme necessarily method max stable infinite multivariate theory hold possible max process fr practice generalize parameter location transform often spectral often negative replicate max process fr margin represent max process function intensity stationary max unit choose family correlation mat ern cauchy modify third order force reasonable environmental paper use parameter serve mat ern form let denote point poisson intensity max stable fr margin variance increment concentrate widely year simulation realistic feature stable marginal bivariate write function statement motivate max field margins joint describe max eq measure location explicitly joint location correspond independence think consideration available coefficient form directly manuscript estimate pairwise unit fr move triplet location distribution close follow pairwise triplet serve bayesian free aim posterior computationally prohibitive approximate area facilitate auxiliary dataset integrate simulated interest exactly point else recover likely occur probability practice summary sufficient limit familiar occur density simulate form uniform favor straightforward accuracy challenge enough computable produce identically draw role draw approximated approximation closely resemble practice highly informative remainder implementation motivation stable margin show relationship extreme parameter coefficient gamma margin relationship transform margin usual fr margin unbiased minimize squared square equal square mathematically ol procedure utilize approximate algorithm ols obtain remain eq integral difference two curve take final suitably collection particle correlation analog mean pointwise pointwise performance study curve pairwise coefficient parameter sum residual ordinary square proceed use summary pairwise composite approach improvement computing method move consider order tuple triplet explore max stable triplet equation basis utilize triplet estimation triplet rapidly increase triplet pose computing summary decrease uncertainty estimate triplet coefficient large natural homogeneous ideally homogeneous heterogeneous triplet group measure triplet euclidean measure two triplet permutation triplet identical length rotation translation two set theoretical isotropic triangle respective length increase entirely actual estimate triangular size distance choose cluster balance maintain homogeneity ensure triplet average method assign cluster cluster choose minimize increase sum center merge location requirement practical dissimilarity computable consuming average begin draw except draw stable fr triplet coefficient absolute collection value filter final distribute collection compute error simply identically draw empirical construct dependence mat ern conduct specify range dependence preference suffice error year location uniform grid dataset composite example approximate computing draw substantial computing need simulate computing around iteration pairwise constraint replication simulation run filter percentile accept posterior distribution threshold specify percentile relative smooth opposite produce abc behavior comparison correlation pointwise h focus comparison region higher great way stress pointwise abc use use handle assess manuscript computing report compute composite abc composite reduction vs composite two follow approach mse low abc essentially tie composite likelihood within five standard remain large conclusion view whole favor outperform composite give composite find great abc pairwise utilize essentially finding section carry computationally approximate bayesian closely performance approach simulation increase adaptive abc produce approximation approximation second abc allow implement algorithm specifically simulation particle filter percentile repeat times filter percentile ensure particle accept call weight produce weight modify estimate parallel initial cost performance expect result three f approach benefit range aim loss united management loss month daily temperature site center united historical http daily sites north daily region jointly four responsible year daily temperature month risk transform unit univariate extreme maximum extreme scale heavily relationship spatial coefficient triplet describe group consider ern correlation criterion find mat ern range selection abc range spatial process scale draw run pointwise accept approximate pointwise credible interval pointwise centroid centroid census http www census www specific extreme value standard spatial krige whereas location fit matching replacement stable mat ern correlation centroid centroid temperature threshold incorporate uncertainty intermediate exposure million percent fall threshold degree provide evidence respect additional support financial product loss show beyond calculate interested calculate expect policy model stable
natural generalization side q minimax sense attain eq I address pre belong occur post grateful useful discussion office nf air office scientific research grant fa national grants california mathematics section definition mathematics california california g department mathematics university california usa mail edu mathematics california usa mail nan composite either embed exponential stop minimize rule approximation rule verify asymptotic test open test test identically whose sampling soon possible favor absolutely solution testing test eq level eq stop call test terminate surely whereas terminate almost furthermore stopping time alternative hypothesis side exhibit optimality measure associate alternative p exponential versus observation likelihood side delay latter approach also ratio statistic give lebesgue bound q bind attain mixture continuous information trajectory favor minimize maximal kullback leibler favorable motion mixture family particular choice density lead attain theorem emphasis hypothesis df ix df ix assume mass required belong exponential parametric setup approximation likelihood ratio discretization main motivation setup sample channel population possibility correspond attain non obtain approximation see even additional may include problem actual control channel typically advance thus case misspecification naturally approximate continuous discrete evaluate point allow mix design attain kullback identical latter attain asymptotically order asymptotically e ratio converge I rule attain natural criterion e kullback leibler distance express minimax mixture highlight show constitute use random time family integer theorem impact develop limit stop walk family stop probability play role one reason connection tool rule theory decompose statistic long change basis order additional rigorous say satisfie growth uniform nonlinear understand perturb walk consequently variety sequential asymptotic normality whereas purpose useful quantify focus rule optimality simulation sequential conclude call stop sided assume refer expectation set e kullback leibler versus walk whose denote walk brevity easily show e designing side sided attain infimum eq eq consequence predefine asymptotic term stop obviously active close assume exclude index decomposition fact slowly able rule break presentation without insight need exist jt ideally like minimize task attain distinguish notion follow use theorem asymptotically positive weight fully main subsection lemma sequence iy ip condition q definition I slowly obtain suppose q asymptotically process due constant j increment mean eq iy I arithmetic therefore measure yield prove assertion furthermore eq decomposition slowly weakly recall complete arithmetic either walk arithmetic follow nonlinear long p n aa integrable satisfied converge consider first lemma inequality first q long chi square variable eq everything obtain negligible either condition eq usually nevertheless case corollary bound order attain content theorem arithmetic finite auxiliary bayesian optimality nan therefore cost observation leibl divergence prior h integrated stop moreover indeed small minimization establish optimality test infimum important particular scenario stop arithmetic asymptotic define mixing way mix consequently remain construction whenever mix test third asymptotically follow weight every q mixture mix q every perhaps choice implementation reduce uniform work criterion reason optimize condition thus mixture rule attain mixture symmetric minimax rule inefficient order unless suppose control attain asymptotically sided meaningful minimax appropriate bound definition ratio write define recall case mix respect lebesgue positive moreover see complete substitute mixture however continuous asymptotic remain minimax maximal leibler asymptotic mixing attain bind must continuous density exact exponential rate imply mix completely normalize close therefore optimal mix mixture computable minimax show continuous use minimax discrete optimal mixture interval take eq corollary asymptotically asymptotically third minimax corollary theorem write kullback divergence q p arithmetic see condition
trick kernel tractable student novel machine arrive carry performance method demonstrate capable distribution reconstruction nevertheless unsupervised task image cubic fast another extend define g graph develop novel kernel methodology investigate limitation decision certainly test frequentist characteristic trick marginal likelihood enable issue potentially include numerical would song helpful discussion le song part grant ep ar college popular nonparametric rely present trick derive regression despite process kernel aim demonstrate wide machine obtain popularity past decade rely chain monte approximation ever grow form approximation would desire many perhaps process predictive exactly method adopt frequentist remarkable algorithmic clarity widely trick limitation apply representation observation product product increase increase computational notably allow popularity bayesian kernel trick rarely tool rare new machine kernel computation rigorous framework reproduce rkh offer hyperparameter mean incorporate model methodology find mm simple observation bayesian inference dot crucial finding fortunately orthonormal generative underlie principal pca show distribution preserve orthonormal transformation sec review discuss apply trick predictive consider sec experiment high bayesian dimensional euclidean task nonlinear ambient lie sample consider estimate distribution importantly require result still amenable kernel trick discuss guide distribution trick term scalar invariant orthonormal space orthonormal want express rotation permutation kernel conjugate gaussian meet invariance wishart gaussian normal wishart restrict two way firstly secondly wishart set spherical sensible restriction ensure define observation q dependence scalar invoke inversion q reality however assign actually make trick product say previously interested manifold geodesic density input space implicitly invert mapping define multiplicative jacobian correction relate input change variable density appendix pt pt illustration estimation embed observation real line multiply give scale exactly apply trick simple unimodal multimodal possess draw complicated model linear fig map dimensional fit inference map student predictive manifold observation multiply jacobian fig final fig remarkably jacobian include mapping feature essentially student predictive restrict indeed simple although improper observation decide ignore jacobian pt student se vary panel generative want require embed hilbert potentially infinitely measure model space address misspecification stem project manifold limitation closely underlie generative bayesian inference effective popular perspective induce sensible recover toy degenerate feature along roughly axis fig intersect manifold twice delta degeneracy recover component clearly degeneracy integrate degenerate topic kernel element observation space exploit way feature interpret therefore characteristic help task gaussian incorporate unsupervised construction trick method rely chain carlo call doubly intractable use intractable unsupervised model simulate convenience improper space point invariant subsequent k condition different setting process chinese restaurant like eigenfunction induce performance label table constant past include deep belief net hard directly unsupervised reconstruction absolute assessment introduce task able model recognition test unsupervise task use detect handwritten digit training label gray handwritten model ten student length median point digit dimension label point threshold consider baseline dirichlet mixture gaussians operate three choose search auc separate ten test compete significantly complex investigate carry sequence average auc size amount becomes consider
unnecessary regressor leave advantageous power insufficient consideration lack flexibility original regressor explain regressor alone reveal concern ratio regressor generate regressor relevance response drop little regressor probably variable higher less regularize build flexible generate performance stable rbf possesse describe may optimality cost variance experimentally rbf additionally indicate first present synthetic noiseless use handwritten character define handwritten character canonical difference target resemble desire like comprise font character difference handwritten original font generate font character handwritten character would font font canonical system use font vector field comprise vector evenly font express handwritten explanatory font response explanatory comprise respective output unit draw drive respective probability feedback connection marginally activation high connection become unstable drive internal discard internal collect row observe mse modeling stage mse generate regressor response variable variance regressor select moderate steady variance regressor select regressor give ratio regressor half contribute vector examine cause weight show half contribute little explain response purely reveal analysis relevance regressor original locally linear regressor train mse stability robustness thus test model half little explain response high locally rbf try performance rbf evident locally figure show regressor significant dimensionality improve robustness dimensionality tested attribute strongly variable add sequence transform training sequence reservoir sigmoid teacher force step sequence use formula trial variance code able compare result period leave difference net formula teacher internal repeat test interesting free locally worth training plain fluctuation smooth minor smoothing beneficial investigate stage always sequence table outperform plain linear fit similar sequence local work well regularize outperform linear ten trial may benefit regularization analysis provide deep present fig flexible explain probably room improvement extracting regressor g extract delay sum suggestion research present locally improve penalization regressor linear transformation rbf regressor rbf helps understand state consideration construction mechanism matrix activation give regressor enable concern mechanism mechanism internal concerned adapt topology rather adapt world free yield interesting activation state space equation besides sigmoid radial rbf activation test stability insight construction mechanism delay detail delay regressor analyze delay importance analysis temporal underlie whether rbf enhance room improvement effectively combine leave automatically regressor automatically desirable recently propose coordinate find significant improve rbf well evaluation present generalization alternative traditional rbf improved importance regressor via analyze therefore straightforward improvement improvement likely design model parsimonious well aid non expansion rise regressor relevance teacher often certain regressor effectively explain teacher concern generate contribution relevance locally depth underlying regularize build may improve relate suitable hand limitation sometimes insufficient flexible parsimonious excellent alternative feed rbf net state local forward variable radial rbf novel network easy appeal attract output dimensional vector internal element sigmoid state temporal teacher expansion carry diverse input teacher name state diversity appropriately teacher traditionally usually trial diverse successively simple construct generalize expand parameter mostly expansion adjusting regressor linear variable desire importance quality regressor generate expansion state generate regressor various relevance regressor variable less regressor hundred regressor contribute instability regression large response unstable usually easily fit overfitte regressor point couple therefore model traditional regressor improvement address issue regression combine pruning test pruning decrease result prune exhaustive square mse regularization delay sum bayesian approach effective delay improve memory deep model parameter smooth regressor relevance smoothed teacher alone fully indicate usefulness regressor experimentally regressor orthogonal individual jointly variety appropriate robustness effectiveness later text term model flexibility insufficient analysis able determine whether usually require linear approximate forward neural rbf nets rbf effect input least iterative base considerably flexibility require training meaningful contrary appealing estimation support rbf considerable popularity decade may rbf easy train possesse recently however suffer instability deal datum perhaps regularize constructed rbf flexible excellent possess paper regressor locally analysis provide locally locally present rbf model noiseless research discussion modeling conclusion forward matrix regressor variable th model express notation orthogonal triangular minimize least triangular obtain computing coefficient concerned orthogonality useful transformation carry orthogonal regression orthogonality regressor equal regressor regressor orthogonal regressor regressor alternatively algorithm sub select regressor reduction ratio ratio alternatively regressor gradually variance selection certain significant introducing regressor cause marginally regressor contribute ill conditioning model build overfitte selection purely appropriately regressor term regularize follow error diag normalize regressor criterion tolerance produce sparse regressor parameter unknown set carry full result sub use use iteratively unchanged two iteration regressor often dramatically suffice parsimonious enhance cost determinant improve combine selection regularize eq stop iteration regressor regressor parameter quick system output follow form eq step time delay lag white identify datum possible approximate rbf regressor rbf centre norm rbf nonlinearity thin spline function choice write I set parsimonious generalize generate iteratively parsimonious centre effectively make centre combine optimality regressor select manner iteration state update equation desire output discard rest store design eq common zero traditionally square l regressor various desire shape govern pseudo weight nonlinearity mostly experience approximate response desire accuracy obviously pseudo rise regressor high regressor contribute instability variance fit noise issue desirable regressor address regularization regressor response noise alone generalization regressor regressor
depict hmm simple break isolated indicate right history p induce observer remain nontrivial induce possibility direct calculation output thus past least positive comprise machine technical arise conceptually similar machine generator history hide markov characterize finite alphabet history generator correspondence finite generator machine generator machine history machine generate machine history follow gm hide markov key prove come study synchronization generator machine terminology generator symbol random x ts x observer current observe machine word machine relation low maximize observe symbol describe symbol formally define cross realization primary exponential constant synchronization differ time shift length refer observer instead observer index work symbol exponentially unlikely observer small generator n exist lemma tt w x x iw ni appendix equivalence measurable measurable ie ne class history generate preserve ix j directly point take I x w w ix generator assumption machine process generator history history generator machine priori clear indeed hmms hmms generator uniqueness long hold machine reverse finitely finite history hide also generator machine note appendix know characterize ergodic hmm generator hmm construction claim throughout n equivalence entirely connect component strongly connect existence x hmm transition define essentially fact separately two case equivalence l sm give ii word nan convention take always analysis well w claim ie set sum I mention take equivalence word x distinct lemma hmm strongly define via contradict distinct class two component process generate generator distinct argument lemma w stationary machine distinct one satisfy demonstrate history generator machine however recently also proof improve intuition equivalence new come recent bound state large machine number result countable generator unfortunately countable decay long unclear whether countable either hold countable machine entropy acknowledgment award nf fellowship establish trivial alphabet sequence nontrivial measurable set martingale hence length measurable equivalence class equivalence alphabet past recall past implicitly follow claim several various word tw n tw x w tw regular symbol conditional eq regular regular equivalence fix x claim eq hold equivalence e equivalence measurable set throughout ergodic alphabet past proceed measurable countable intersection p w p class definition w w measurable establish probability equivalence claim proof proof theorem assume ergodic alphabet define set length e x length equivalence nan class prove bind guarantee need claim application expectation version claim w x e e e claim finitely finitely alphabet probability past class class symbol finally equivalence stationarity fact transition hmm pt machine stochastic process originally machine hide analyze though alternative definition hide state key difference machine generator machine let whose infinite sequence correspond structural dynamical system subsequently context thorough definition original history discuss machine oppose history derive generator process hide obtain initial underlie establish history generator formally generator implicit technique substantially somewhat section argument implicitly concrete directly result fairly elementary state thus useful providing intuition terminology generator show consequence helpful order parallel generator machine study machine machine alphabet generator stationarity history definition stationary process introduction machine come physics substantial symbolic dynamic review development reader refer overview symbolic model help machine understand relationship reader skip type first shift present direct label alphabet consist walk present vertex edge restrict essential vertex along thus graph occur present state accept clearly essential accept thus unique say word shift irreducible irreducible strongly present say symbol irreducible shift unique present graph symbolic dynamic presentation refer irreducible shift minimal deterministic irreducible label allow future x vertex consist equivalence vertex vertex past necessarily minimal deterministic arbitrarily start state irreducible graph cover recurrent minimal three closely present irreducible irreducible necessarily recurrent irreducible component cover cover extension purely history analog analogously cover infinite sequence allow history probability cover recurrent history equivalent abc process example exist ergodic process support irreducible infinite even example shift right leave cover associate generate shift extensively rich theory many characterize stationary shift induce past finite length positive probability future natural machine exist infinite latter alternate process unfortunately quite well despite alphabet sequence sometimes name corresponding normally assume theory strong old particular relevance g restrict show requirement surprising perhaps construction irreducible associated infinite give may past sequence equivalence history machine g induce probability symbol infinite future converse necessarily bad nice history machine derive finite complete stationary hmm output word state overall next symbol q respectively associate hmm markov irreducible transition word choose process ergodic irreducible ergodic ergodic x algebra bi infinite stationary measure stationarity uniquely finite consistent measure two preserve edge label edge irreducible hmms converse hmms also concerned generator machine irreducible hmms important property distinct hmm strongly connect symbol symbol iw check hmm distinct condition separate distinct check distinct states generator associate generator machine ergodic process probability define generate transition generator symbol hmm path nonzero long start symbol equivalence history model whose class past consider induce probability take specify clarity verification defer reading overview separately entirely self none note ergodic parallel generator although requirement stationarity actually need ergodic alphabet past sequence w symbol tt trivial word however probability well equation past normally define conditional like uniquely intuitively keep mind understand conditional probability intuition justify construction history set regular equivalent prediction simply precise drop subscript infinite see equivalence equivalence appendix equivalence transition relation well x regular also equivalence symbol independent equivalence class formally distinguish function history machine appendix say equivalence finitely characterize slight
note yield represent term price deviation term second return long return minus convenience parsimonious perspective term long price factor unobserve variable term equilibrium growth price follow additional assumption speaking point convenience greater discount price cost additionally introduction level review demonstrate issue remove potential positive additionally admit time vary yield analytic contract unobserve develop convenience proportional square root instantaneous yield cox yield proportional instantaneous convenience yield level analytic price multi several extension question relate give neutral time return convenience yield mean coefficient term rate volatility modify function exponentially affine family derivation price consider exponentially affine neutral allow additive account volatility imply volatility remainder xt dt dt dt dt v correlation structure addition risk neutral ensure discussion real neutral develop without variation respect factor short dynamic price contract time contract tt price neutral detail derive space formulation bayesian novel parameter specify panel denote addition parameter use process neutral discretized formulation achieve scheme process order finite taylor otherwise scheme euler two firstly discretization secondly process euler increment scheme mix variable innovation bivariate wiener discretize space truncation specification take develop sense generate transform sir filter methodology zero normal noise long linearization state approximate extend kalman filter formulation develop novel introduce formulation convert neutral pricing proceed posterior distribution real neutral g parameter specify prior world neutral specification random choice hyper uninformative specification section though consider specification specify give deviation space equation form addition admit affect observation ft b detail equation simulation recent sampler state jointly latent methodology particle filter parameter embed fashion construct adaptive scheme use static sequential rao via kalman introduction markov distribution adaptively learn static significantly factor develop involve kalman importance resample sir variance optimally latent sir version filter particle mcmc utilize approximate one interested self well set static latent hundred thousand depend frequency ideally suit simple univariate component sampler achievable especially problematic chain achieve typically lead markov large optimal distribution carlo static proceed metropolis probability require acceptance metropolis hasting recognize particle mcmc mcmc proceed obtain candidate previous chain acceptance key advantage design problem design carlo monte detail distinguish kernel kernel proposal depend past particularly important paper propose must ergodicity adapt markov ergodicity adaptive mcmc scheme use static know ergodicity present metropolis mcmc factor specify static give comprise component present methodology associate particle covariance I aspect stage bayesian adaptive sampling rao consider study demonstrate calibration represent panel detailed context aim systematically assess estimate perform prior static prior datum parameter utilize process utilize noise path day present panel present day generate accord curve contract day time follow contract contract latent curve ft additionally state run sampler present discard burn interval daily discretization hyper setting additionally assume contract observation around variance jointly different note framework risk neutral parameter estimation filtering latent day simulation rao particle mcmc estimation mmse interval neutral estimate true day contract panel consider day contract day contract improve panel contract panel separate trade neutral trace plot uninformative ht figure kalman filter filter long dynamic trading volatility well include within ht log price versus mmse contract panel contain trading day rao adaptive truncation particle simulation study generate equivalent varied ratio observation section trading day day contract day contract mmse posterior four sir proposal calibrate deviation estimate trajectory estimate four day panel contract day panel ts mmse band next result top panel right price latent state result gray correspond interval price predict contract row contract close contract middle series daily price market sample mmse mmse price model secondly estimate forecast mmse estimation extend short firstly allow short factor secondly develop structural model develop allow discretized volatility derive closed price novel jointly filter long volatility regard rao chain methodology calibration real price neutral process need free trend account price contract time contract denote expectation neutral process must first denote v obtained write backward equation drop condition price multiply backward refer allow express pde enter pde fashion substitution ode substitution grouping divide price still keep option ode constraint get solution ode q equation nd kind obtain solution ode turn ode integrate explicitly expansion dimensional remainder adopt book component j useful integral equal easy integral p gaussian variable study recommendation provide positive bivariate e specification discretization I standard mixed expression series perform strong taylor aspect proposal mechanism sir htb simulation bivariate scheme discretization evaluate approximation obtain sample j give adaptive metropolis rao sir conditional metropolis acceptance increment j perform di v I I py x run sir filter cm lemma mathematics sciences bag north statistic work multi price long volatility secondly additive discretized volatility develop equation develop advance sequential carlo monte calibration jointly regard methodology adaptive rao deal accurately synthetic price stochastic carlo filter rao history economics instrumental introduce attempt inter price price account cost basically inter temporal price storage
identify bi directional arc connectivity construct datum edge group dyadic neighborhood vertex week dyadic arc strength count call make number call arc strength call neighbor neighborhood feature include number neighbor common indicate neighbor call make neighbor consider exist look feature dyadic capture indicate call mark window small indicate old temporal indicate higher active long well consider predictor see predictor short tie could indicate edge persistence decay partly relatively short memory build model predict period time conversely operational decide divide period define decay occurrence period evenly divide build final persistent criterion communication method mine class persistent note task build take derive feature building validate divide subset line build test effectiveness period randomly period set remain training ht model features predict simple decay easy tool ease interpretable classifier social discussion discuss strength network well well mining output disjoint rule assign edge persistence reader familiar approach present result reader familiar discuss decision tree divide subspace axis show circle tree series split dimension attribute good gain formally first ht recursively divide subspace criterion leaf meet generate split classify appropriate branch leaf reach assign classified else circle primary task course reasonable examine classification leave additionally importance define feature description network range call number call call call go proportion call go level neighbor neighbor direct call nd call nd order temporal call time choose correlation network predictive membership persistent adjacent window build predict window predictor question examine among question theoretic predict address edge decay week call median week time substantially lower note early omit great eliminate range edge edge asymmetric call therefore view indicator indicate call proportion total make middle call neighbor turn vertex median share neighbor appear edge direct edge call finally temporal week occur one week median call call call neighbor number neighbor number neighbor number call show time blue correlation figure indicate vertex among dyadic raw direct raw call neighborhood feature indicate simple common neighbor good look note two independent old time look category feature dyadic temporal one exception normalize edge neighbor go neighbor must range trade exception pattern correlation correlation second order agent going connect essentially reduce indicator vertex edge value old edge weak old edge strong indicate make appear really relatively independent dyadic multiple indicator exception temporal remainder wish determine persistent quantify usefulness several approach importance theoretic feature quantify fine grain structural feature link problem gain track decrease conditioning randomness quantity persistent perfectly entropy priori informative conditioning dataset proportion among take similarly negative achieve return feature instance instance log ix appeal intuitive class feature feature edge attribute reveal gain calculate direct extent communication concentrate send receive along drop produce feature dyadic time edge predictive follow predictive ability neighborhood number frequency among vertex feature vertex minimal unable strength tie critical factor raise important merely surrogate strength important concrete c four metric correctly three classifier persistent tie proportion tie belong persistent extent theoretically recall classify tie persistent could perfect classify confident define harmonic precision recall question classifier expect majority decision tree classifier predict tie tie classifier decay precise tie predict decay tie predict tree job persistence term table use regression decay persistence task decision logistic tie model correctly tie big job slightly job predict however precision decay class decision logistic indicate persistence decay pattern model yield fairly prediction relatively present logistic decision predict persistence network ht odds call make call call proportion proportion neighbor neighbor call neighbor call parameter odd odd ratio standard statistically parameter begin gain direct strength persistence additional make odd net call call odd effect member sign vertex chance decay edge vertex straightforward actor persistent effect actor vertex effect adjust raw appear process edge persistence decay turn neighborhood effect measure note early general odd edge slow rate common increase direct call j odd persistence indicate tie value tie positive persistence activate likely inactive mention insight recall subtree implementation attribute subtree mean attribute great choose branch call direct another membership early table stand tree direct weight dyadic helps predict level neighborhood level factor neighbor hand side edge cutoff persistent actor week strong odd classify follow level activate recently persistence drop direct relatively non reciprocal incoming direct weak strength chance cutoff period persistent improve substantially explore decay phone determine determine time adjacent million cell phone decay prediction structural predictor observational range take account total dyadic temporal incorporate relative associated phone emphasis weight find metric total predictive decay temporal indicator edge weight reasonable edge build predict short decay phone contact perform know persistence tie likely activity actor actor something stability network instability evolution degree actor result email tie likely decay recently likely show tree classifier dyadic determine network high decay classifier give combination information gain strength edge range persistence logistic say persistent phone characterize couple activation edge high finally persistence edge gain flow persistence evolution boundary direct old relationship early stage balance relationship interaction guarantee persistent case relationship long survival stage balance dynamic consideration effort framework quick component reference anonymous helpful comment suggestion edge likely address people extent embed age decay large scale phone call week determine importance decay relative power assess active continue period direct type persistence prediction strength agree network fundamental obvious centrality essentially classic behavioral exchange theory edge classic balance analysis suggest observe shorter likely party similar behind influential weak edge argument rare embed fully likely receive process dynamically selective relationship embed transform accounting edge empirically edge consideration make persistent level whole network community actor level volatility relationship major relationship identify circumstance weighted think false positive behavioral email phone rely exactly unclear able predict decay circumstance edge increase availability longitudinal social beginning receive increase recent development actor analysis longitudinal couple evolution behavioral review centrality link phenomenon pattern decay interaction report connect need constraint design limitation validity strategy limit relative site limitation flow use think formation relationship limitation importance contain thousand million actor human communication examine analytic primary second actor produce survey selective reporting significance comparative draw conclusion effort characterize either level analyze formation link little actor leave individual edge decay social prominent prominent within organization ask year year organization frequent substantial business contact year
large dimensionality million recovery classic orthogonal pursuit choose support easy method indeed run omp rip show omp recover restrictive minimization compressive pursuit sp inversion pursuit family hard name suggest stage htp decide one support size notable sp back desire solve algorithm typically exception add remove restrictive rip locality replacement however algorithm analyze unified family hard family positive algorithm htp novel replacement think omp instead replace element prove sparse rip provably use locality lsh provably sub require rip local optima family sp able guarantee sp hope unified light kind iterative modify add element set replace set surprisingly method minimization recovery orthogonal locality sensitive hash lsh element support advantage unlike rip default step optima though direct omp relate extreme member partial operator htp end light thresholding include enjoy omp locality hashing noiseless setting provide least condition guarantee sp omit body paper find omp classic algorithm recovery inner residual square current thus also briefly eq thresholding decrease absolute formally next denote denote matrix denote denote index support vector orthogonal matching pursuit replacement omp instead size control relative two coordinate replace correspond combination iterate iterate note satisfie rip large well solve reliably iterative solver tb input size z j input sparsity level b omp recover appear restrictive condition pursuit sparse provide compare heuristic compare rip take rip say restrictive rip condition recover noisy case iteration converge fx ax ec theorem convergence algorithm turn attention family mild ensemble soon large constant particular integer member replace current correspond connect cardinality hard thresholding operator clear operator reasoning hard search partial fact precisely give operation elementary conceptual current ta ta ax ty solving nice guarantee rip note restrictive experiment degradation recovery increase recover step satisfie e fx ax k dd sketch appendix ta residual proceed element name lose use f ax choose expression monotonically optimum rip satisfied bind rip normalize need need sufficient lemma reduce function hence obtain within least element special immediately time iterative newton thresholding pursuit htp call view newton objective satisfie satisfy newton namely stage element support solve drop form iterate least square support technique proof algorithm hard size recover measurement two provide rip condition measurement provide subspace pursuit pursuit require supplementary corollary discuss hash intuition behind find column view sensitive lsh well near neighbor retrieval lsh near guarantee neighbor able neighbor lsh lsh hash randomize hash create hash hash processing independently stage index hash key function retrieve index state lsh lsh lsh directly guarantee hashing require find setting lsh weak rip hash sub however lsh transition diagram omp experiment robustness exact third lsh implementation require omp pursuit inefficient independently normalize unit select optimize omp newton matlab hash routine file ccc omp phase diagram commonly compress sense literature size problem generate apply recover vector diagram omp newton show code different success blue success gaussian rip universal fraction diagram successful recovery phenomenon respective diagram omp plot next empirically compare diagram recovery newton fairly sample recovery measurement binary gaussian figure level outperform perhaps guarantee fixed varying finally show difference newton level error around ccc pt error various incur provably incur increase ccc kb b incur lsh hash measurement construct offline hash goal signal accurately report require method run increase measurement vector minimize hash bit hash incur hash newton input newton hash perform able newton number recovery newton newton monotonically least minima contrast hash increase table figure compare incur hash converge hash slightly time proceed element name lemma square third line assumption schwarz equation least element case let furthermore every md fa fa add get along get imply element add use since element case x f lx furthermore large multiplicative e neighbor neighbor neighbor nearest simplify hence approximate neighbor follow simplification md md md still constant decrease probability iteration lsh ensure lsh neighbor lemma text noisy q assumption c next value term ax ax dd low q last imply eq use cauchy schwarz one new find furthermore cd analyse exhaustive lemma get define imply definition top hence eq provide lx furthermore choose base entry plug use multiplicative reduce thresholding
trade steady behavior serve template briefly q eq order group regularize square approach mix norm wise vector norm eq index norm encourage sparsity group group coefficient penalize group recursive update approach work use homotopy problem solution eq fix accomplish regularization homotopy recursive group solution ease propagate homotopy apply provide group complement active two part index finally differentiable reach sub satisfy eq notational convenience drop leave implicit eq subspace definition sign comprise comprise equivalent therefore invertible eq directly class set write constant share eq correspond sub theorem provide remain path contiguous piecewise segment propagate segment segment seek maximum ensuring remain within increase condition come condition exclusive move q update relate form determine provide great point homotopy calculate calculate cost point table bound solution action per critical number critical solution critical group lasso zero call recursive lasso sparse vector coefficient location shift indicate gaussian matrix create recursive group boundary average square implement standard sparse regularization also achieve low steady outperform steady sparse superior mse occur across group system cluster shift group lasso recursive critical accounting trajectory comparison method homotopy base trace critical yield complexity respectively develop homotopy solution acquire use previous warm homotopy streaming measurement predictor steady square non partition future overlap derive remain q calculated formula simplification define define definition proof ease range monotonically ensure calculate one one q eq q therefore critical value concatenation comprise part q condition indicate critical value accord critical eq solve develop fast require c positive critical general q exist denote slope segment piecewise generally denote base linearity show sort x n ix nh x seek edu group penalize predictor compute exact penalize recursive minimize develop homotopy simulation outperform regularize group identification direct homotopy system filtering prediction processing field acoustic wireless interference
compute ep end minute result cpu assess condition generate sample compare response stimulus separately position compare successfully answer function sample glm link predictive confidence fit reaction versus reaction grey dot line offset right dotted capture data reaction look mean distribution reaction indicate abc posterior cause concern behaviour abc explore suggest problematic shape posterior bad problematic begin toy multimodal posterior iid hand variance line distinguish thick thin versus ep abc ep abc course vs pass vs st correction ep abc ep abc iid obtain thick pass stop execution completion site negative definite slow update line figure slow show type assess correspond site perform pass site seem slowly try run pass invertible behaviour thesis run ep abc fail cause failure definite problematic principled thing could correction ep relatively distribution use build correct toy modal example correct serve third gold standard available manner goodness distribution answer much separate ep produce estimate manner ep might lot mode exist might previous unimodal behave theoretical convergence ep behaviour adapt reaction would scale probably reaction model imagine pick reaction category high accumulation speed create speed vs threshold generate plot still mcmc inference accumulation rate two category uncertainty uncertainty run strategy ep abc probably condition matrix arise eliminate choose million acceptance pass minute stability abc reaction mark kernel density check summary statistic reaction time space composite also context replace posterior marginal observation negative exist straightforwardly abc treat make describe make concrete class likelihood intractable hence marginally may take illustration alpha tu stable fix skewness scale parameter gaussian suggest transformation fourth create successive block composite approximation composite take possible subject time alpha average abc abc posterior constraints metropolis abc filtering running time three six hour iteration particle alpha stable especially reasonable source consider gold approximation different set constraint ep abc cl apply size prove model could likelihood version ep composite likelihood spatial distribution tractable ep abc deal marginals ep approximations block dot histogram posterior limitation likelihood posterior poor multimodal one scenario mathematical ep assessment understand start recently address error certainly important remark first ep empirically free result base abc designing site summary must ep away completely take seem quite application summary limitation abc future ep helpful author author ep generic transform p exponential family distribution describe simply compute p create hybrid natural site moment generic exponential family stable update list family essential derive hybrid h product case hybrid hybrid calculation yield hybrid dimensionality correction ep approximation henceforth correction correction derive expand correction truncation might everywhere although arise application un normalise correction particularly ne il integration easily obtain product quantity ep converge happen site matrix correction expectation improve correction nearly abc expensive discuss social science efficiently bayesian still thank approximate abc abc routine often rate introduce choice make quite user ep know fast magnitude standard produce ep abc replace statistic iy iy point possible summary entirely ep likelihood free novel abc word bayesian free quasi science example include biological choice economic evolutionary biology environmental intractable like usual statistical traditional tool directly introduce explain often quantitative analysis reproduce free context version abc conditional keep otherwise reject vector summary statistic quantile sufficient vanish p suffer two error density play small induce increase lead somewhat although increase acceptance e g matter analysis may take tune real problem automatic recently problem still one sometimes pilot introduce ep abc adaptation advantage ep may hour day ep requires decompose possibly way sequentially ep abc convention way operate ep replace possibly abc simple dimensional summary whole iy I iy small course amenable least detail paper discuss genetic collect recommend construct theoretic figure supremum equivalent impose simultaneously abc simulate dataset abc resp minute minute number transition obtain ep density obtain ep run apply use pseudo actual sum perspective equivalent process indicator replace abc resort mcmc sampler design walk marginal approximate marginal sampler big situation approach markov report day cpu simulate transition correspond practically ep abc black latter difference abc corresponding bit close reaction behaviour subject subject stimulus move move alternative observe measure reaction random reaction time information decision reach accumulate variant evidence illustrate reach boundary determine response wiener independent wiener process reaction corrupt represent time subject execute answer r r credible capture reaction basic experimental describe allow vary trial trial mechanism reaction assume boundary stop high determine need account subject
g institute brain regularization model combination approximation effectively even numerically lasso recently pmf show pmf yield reconstruct inconsistent input correlation globally fast pmf problems input scale cubic introduce multiplier formulation cubic close linear dimensional inverse variance least square ol vector covariance problem approach typically uniquely addition ol well get replace rank optimize validation improve interpretability lasso linear ols interpretability regularization prior shrinkage ridge result optimization problem long approach variable narrow spike center distribution high thus mcmc bayesian approximation posteriori tend slow propose integrate binary selector remain analyse form absence prior motivate organize variational variational term solution define control prior insight solution compute close resort approximation variational variate sample sparsity phase plot solution behavior close argue variate map lasso regression pair pmf recently pmf outperform example correlation level irrelevant globally fast pmf tends regression p input x factorize eq q since affect identical spike contain representation detail compute due variational pd likelihood spike selector selector bit n spike substitution w use identical parametrization parametrization require prior equivalently involve quadratic bp solution base bp variational approximation energy find bind width simplest factorize specify expect term first term line prior give set equal fix eqs algorithm variational procedure non solution minimize n non eqs expect replace variational equation would expect ols normal ol I variational mechanism depend dynamically adjust thus rank rank rank remain note fix choose validation prior help step obtain increase implement anneal sequentially empirically minima far low energy eqs well alternative dimensional eqs require repeat summarize minimal cross validation pn I pass cross validation input uncorrelated map without resort sort otherwise optimal eqs interpret variational eq explain total reduce explain interpretability factorize although coefficient eliminate correlation number sparsity illustrate appendix transition solution unique line exact right bottom low corner corner unique dot solution solution minima stable well low indicate order variational solution co separate variational result effect compute decrease solution variational easy modal small negative inaccurate multiple minima region dash dash interesting variate ridge lasso ignore deviation depend ridge prior negative solution depend difference ridge shift identical behavior interpret soft variable identical plot extend variate orthogonal breaking except term increase except depend way plot qualitatively variate compare regression pair pmf input variate gaussian specify structure optimize optimize ridge quadratic lasso pmf regression pmf optimize dataset input pmf ensure pmf take input teacher minimize correct range solution fig plot minimal variational run solution low error fig component correct plot versus minimize fig feature bad remain large sparse row bottom row regression middle b solution ridge versus r r train lasso significantly outperform ridge prediction parameter non difference lasso pmf lasso solutions pmf input multi gaussian compare pmf instances r ridge pmf pmf outperform lasso ridge prediction pmf lasso pmf randomly correlation weakly input fail pmf limit pmf keep constant correlate input get minima instance yield instance generate randomly component two strength correlate correlate pmf initializations lasso inconsistent consistent consistency different respectively fig inconsistent versus range bottom leave bottom right r b optimize trial inconsistent yield large quality lasso bad example suffer find remarkable might sub minima pmf pmf approximation consist predictor include value pmf pmf denote hyperparameter see pmf perform fix initialization random pair sampler pmf pmf pmf approximate truth error pmf together confidence initialization show obtain small observe pmf two solution soft initialization showing initialization agreement contrary pmf always evidence pmf analyze input practical relevance genetic scenario active active run plot area roc definition method except pmf train pmf low theoretically generalization target figure operate roc calculate weight inactive roc positive fraction threshold area classify perfect function number pmf regime poor regime column shift value pmf regime pmf slightly well area roc pmf ridge lasso pmf correlation average run plot roc mse pmf validation pmf inputs nucleotide nearby highly correlate distant snps strong prevent size pmf preferable comparable pmf interestingly difference pmf minima pmf feature conclude analyze method scale cpu figure show pmf describe appendix fig pmf constant
estimator see proposition give influence q remark influence function ht unbounded limit reliably form calculate generate normal distribution sorted iteratively repeat value value mle parameter robustness influence clearly greatly example involve nice unimodal excellent exhibit histogram maximum ht conduct explore generate robustness median well divergence hellinger estimator include huber location show analysis model estimator case good mle term theoretically note asymptotic size perform mle comparison various table high contamination small outperform mle decrease dramatically aim investigate divergence univariate exhibit behavior evaluate show solid normal keeping demonstrate amount contamination provide good maximum paper divergence theoretically robustness leave study open estimator divergence location member offer attractive maximum robustness key phrase divergence estimator robustness estimator divergence widely model framework context divergence appeal bandwidth way continuous smoothing benchmark condition full divergence robust test approximation divergence censor introduce estimation copula refer divergence estimate function location concern article suitable capture robustness insight attain huber without efficiency background concern devoted deal remark divergence absolutely divergence kullback give divergence divergence chapter class overview origin interested divergence recent simple I interest underlie dominating function exist eq finite mp define property mle coincide mle leave divergence keep parameter divergence contamination crucial outlier illustrate empirical contamination shift contamination parameter maximum remain close divergence associate divergence mle affect outlier robust divergence equation improvement estimate robust estimate ht contamination outli outli term stable evaluation value situation assess robustness estimator
way counter multiply bootstrap get conservative large come quantity weight create bootstrap weight prefer yield thing equal prefer bootstrap variance well weight uniformly nz nz naive simplicity observation implement replicate quantity weight index replication represent possible stem delta usual iid holding fix nz nz delta method bs double stability proposal give example get share pt reweighte mean delta z refer take product derive exact introduce k z u un fraction match exactly triple match matching effect model coefficient ideally would bootstrap biased typically conservative sometimes interesting instance trivial negative bias column resample bootstrappe name page method way paper noting model call weight exact gain interpretable netflix worth point arise form sample level factor greatly dominate internet factor country aid interpretability construction match netflix datum dependent people name phone many even phone people phone fraction simplify may expression retain explicit constant gain find bayesian ordinary bootstrap variance setting reflect additive nonzero coefficient reweighte bootstrap variance hand ed bootstrap gain coefficient nonempty mod term netflix variance extent differ variance bias inference unbalanced entity systematically realistic produce error tend un next development effect mod give exact gain interpretable bound gain term q gain reweighte q bootstrap conservative dominate every analysis use small sum exist fold random share index index replicate nest within index nest outer ordinary word include outer nest another subject outer contain repeat measurement ratio outer length facebook user highlight substantial presence additional effect three reject would quite interval portion conservative worked conditionally hold clear informative thereby introduce mean correction netflix company rating unobserve rating people netflix try rating make pair predict bias facebook difference comment comment accounting involve infer comment make comment conditionally describe stability comment length adjustment kind mechanism necessary value sampling bias adjustment build bootstrap adjust alternatively statistic bootstrap seem effect bootstrappe identify suitable bootstrap plain practically important example click rate usage bootstrappe biased effect set product weighting replace acceptable properly calibrate variance multivariate replace variance covariance consider bootstrap extend smooth expansion use confidence interval estimate central properly calibrate correct bootstrap percentile confidence conservative percentile way via estimate define mx mx practice histogram set loading depend let describe l loading make could contribute generalize svd extremely contribution handle bias bootstrap phenomenon bootstrap inference accurately appendix contain easy theorem statement preserve proof q numerator z z model u naive bootstrap naive resample theorem stability naive variance effect x z n effect expand reduce z w z sum vanish establish variance hold nz nz variance bb bb w n n theorems approximation coefficient theorem show factorial line eq un un establish interpretable variance reweighte z w ii w distinguish next turn proof k effect ed gain nonempty exist contribution similarly u theorem begin analogous expand assume prove n z establish proof theorem last z model coefficient product reweighte next r reweighte dominate j eq j r j comment support nsf dms bootstrap estimate take set independently observation independently conservative biased variance unbalanced apply computation parallel illustrate comment facebook effect commonly arise internet service automate draw netflix rating neither iid interaction internet easily two factor account string document place web page measure spend read load page version rely balanced necessary might mechanism specifically independent mean random broad jx ij pt approach bootstrapping independently bootstrap bootstrapping put level factor w r j j j bi j get bootstrappe bootstrappe row usually relatively balanced missing remain conservative sample unbalanced random effect every column interaction resampling analyst variance product I ji scale computation bagging reason fashion multinomial bring substantial synchronization cost expression dependence distribution easier develop effect main variance easily computable give explicit formula naive bootstrapping estimate instance netflix resample column close n comparable acceptable contribution naive strategy fact resample generalize become reasonable component roughly simply describe dominate random every nonempty random bootstrap effect distinct variance bound away outline introduce observation define seek naive bootstrap unless high case bootstrap factorial closely match resample interpretable consider combination variance effect dominant bootstrap closely match variance repeat nested one reweighte facebook uk long comment mobile device reverse interface significant factor structure appear product effect discuss among reason variance random effect contain interest integer notation write mind level internet ad priori compose value observation practice order estimate apart brief subset sum name hold random depend u u x v writing general expression denominator goal resample different observation index convention quantity important dependence pattern j j match say highly perfectly customer phone specific motivating define pair match
purpose must hash orthogonal hashing projection rely pseudo practice hashing replace hashing modification time trivially conduct line pure research notice truly loading dominate memory testing svm benefit hash gb preprocesse cost loading hashing reduce bit fraction loading cost still load subsequent report experiment certainly would experience mining community hash especially similarity mainly work view either two set use one apply hash bit storage prohibitive truly application efficiently express inner product immediately hash inner extremely concatenation total indexing use development bit hash simple solution store low bit convenience datum highly example small highly large replace bound extensively independent permutation encode often e provide logistic popular package regularize solve optimization regularize important penalty purpose demonstrate bit simply conduct approach store bit value way store merely bit run suppose originally binary digit expand feature fed solver experimental follow hashing randomly select testing sample gb dataset expand dataset product combination product choose tool effectiveness experiment conduct cpu ghz gb ram windows l gb testing expand datum also test expand training wide fine mainly accuracy hope entire gb resource accuracy accuracy plot result one figure verify svm merely accuracie h dimensional sketch algorithm variance rp review rp convenience vector product general multiply e g generic ij er q satisfy small elementary know point distribution probability refer count sketch algorithm vector introduce write biased inner correction multiply entry situation multiplication hash er er er vanish essentially option pre multiply course variance become number dense probably bit need time hash square norm inner size gb logistic solid representative bit hashing achieve substantially bit dash panel plot bit advantage bit hash hashing training hashing suggest bit hashing far apply additional bit indexing extremely large number conduct notice indeed reduce hashing widely hashing require practice well understand use good hashing simulate permutation preprocessing require time gram trivially hash substantially consumption store example hash disk task near etc example learn logistic truly large datum loading dominate loading gpu loading without gpu preprocessing time magnitude train gb expect loading time preprocesse gpu preprocessing cost loading gpu gb conceptually hash require mapping store mapping application however store permutation would infeasible permutation require store popular hashing uniformly store store hash e avoid modular gram gram location walk parameter deterministic simple permutation store permutation practice permutation hash solid svm right curve simple lot bit applicable hope draw interest research search microsoft microsoft microsoft com work use bit gb may compare access paper study expand report merely per
contrary seem execute effort routine mathematical device teacher truly system fully supervise supervised reinforcement ai ability attribute new teacher letter suggest architecture way prototype dynamical shape external process biological knowledge scalar time represent certain origin visual sound pattern algorithmic adjust strength weight pattern profile form phase space nn fix new occur nn evolve towards technical occur formation spurious unsupervise considerable propose construction dynamical shape external without supervision stimulus ergodic dimensional probable new like place profile move possibly affect space base loose analogy slowly shape pressure assume initially flat profile fig drop distance elastic factor capacity forget deeply fast try forget online memory external stimulus moment drop position stochastic nature arbitrary derive profile consider interval shape describe g function q however show shape condition leave behave decay fluctuation trend sort change evolution profile b circle apply delta combine go infinity tend dirac delta kind peak shape line consecutive correlate actual circle eventually uncorrelate fast dynamical algorithmic restriction evolve recognition phrase child song six song involve three choose illustrate principle wave file agreement speech slide sec roughly peak extracted note frequency slightly frequency respective note introduce see individual automatically hz save stationary ergodic consist song finite line automatically probable frequency show hz hz hz hz hz hz phrase phrase signal phase ft ft ft ft purpose finite multivariate visualize way fig take coincide axis connect feasible pattern point whose probable five scale also wave file create machine digital computer neural device close propose unsupervised traditionally however supervision I regime ai device pattern combination major curse complicated ai device grow become connectivity curse work duration calculation device curse acknowledgement critical comment play approach construct
method tree interaction associate associate freedom though forest offer construct tree quasi random property underlie causal causal predictor adopt subset ridge limit penalty arbitrary alternative preference reflect pick category interaction contribute relationship accurately describe decision model identify logic spline restriction product fall category place restriction convenient formulate consider relationship f g kx disjoint predictor allow restriction partitioning expand copy accordingly require belong useful order group f determine predictor nan group interaction term vast hope relationship determine fortunately detect contribute usual formula partition nuisance pt base construct g association multiple copy keeps assign describe multivariate response trait response logit link marginal parameter require exhaustive partition infeasible sized use chain monte carlo statistic propose component propose processing spend posterior almost speed methodology provide section material carry design various comparison exist detail simulate remain material typically predictor causal response ask three examine predictor scenario concentrated underlie relationship ii iii particular frequency predictor multiplicative third involve present relate underlie average causal different predictor figure interpretation reporting successfully detect causal well clear increase essentially maintain drop second iii come perform simulation prove robust relationship violate generality performance pt previously study individual examine set http university www ac uk supplementary individual nucleotide expression level dr di interaction snps million present snp display bottom report circle run triangle dash vertical mark snps interact provide threshold top association test detail text promise rs rs association search highly correlate often fine genetic snp rs former snp latter posterior interaction rs indicate project pilot trait focus gene decide eight original phenotype correct population choose miss approximately unobserved plot bottom plot main text pt method strong association suggest single appropriate mark vertical suggest strong association allow contain explain variance variance examine deal encounter project expression know snp association within region mark achieve genome association approximately suggest spurious gene association rest top truncate project release across five snps subset g contain reflect event snp allow miss infer reliably et decide adjust true plot figure result vertical find association give great recognition possible causal allow identify concern know case hope verification count heterogeneous stock snps length using design value unobserve provide gender code genome decide cd link g decide copy demonstrate top due linkage rs indicate top snps repeat show run offer plot snp pairwise gender association significant test interaction snp gender snp act gender specific agree allow multiple copy option material fitting predictor group interaction unable distinguish say way interaction copy predictor recommend tune superior power iii try believe simple model method compete fairly strength demand many search interaction benefit consider existence strength derive generality perhaps suffer complicated disadvantage underlie certainly mcmc estimate scoring visit error seek estimate use bayesian mechanic generality posterior correct space partition unfortunately small force visit force instead able try variability introduce compare expect strong association repeat random highlight obvious trace plot additionally permit repeat association scale linearly require base true suitable predictor pick high certainly true association standard pick perhaps top biased predictor reflect simulation five supplementary almost heavily demonstrate drawback treat predictor natural overcome application pursuit develop additionally influence inclusion neither fit chance fit offset inclusion imply unlikely predictor appear consider would moves resort exhaustive predictor suitable transformation reduce continuous measurement neutral increase decrease hope whole detail let nan belong unique order irrelevant ordering single set model predictor expand copy predictor alternative keep predictor relax alternative discuss interested observe fully predictor calculation nuisance pt partition construct probability predictor equivalence contain partition predictor ensure marginal assign equal calculate predictor way th contain calculate recursive fashion bs bs q weight member place even straightforward could favor interaction must largely due couple reasonable allocate calculation ensure extreme size practice vast amount unnecessary assume summation equivalence achievable condition hold involve prior ks ks g p ks sp g ps ks sp n cumulative value find copy predictor allow association copy set copy associate multiple copy predictor element partition partition copy effect prior relationship increase necessary specify minimal recommend therefore underlie write regression f k functions assign penalty categorical belief control extent smoothness residual integral incorporate reflect preference small improper binary logit
algorithm contrary value jump opponent opponent jump drop slightly finally next jump approximation last opponent tracking iteration two behaviour opponent approximated number iteration factor play become identical fail adapt equal play opponent jump play drop account lead opponent article performance geometric play player hill hill game risk dominant nash c ccc episode payoff end replication parameter track pre ia simulation smooth allow factor whereas play play nash replication concern geometric play payoff factor play nash equilibrium difference depict propose play geometric target assignment game agent coordinate common total target specific action target target many choose action target value utility target product express utility produce life receive target greedy simulation inverse target target sample fix end information update strategy target action depict instance geometric play instance step score instance score good equal need reach reward management simultaneous occur area people collect people capacity capacity formulate player one allocate variant utility objective save express utility add people player one player help follow available scenario reach generally action time need reach one people trial scenario easily unnecessary approximate response solution algorithm branch base ratio could variation play gain mean variation bad percentage overall percentage people c c c scenario geometric play coordinate percentage group play regard percentage trial performance adaptive play decrease observe play scenario include structure one time penalty great utility variation play initially search utility need local optima influence instance stop play geometric c opt stop play c opt time geometric factor play percentage collect step even people geometric area iteration result play perform adaptive factor play perform rnn percentage overall people percentage rnn collect percent play percentage factor play percent evaluate penalty penalty inefficient allocation exclude ratio evaluation especially play classic form incorrect give recently stream datum examine impact play choose carefully induce poor combine low weight estimation drive adaptive factor hill assignment management play play planning use empirical play intensity play david department mathematics uk game learn game play play implicit variation allow realistic opponent streaming literature observe geometric play variation theoretical optimisation optimisation sensor traffic scheduling domain communication complexity optimisation optimisation term nash versa maintain belief belief choose expect reward strategy action equilibrium game practice assume particle predict filter difficult paper adapt action empirically step play need hence communication overhead optimisation demand propose classic start description game section present result hill game target management scenario finish games optimisation optimisation element game joint player choose accord select action deterministic choose act strategy action strategy mixed element player gain choose resp resp decision player choose action expect belief strategy strategy response show least equilibrium equilibrium equilibrium imply possible player utility change strategy select equilibrium action pure pure nash equilibrium particularly category game agent game utility follow stand player player game equilibrium life design individual utility potential act utility obtain select obtained arbitrarily choose eq optimisation game prove equilibria player increase play game play choose response belief opponent player belief choose action player update belief good belief beginning maintain arbitrary weight function formula opponent follow belief play nash classic play strict play game steady play payoff game game nash equilibrium become period use use response action choose action strategy case player rare player mix variation originally poor serious change filtering mean depend descent respect residual error online stream generalise factor adaptively change ascent log new sequence stream parameter write arrive factor express fitting update instead classic play maintain belief place time action discount weight belief play similarly geometric play close action classic rule consider opponent form identically independently opponent opponent strategy respectively update recently observe eq order derivative j adaptive ensure leave player action belief response play player carry weight j accord response opponent factor lead poor ascent lead algorithm area result big jump solution evaluate play combination opponent stream opponent strategy repeat time time measure square range combination depict dark less value great approach wider observe great square suggest value behave play game easily opponent independently poor estimation opponent
exactly none fix true marginal algorithm bp perfect marginal exploiting assume correct turn gaussian match one component performance ensemble accurate still available hull reach equilibrium bp consistent increase bind observe equilibrium propagation learning proceed bethe algorithm cause converge learn marginal project cycle circle onto space color inconsistent black belief precisely marginal red even amongst fix marginal bp red concentrate comprise h statistic first ix positivity appear ij interaction bethe hessian figure ise couple bethe hessian width pdf marginal bias marginal measure bethe select bar median bp performance encounter bethe iv last gaussian distribute statistics iv generate ise target select bp ensemble bp method parameter run message time step message change belief bethe learn performance ensemble iteration bethe learning belief poor target loop get marginal guarantee selection target belief give excellent use give order limit gaussian ensemble make bethe belief ise unstable bp hessian throughout bethe convex bp fix yet might hope belief propagation systematic bp hope provide work fix point average preserve locality scalability bp raise possibility inference generalize novel novel input belief use fail example happen bp failure thus address marginal available marginal exist variable hide phase engine brain inference perhaps belief undesirable circuit big blind reasonable inference draw precisely occur bp blind eliminate mechanism perhaps fluctuation average thereby conclusion mechanism acknowledgment university york ny york definition belief loop one might hope adjust purpose explore previously claim marginal belief probability marginal propagation whenever hessian bethe energy definite produce inaccurate belief belief parameter perturb learn mean calculate generally require sum exponentially np hard one belief bp product pass algorithm efficient merely approximation model loop appropriately achieve belief case ensemble marginal p write index interact variable sufficient statistic irreducible write sum linearly depend collect match marginal algorithm energy use converge belief must graph equal graph belief guarantee ib globally joint belief marginal belief simply belief circumstance belief bethe energy kullback leibler energy optimize energy appropriately depend entropy minimize recover energy entropy entropy neighboring factor structure graphical bethe entropy bethe energy bethe marginal nonetheless bethe energy often enough gibbs free minima approximate marginals minima bethe free successful bethe free write satisfy positivity consistency propagation matching marginal simply condition bp reach bethe span belief bethe hessian width pdf slice bethe free energy line axis space parameter energy add minima second free energy identical parameter slice bethe colored bethe hessian bethe learn belief blue along point region jump region exist binary energy symmetry target marginals bethe eigenvector thus bethe marginal occur minima bethe marginal exist actually quite multinomial loop definite small correlation definite bethe pseudo moment matching fail descent procedure descent leibler free bethe divergence energie bethe energy optimize propagation bethe entropy energie belief propagation obtain belief current change opposite direction generally bethe free belief decrease allow draw
issue action policy round subroutine application sensitive oracle work context modify argument use probabilistic net cost thm sensitive require tractable would common reward feedback delay round straightforward modification proportional delay policy definition domain let arbitrary round choose reward action round choose learner choose reward short learner set enumeration policy impractical general illustrative second option oracle correspond learner policy reason sensitive interpret negative cost policy denote maximize learner produce nature history take policy follow relative learner internal randomness regret policy denote context shall give context shall action etc correspond randomize x b tx choose ta step policy induce value theorem find specified method algorithm step project onto smoothing finally determine analyze first game player produce find allow randomized policy correspond space randomize policy hull feasibility randomize inner expectation convexity expectation max without value hand satisfy non existence see constraint policy elimination quantify show policy yield eliminate yield second fix inequality denote conditional obtain analyze monotonicity yield least round history approximately value satisfied slack choose reward minimax keep explicit good policy version eliminate chance recover perfect address short present ucb keep action choose explicit tracking avoid implement suboptimal frequency use previously effect follow round appendix variance policy quantity incur round induced quantity draw high probability bound value round direct round mostly policy bad bad policy allow estimate analyze show relate reward defer section solve describe polynomial optimization ellipsoid method solving equip separation separation oracle produce side ellipsoid standard center decide empty give two empty processing use ellipsoid oracle fix contexts kx px convex program eq equivalent optimization optimal follow require interpret distribution equivalent solve search testing feasibility detail complicate need requirement lemma leave oracle separation oracle first consist separate hyperplane constraint convex set separate separate point oracle check expectation least section show argue class close result positive write average delta function approximate choose drawing denote least let q v follow gm fix eq eq q distribution z x mean z mp chernoff third fourth final display observe maximum second display fact inequality follow display lemma respect add existence proof determine fix guarantee eq factor bound reward quantity check index round effect allow slack constraint constraint slack e kb state slack somewhat arbitrary slack appropriately eq bound pick follow deviation pick least let easy since union bind pair next lemma imply since lemma imply satisfy slack combine k inductive deviation suppose contradiction fail fact contradict must immediate show policy large round round assume pick third inequality lead existence feasible non small eq compact condition otherwise prove round trivially sum condition hold lemma union approximately ellipsoid fix drop subscript become relax ensure region constraint set point cauchy schwarz fact etc b form find point call relaxed program ellipsoid recall requirements radius radius ball region contain feasible feasible region feasible consider point assume eq thus hence give ellipsoid perceptron one candidate point feasibility check constraint constraint separate violate hard recall follow follow iteration hyperplane separate policy satisfy hyperplane put algorithm candidate need shift origin perceptron update separate hyperplane assume perceptron separate hyperplane hyperplane fact run algorithm convex hull perceptron step policy euclidean projection quadratic finally give else rewrite define xx check constraint ellipsoid run ellipsoid program hyperplane separate separate ellipsoid fy ellipsoid conclude infeasible feasible notational convenience define separate ellipsoid start ball solution constraint check construct separate hyperplane else check separate else point perceptron policy lemma ellipsoid lemma every lead algorithm lemma thm corollary claim definition problem observe receive optimal oracle setting enable delay previous bandit follow loop context information action action world present reward bandit reveal choose several feedback prefer essence set difficulty avoid reinforcement bandit half half motivate contextual interesting news article ad internet naturally model bandit medical generally imagine future essentially bandit consist distribution choose algorithm set success compare policy regret computation policy policy space computational hope hand reveal classification sensitive regression classification efficiently bandit similarly large bandit sensitive oracle
number write exist similarly write let let correspond shall assume self many time denote embed throughout fix positive distributional fall depend manifold noiseless allow manifold nontrivial fix compact set hausdorff I measure gm attempt idea model model noise estimate homology group minimax case noiseless fix satisfy tangent plane uniform measure item g state manifold proof use sphere top result manifold sphere around bottom construction make rigorous q tight case omit point seem seem actually occur estimation circle indeed rate smooth boundary remove effect recall clutter model give denote direction tangent positive mod tie maximizer vc hyper least let high see radius empty contain ball hence lemma sm ns mx sm constant support hausdorff compact dimensional gaussian gm since manifold hausdorff could unbounde truncate q enough tm u u picture density bound identity g g I dc vector c du du u du kt dirac delta generalize corresponding integer kt kt hence le ball center origin cauchy schwarz q u e conclude conclude eq special function estimate agreement however technique quite proof technical deconvolution zero around origin simplify considerably thought estimator let define calculation j define define let dt dt dt dt dy kb bx gx gb fix h u ga pt define complex dm nn follow combine h dy lm lm lm lm n prove deconvolution particular slow consistent wasserstein case form regression deconvolution rate conjecture estimator might rate slow manifold deconvolution error purpose section recall model singular manifold recover usual deconvolution support dimension regard deconvolution favorable bound ordinary versus show typical favorable bind ordinary deconvolution top top plot distance density bottom plot nearly indistinguishable favorable pair need density favorable manifold density rather plot perturb circle hausdorff density nearly indistinguishable fact variation support relate manifold regression measurement observe usual convergence deconvolution indeed nonparametric measurement error let however moreover manifold lower favorable follow convolution refer convolution generate favorable pair use case favorable fan subtle way eq choose q drawing reduce error special like usual upper construction manifold except try methodology realistic additive deconvolution know least seem realistic proxy work remark nsf dms national support nsf dms air fa risk hausdorff several connection class area machine yet basic manifold minimax risk hausdorff estimate assumption riemannian include compact nonempty noiseless noise far usually call study singular put mass
separate treat calculate I kde represent statistically source heterogeneous population kde diversity support diversity still support variation heterogeneity power dependence help determine composition variation likely versus contribution diversity important concern f exclude pdfs differently relative bin study evolution vary important like population include also convenient keep point represent pdfs graph gauss quadratic gauss interpret diverse variety circumstance map understand unit gaus non domain set include quadratic series set gauss set baseline pdfs within versus totally effect bias calculate quadratic support diverse amongst pdfs identical thus index save computation analyze l l l c pt source quadratic gauss map value gauss decay expect support variation detecting support diverse support representation disjoint set six table notably maxima homogeneity support totally furthermore average aggregated different conclusion sum aggregate base metric diversity uniform purely location information compare intra nearly inter agent intra highly disjoint intra show case nevertheless detail pdf aid first consider variation pdfs sensitive sensitive considerably pdf metric slightly interpretation component population scale versus population dominate overlap instead disjoint much close compare identical system support normalize useful proportion diversity diversity var boundary range analysis variation support contrast support variation especially quadratic gauss aggregated set homogeneity heuristic detect pdfs meaning detect pdfs moreover like diagnostic expect none mix effect support effect compare map set point metric diversity homogeneity quadratic end metric diversity show homogeneity perturbation especially one however primary set upon contrast diversity among pdfs entirely surprising great pdf estimate size highlight display motivation base moreover converge analog rather decrease mean sense add always relative likely datum considerably base therefore aid point analog analog target datum induce add infinitely string existence introduce divergence estimator true thus exist sample say string bias bias aggregated datum homogeneous structure represent infer whether population ref presence correlation simultaneously population composition difficulty previous datum particular patient repository patient measurement range measurement patient collection patient least visualize normalize pdfs population pdf plot wide pdfs fig motivation choose moreover value likely homogeneous compare snapshot difficult situation existence extremely broad patient likely substantially say second representative small effect kde considerable patient measure see time consider conjunction say differently relatively member fact reflect variance order versus range value pdf diverse less sensitive pdf variation also population nonlinear evolve measured population must care constant know representative temporal five notable peak interesting population outside window informative hour peak presence human appear less scale long drop work third comparing observe difference homogeneity amongst combine qualitative somewhat homogeneous population enough resolve signal considerably fourth aggregate peak usefulness aggregate measure evident compare histogram kde estimate fig identical aggregated calculation display strong error evolution valuable otherwise interpretable context hour day fig kde extremely periodic evident fig bias kde imply presence daily clearly evident kde average time theoretic reach population hypothese seven heterogeneous homogeneous seven patient regard population code act proxy composition assign patient two frequent member six occur pay patient drop patient frequently code drop code drop homogeneity broadly speak code conclusion draw theoretic static reveal heterogeneity algorithmic computer diverse non measure length diverse likely interpretable population diverse picture particular calculation imply information homogeneous filter frequent measurement yield homogeneous correlation patient likely population require nevertheless analysis result paper address within framework I multiple iii distinction stationarity case handle section ii difficult detect state ask amongst fail multiple issue need multiple piece standard paper series differently single source appear know single state amongst individual support relative paper mean whereas likely heterogeneity say pdfs point per one upon variation source perform calculation future generate individual eventually integrate scheme mathematically population nevertheless detail remain list include regard technical claim etc relationship imply propose practically complex series leave interesting question thank carefully read useful comment acknowledge financial grant lm begin recall recall pdfs define abstract pdf pdf next convenience ip ip write term collect form begin recall average abstract relative pdf convenience define follow never value summation collect collecting support diverse well represent support diverse disjoint support homogeneous determine heterogeneity well population etc contribution quantity source set delay composition origin primary delay mutual delay aggregated tool detail delay understand heterogeneity population nearly time university health record repository picture composition aggregate set pdf understand series normalization deviation demonstrate study human series practically theoretic insight composition time dependent degree homogeneity human measurement aggregate measurable system measurement way treat lie average ergodic require gain much aggregated population element advance make physical statistical mechanic aggregation context fact often average quantify problem homogeneous produce aggregate whether aggregated population mind delay mutual understand density quantitie iii diverse manner apply possibly diverse need aggregate ref short focus receive care center clinical measurement human population analysis limit broadly split primarily background focus characterize second characterize propose metric demonstrate read theory devise interpret delayed series motivation come understand health complex biology record repository university datum collect future human contain regard million contain laboratory difficult particular correlate state population type disease phenotype population understand disease disease evolve define completeness medical record benefit wide spread carry practical gain wide biology represents term physics motivating complexity laboratory science nearly datum control many easily dependent context g begin assume discussion pdf note approximate estimate density specify support pdf lie always exist pdf may seem critical pdf auto delay average aggregate member interpretation employ kde calculations develop ref kernel bandwidth estimator relatively insensitive qualitatively moreover use point amount permutation order pdfs detail address sum individual represent essence integration perform define note random want hold interact statistically necessary interact copy independent verify merely apply conceptually correct note product dimensional orthogonal thus theorem conclusion individual average understand pdfs value individual element aggregate pdfs length treating series calculation add mathematics source collect individual specifically pdfs individual include point individual specify specify delay calculate intra column denote dedicated quantify implication aggregate compare aggregate sample versus aggregate methodology numerically iii bias aggregation estimating density proportional poorly measure compare understand datum point broadly average aggregate computationally calculation calculation suffice pdf style bias follow note datum carefully quantify estimator versus bias pdf aggregate population differ evenly time average consider scale linearly contribution eq obeys law difference bias estimate say satisfied element poorly population population importantly converge cardinality aside overall effect presence kde name estimate close kde contrast histogram estimator probability empty yield finite simply kde histogram working understand context poorly measure aggregate population estimate carry calculation minimization bias attempt fundamentally estimate random permutation mix data replacement random population thus preserve inter individual finally exist bias exist aggregated context randomly regard individual replacement vector population correlation relatively pdfs think estimate support pdfs population individual bias estimate discuss dependent point general paper dependence apply use pdfs collect exclude use pdfs calculation whose frequency filter consider example population differently individual population hour second subset sample month represent patient month calculate month represent graph two population month complicated patient particularly health care patient patient possibility filter calculate estimating pdfs change pdfs population estimate notation represent pair individual total pair iii individual population monotonically note normalize square quantify composition member uniformly one composition entire population close composition subset possibly individual quantification percentage individual contribute homogeneity sometimes always correct really give specifically bin representative population resemble median among among abstract definition mean relate get focus q q explicit go relative support population represent write briefly auto information calculation series define thus tend toward understand aggregate ideal stationary source circumstance obey represent intuitively say create aggregate pdf pdf closely resemble helpful concept relative actual population separate naturally conceptually abstract support aggregate one spirit pdfs roughly imagine achieve goal relative severe aggregated define abstract quantify relative abstract pdfs difference thus difference define arrive drop explicit appendix go zero width band pdfs decrease overlap individual homogeneous population band pdfs disjoint population say aggregated population follow second difficulty interpretation calculation yield explicitly completeness define aggregate analog q contrast tend information individual pdf mean information section speak broad practically ii least element leave conjecture accurately represent individual population identical statement population briefly prove conjecture stationary rely eqn homogeneous essentially one population make understand understand source tie population important differ contrast population split broad support iii difference estimate support pdf aggregate collection understand graph pdfs difference permutation population roughly estimator regardless support small next individual permutation plus estimator contribution bias approximate wise zero order visually become gaussian amount primarily drive imply support circumstance estimate average versus aggregate interpret estimate overlap support similarly element instance support lead homogeneity poorly population define less diversity close great diversity worth compare difference reveal principle behind depend estimate time diversity particular variation collection due intuition g offer justified inspection remain happen note act extremely unlikely concavity depend nature whether general less clear diversity amongst situation understand pure individual graphical pdf estimate size well estimate static section dependent contribution due entirely limit aggregate component intuitively intuitive diversity support contribute induced fig begin relative average pdfs p clear average pdfs relative relatively trivial support support primarily moreover individual vary event variation support affect individual pair point effect likely effect correlation time population reflect diverse population establish interpret three homogeneity homogeneity address population ii support address support iii homogeneity address variation pdfs graph pdfs quantify homogeneity moreover homogeneity exhaustive simple intuitive method right moreover least quantity supplement
dependent assign every alphabet string family sequence computable outcome ergodic markov symbol source bernoulli induce variable outcome consist sequence sequence divide probability every note large outcome essence computable repetition contain computable go definition occurrence character string character unless occur order reconstruct induce bernoulli th computable entropy entropy stop string copy pattern reach sequence copy outcome concern bit family alphabet state reach process state state state require program bit bernoulli process frequency binary typical sequence bernoulli computable outcome string know bit generate successive bit variable string bit compute program logarithmic consider case ignore english experimentally compression technique modeling set symbol stream next symbol stream shannon consider alphabet character alphabet since hx high knowledge kolmogorov entropy entropy knowledge kolmogorov complexity observation computable shortest compute add program short program compute sequel extend notion alphabet string computable probability outcome finite respect joint mass kolmogorov complexity theoretical direction reference metric normalize proper manner free measure great impact parameter distance variant test database major turn heterogeneous detection range forecasting music paper string machine like kolmogorov upper metric distance satisfy normalization binary minor ignore string family variable computable computable probability mass function moreover substitute kolmogorov complexities entropy family computable mass family computable probability mass hence ignore computable computable variable singleton mass markov computable clearly function entropy family bernoulli gaussians process consequence bernoulli string identify relative source stre static bernoulli total leibler mean comprise header header ignore joint sequence shannon probabilistic version e hx since mutual variable entropy cite connect two empirical entropy definition string computable appropriate family computable computable variable outside framework approximation entropy operational formal definition intuitive notion domain integer value extend argument rational argument call upper computable countable recursively countable computable example computable kolmogorov recursive example recursive informally string string reconstruct machine constitute program program deal kolmogorov matter plain call formally kolmogorov short input output unconditional kolmogorov machine additive absolute character thesis ability machine simulate execute kolmogorov finite object kolmogorov absolute quantification shannon hand deal average produce former theory much surprising complexity somewhat weak variable string view indeed input another similarity mutual finite string remarkable information plain kolmogorov complexity conjecture national center mathematics science computer support address email compression version empirical alphabet previously consider possibly description induce involve relate entropy kolmogorov basic shannon alphabet interested message receiver receiver consider message message entropy source notion author intuitive traditionally notion finitely outcome string alphabet message entropy receiver reconstruct therefore draw encoding
continuously ny constant nf appearing contribution product w recall multiplication r deduce satisfie simply lebesgue continuous couple weak twice vanish expansion yield deduce mse variable risk regularity estimate unbiased replace reformulate chi put everything together specific transform representation notably wavelet block discrete cosine l via denoise strategy long prove reduce various type possibly redundant pass treat procedure generic j arbitrary perfect submatrix channel complementary channel w coefficient free l actual l rule sophisticated processing pointwise continuously differentiable piecewise differentiable partial derivative introduce hadamard element pointwise proof straightforwardly develop transform denoise prove band need shrinkage minimum pointwise freedom choice replace mse recent degree optimize generalization square estimate directly close yield equivalent potential realization display transform estimator previous square orthogonal explicit remarkably particular haar derivation mse possible unnormalized haar wavelet channel resp channel implement unnormalized haar implement filter z em give unnormalized haar scaling wavelet haar wavelet chi chi square random degree freedom empirical coefficient chi jk since filter wavelet process j jj unnormalized haar scale variable unbiased ease ii involve ii successively equality omit natural wavelet representation introduce justify contrast case removal wish poisson unbiased requirement continuously differentiable approximation show good value pointwise cc right thresholding sophisticated denoise dependency wavelet call show quality poisson unnormalized haar wavelet unnormalized haar wavelet delay sign magnitude coefficient scale advantage persistence let account similarity neighbor thus intra dependency naturally arise haar wavelet parameter optimize via advance process k sense linear finally jk magnitude image image magnitude measurement magnitude n dimensional datum freedom magnitude mr necessary technique magnitude mr rescale eq three quality metric signal visual metric introduce see variance reliable signal background sophisticated various level set mr cccc cc intra inter unnormalized haar unnormalized haar art mr evaluate pointwise apply transform domain intra inter unnormalized haar wavelet baseline provide unnormalized haar intra scale shift invariant sophisticated variant unnormalize haar wavelet cycle technique fig show improvement average cs observe cycle invariant transform cycle sensitivity optimize appear address common choice mmse choice benchmark retain mr denoise code spatially mmse filter mean mr respective mmse optimize mmse support variant cycle intra haar wavelet thresholding pointwise overcomplete haar block discrete cosine note span overcomplete basis previously invariant table observe basis overcomplete consistently denoise relative state condition dependent obtain consider overcomplete dictionary e em cs cm haar haar haar let cm output average realization c haar cs haar let em haar let average various denoise denoise fine confirm higher obtain denoise algorithm execute matlab run os equip ghz intel processor quite image within transform dedicated reconstruction haar cs haar cs file denoise magnitude mr raw image acquire mr weighted signal n treat fig denoise various noise efficiently bias cc em via haar haar let derive magnitude mr denoise comprise continuously differentiable focused transform domain pointwise coefficient consider specific unnormalized haar wavelet multiscale allow adaptive joint inter intra simple soft algorithm test image qualitatively finally mr efficacy mr image chi estimation datum implementation online rgb theorem ex wolfe chi unbiased magnitude mr denoise article derive unbiased expression expect square associated chi chi degree broad class shrinkage transform unnormalized haar art simulate secondary fundamental medical imaging provide contrast ratio acquire mr physical structural strength average encode development focus primarily inherently high image reduce system pursuit objective acquire post denoise essential visualization meaningful acquisition fourier sample random primarily fluctuation degradation pattern straightforwardly inverse discrete space resolution reconstruct image determined frequency work accelerate mr acquisition trajectory nonlinear consider axis work image consider apply wavelet mr real article chi unbiased estimation denoise mr image two
cause thus algorithm construct dataset construct coordinate synthetic random tend irrelevant use try lipschitz repeat normalize observe rather across difference normalize ht lin sim concrete value report squared error mean glm lin sim community normalize perform uci choose dataset glm standard simple heuristic sim index procedure problem fix via perform fold fold table square across ten fold normalize variance fold many slightly illustrative transfer function plot fit lipschitz result piecewise intuitive smoother find reader notation show error current x x w u literature rademacher rademacher entropy follow class standard utilize square rate union particular element sense union easy right pick verify f hilbert integrable inner yx verify prove minimize therefore convexity require establish inequality statement inequality convexity square w dimension begin lemma note u well least simultaneously invoke get cover analogue somewhat bind dependence let x tell empirical risk minimizer follow lemma lemma argument combine invoke usa microsoft usa microsoft generalization regression target dimensional problem often provably efficient glm provable provide efficient practice linear function value assume flexible extension existence link expectation immediately include regression aware rate provable square challenge practically question non achievable simultaneous estimation former continuity possible effectiveness appropriately attempt good note problem significant body heuristic improper learning type sim agnostic setting flip complexity polynomially provably recently provably efficient common assuming model computational perceptron regression unfortunately datum request empirically assumption glm hardness theoretical provable seek address theoretically algorithm monotonic lipschitz practical parameter free provably computationally moreover feasible original ran case lipschitz generally assumption glm efficient despite non assumption learning sample support feature decrease role lipschitz equivalent restriction measure equivalently counterpart write construct decrease algorithmic condition presentation result constant problem properly simple perceptron algorithm close arbitrary decrease appear algorithm input tx show iteration nearly independently decrease iteration appropriate pick line somewhat rough idea iteration square within reasonable hypothesis iteration accurate unknown correspond parametric single index model main difference compare transfer must track iteration fit monotonic maintain get training decrease lipschitz literature method large reader one randomly subsampling argument beyond input tw I I turn formal formal subsection difficulty simultaneously plausible sharp theoretic reason dimensional low setting independently support unknown lipschitz follow iteration dimension sufficiently minimize base formally lipschitz entire linearly sort function suppose decrease value constant optimality easily formalize kkt constraint I must bit additional training input define clearly call ti relate actually value conditional somewhat require result need space probability heart
monotone incur error ignore negligible care take restriction naive approach generic monotone learner inefficient require closeness sample complexity monotone complexity fortunately monotone distribution learner agnostic handle monotonicity naive give monotone distribution monotone overall show suffice handle additive besides term hypothesis overall bit learn total modal run hypothesis make modal completeness far decrease yes analogous testing whether modal correctness learn access modal fix sample distribution partition j j r j ki ia j I ia repeat start interval atomic continue interval bit ready explicitly exposition neither non suppose extreme fact either first fact failure atomic atomic step satisfie atomic kb case interval thus reason every step invoke contiguous interval enough atomic otherwise atomic k sample standard argue atomic contain ball bin desire ball completeness execution step run contiguous atomic interval contain extreme return failure leave extreme point modal failure repetition moreover claim increase orientation follow increase contiguous output yes far every union interval partition either bit negligible point expect event discussion precede lemma partition negligible total variation ps ps tv bound rhs negligible contribution contribution close monotone hence observe monotone interval hence desire expectation ps ps ps suffice term claim binomial claim expectation j q monotonicity mapping claim apply e ps claim rhs old j ps ps old use j consequence concavity require clear claim easy polynomial analyze testing access modal use sample property yes vote quantity capture iff average average distribution pass consecutive sum test sound modal ingredient procedure convert input access modal distribution output make long option triple point either neighbor theorem establish correctness use property start completeness say every collection interval good henceforth b write consider similarly I term sum term e completeness imply I say yes probability least crucially let modal non I first show let modal far exist I latter assume rhs decrease domain point I I c achieve sample fall empty iv combination desire thus prove let modal distribution n c high stage stage reduce mode one modal appropriately show triangle axis domain informally identical left mode maximum work height equal right flat reduce proceed convention mass cut quantity mass interval point unimodal decrease mode right interval indeed define mass exceed mass unimodal interval mode outside decrease latter interval recall decrease argue b construct recall lie interval continue iteratively remove may conclude establish complete step trivial order even naive would programming run want decide use dynamic jj running completes run low think accuracy sample weight region accuracy need least level strategy develop namely decompose modal simple monotone context efficient motivate kind log concave monotone hazard sample would construction proper modal priori anonymous agnostic learner also agnostic claim explain follow special theorem sample require arbitrary let conceptually three partition carefully interval weight assign entire give reduce correspond obtain hypothesis follow inequality output hypothesis I interval combination third tv entirely explain show fact semi agnostic claim clutter description non algorithm succeed average satisfie close show assume desire distance imply distance deduce complete test routine choose winner candidate hypothesis hypothesis high routine select winner candidate close run candidate chapter build difference approach competition hypothese symmetric compete hypothesis support competition competition carry draw return mp draw return check competition competition competition winner intuitively winner winner intuitively moderately winner variation let indicator iff mutually chernoff p h winner competition ii competition winner winner theorem require I take account probability learner binomial hypothesis place resp increase increase provide boost algorithm hypothesis distribution candidate generate would boost success appropriate distribution formally let suppose collection use output chapter distribution cover notion competition choose cover algorithm choose achieve cover instead run choose output never exist output argue never competition consider draw union never competition argue competition failure suppose perform construct hypothesis generate true variation conditioning assumption time run additional run remark run produce run description distribution pair description former case description output consist monotone interval construct candidate pt claim corollary conjecture claim mit mit edu university ed uk edu modal discrete peak generalization unimodal study perform modal probability give run factor prior case crucially subroutine easier consider modal section precise g apply naturally simple unimodal distribution study discussion aim modal variation give access theoretic goal bind computationally run size input contribution nearly rich restriction researcher unimodal among probability mostly deal theoretic monotone unimodal central learning cite mention monotone unimodal complexity asymptotically discuss detail ingredient relatively develop algorithmic aspect rather contrast interesting issue arise learn modal modal output hypothesis give distribution argument adapt discrete case show generic distance monotone sample copy length monotone nearly precisely bit running time factor aware modal know simultaneously fix exponent depend section give efficient distribution concatenation monotone increase learn modal indeed guess run inefficient moderately naive roughly separately since interval monotone used interval contribution failure efficient totally give polynomial turn polynomially roughly moderately naive unlike instead successfully distinguish property versus variation property testing output yes crucially use able identify ii handle thus roughly run algorithm algorithm understanding note testing preliminary diagnostic whether expensive differently modal monotone monotone aware successfully property decompose learn may find elsewhere element vi correspond increase decrease nonempty point min extreme extreme interval resp denote resp modal distance ps kp kp tv capture subset variation identify recall interval vc deduce several sample vc value successful partition domain mass next step close distribution monotone monotone interval mass contribute though interval affect focus monotone overcome monotone algorithm testing interval confidence distribution satisfy probability boost confidence overhead give bit description distribution hypothesis decrease detail algorithm simple bit modal input access modal domain atomic j j light interval conditional easy claim draw sample run simple step failure disjoint rest simple good interval except potentially interval construction step light point heavy interval heavy sum proceed contribution modal
arrange triplet visualization pursuit long really macro entire inside micro interesting work normality prefer experience index micro attract micro pick cube know analyst come macro cube structure subject confusion unfortunately subject matter knowledge criterion important really exploration process may decide guide projection towards meaningful similar automated parallel pursuit structure like ever half measure discrimination mirror normality capable really general question place answer incorporation modification index possess benchmark even yet asymptotic constitute coherent applicable organize section recall traditional pursuit algorithm design overcome limitation three real pursuit literature clear precede flexible exploratory begin try exactly usual exploratory projection break user index projection project substantial optima index specification constitute interesting implicit projection normality specialize like apart scope intervention direct incorporation pursuit dimension benchmark index benchmark possible essential feature notion user without necessarily thus input incorporate require specification though number differ search similarity dissimilarity search support structure way occur use previous could arise old drop expand could start explore analyse case subject available two arrive factor level easily interpretable benchmark exploratory discriminant always possibly thus less powerful look discrimination neither benchmark traditionally find projection pursuit fairly simple contain naturally still comparison synthetic reveal original close meaningful synthetic dataset ad principle benchmark wide utility split distribution hope projection remark obtain cluster highly structured difficult benchmark otherwise reason motivate utilize course investigation second important projection extent similarity projection multivariate dataset exploratory analysis kind distributional principle adapt index requirement power scheme yet projection appearance cloud dramatically cell scheme choice generalization multivariate kolmogorov multidimensional multivariate von statistic rather serve index index nothing absolutely like cumulative characterize however advantage continuous statistic interest propose estimation plug evaluate numerically region integration convenient estimate location serve circle centre spatial fit parameter take parameter experience preferable least fast requirement possess easily apparent none possess easy hundred take time whole search search require kind hardware available computer naturally useful statistic preferable adaptation preferable essentially exploit relaxation integral accuracy evaluation way accuracy also kind quasi stability search finally functional point evaluation ks type multidimensional thus evaluation also involve kind sort easily computer three spatial next index projection relate trust package carlo package optimize throughout algebra adapt package encounter wish illustrate matter come number generator package benchmark rarely deal benchmark suggest number sophisticated generator however always attention dataset generator uniqueness different seed generator view different seed simulate optimize initialize allow start radius multipli solution though projection reveal subtle generator part primary interest though clear either exploratory investigate could use relate relate intensity dataset quite high would build argue two kind result vast majority solely gene individually separation need benchmark comparison class demonstrate interested projection figure separation appear solution projection exploratory analytical article come vast identify ratio present gene retain largely exploratory projection pursuit entry gene order row gene entry magnitude could importance gene less retain change bit separation albeit important identify separate exploratory pursuit separately exploit linearity pursuit quick follow solution drop euclidean criterion reproduce original well preserve figure obtain separate plane draw however misclassification something great concern neither projection encourage pick subtle fundamental pattern relate produce contain production region identify broad south east west four south north south section none available arrive benchmark recommend identify obtain pursuit plot finally isolate comprehensive structure graphic static understand h pursuit algorithm qualitatively projection display cluster degree separation symbol break south west readily symbol break cluster bottom remove region reduce none show shift three achieve well projection show north precisely method box modify interact exploratory even currently level interaction incorporated hand always dynamic graphic drive exploratory pursuit balance highly usually however black
q every without generality p together require reward state reward draw adversary however subsequent round adversary force reward need rapidly amount adversary market maker call bandit maker vanish previously maker know distribution market maker asymptotically cost know advance fix function market maker achieve reward obtain performance maker uniform belief initial first period switch draw price towards half potential adjustment direction lead market meet price attractive bad maker asset trading preserve guarantee market literature optimality relative class market understand work far ever grow work explore suited market environment substantial evaluation market life market interest modular combine bandit algorithm obtain free space well understand practical behaviour market market claim free case automate market free analysis outcome cost function price world pricing payoff decompose two proportion occur price market maker first handle cost providing provide overview mechanic explore vs price reduce demand conversely reduce contract bundle demand maker demand select sufficiently exploration tradeoff well literature maker use cost mapping size bandit reward bandit cost modular grow bring development market motivate price extreme situation maker situation period asset occur quantity exposure market market maker since incur loss perfect knowledge outcome tradeoff price draw know centre maker still price extract belief centre observe still trade trade optimal maker extensive automate market market expert made market mathematical base market follow learn stochastically draw belief mirror descent see market maker outcome reason onto belief market maker motivate extensive separate prediction market make maker balance extract automate market sum market extract market maker case rather narrow equilibrium criterion function market sequentially proper scoring compatible player belief action proper reveal interact maker act enough compatible mechanism paper arrange interact maker proper compatible bandit price back economic consider case modern method specific online price adversarial overview component subsection exclusive period period maker cost pricing behaviour sequentially market maker sequence period shift vector describe introduction exploit trade maker round view bandit formalize market pricing behaviour maker assign vector represent market maker component case event market position want portfolio share price pay security vector detail construct important bad bandit vanishing regret fix market maker attempt receive face market obtain difference market maker outcome abuse maker cost outcome use q complementary maker condition also function offer price exist crucial potentially never offer price market maker c imply consider lipschitz continuous continuously price family cost fair price discuss logarithmic scoring construct single price market price cost straightforward verify condition satisfy monotonicity boundedness hold treat game closeness metric supremum cost indeed pseudo triangle carry difference show market make outline salient motivate allow cast adversarial armed market play round player arm predefine choice continue large total reward crucially round player force balance action repeatedly play game central action receive reward play typical analysis derive sublinear bad adversary function adversary choose play result reason latter set available compact subset mean choice make difficulty relatively work action space tend place strong restriction require require technical broadly vary say restriction follow definition state lipschitz adaptive interpret reward make locally view impose difficulty inherent make realistic scenario trade two choice round mean sequence necessarily history function thus interpretation market must assume adversarial non market maker cost function meet condition rich
robust pearson correlation nonlinear relationship view especially suitable situation relationship speak extent tend require relationship coefficient consistency covariance establish matrix via hard predictor eliminate explanatory irrelevant actually significantly independent feature marginal utility frequently correlation propose marginal select screen additive generalize model implicitly depend explanatory change correspondingly relevant regressor fact diagonal equation conditionally average also reflect permutation cluster explanatory variable utilize score decrease order kx k k kx generality assume form perform variable construct overlap construct selecting continue index step equation grouping weakly correlate new predictor actually dimension reduction pearson screen second explanatory second feature thresholding try discover multi could utilize discover benefit order economic financial construct index science break edge keep correlation show np hard sample covariance correlation order matrix permutation simpler regularize limited nonzero r simultaneously explanatory combine especially suitable modeling method financial thresholding regularization remove irrelevant remain one avoid np hard clustering base sign estimation procedure implement table rank correlation unlikely economic disjoint give ideally correspond w motivate refinement consider x correspond lagrange contradict requirement role increment extract simple sign require minimize choose otherwise value iterative achieve step present minimization exist also model dimensional group third approach fit great deal fit face fast macro covering include production capacity price start stock price rate exchange time except rate nd difference raw denote variable index mat component u item less care pr pr u services pr service correlation threshold interest price indicator percentage effect price magnitude value besides economic adjust economic model display relevant detailed digits index provide forward backward construct overlap overlap come th originally correlation th variable th item medical implicit consumption identification mainly concentrate link function space price implicit consumption respectively interpret service actually interpretability economic view keep parametric coefficient table rd economics correlation parametric price unlikely sign corresponding parametric coefficient column table explain variation positively mean eliminate except lar estimate least angle include economic indicator economic activity estimate variation lar improvement lar already perform paper spatial quantify consistency dependence level resample cross estimate utilize link graphical spatial backward structure method spatial panel economic financial series superiority function directly correlation define except use introduce address pearson zero correlation zero brownian independence correlation functional mutual correlation detect dependency want point step screen explanatory descent residual square regression construct thus could f jx sure correlation screening define error help rewrite study focus cumulative investigate motivated grateful li xu discussion particular thank stay california berkeley subsection definition assumption inequality see dependent yield tm go complete bs tm jj since v b lead eq q integrate rhs minimizer imply jj condition lemma well spatial panel economic financial first spatial hard latter j quantify consistency discuss validation consistently utilize screening implement forward relevant modeling apply large panel economic financial price superiority technology create economic financial genetic panel economic begin variable view assume estimation play role important numerous finance limited management interval forecast grouping via reduction recent tool low frequency variate good eigenvalue support thus attract lot attention recently one construct impose structure model among order weakly panel array invariant thresholding covariance high extend iid series setup important go recent understand especially serial correlation temporal dimensionality begin complex parametric adopt nonparametric curse disadvantage maintain curse however prior work carry discuss one specifically explanatory high single perform selection technique eliminate gx unknown assumption additive actual structure sample size specific class approach character economic financial near expect unstable sensitive minor perturbation expect selection economic principal component regularization method weak matrix compose requirement denote set index eigenvalue gram cardinality submatrix gram satisfy integer derive upper view fact find penalty ideally penalty coefficient shape nonparametric high among end develop speak try could cardinality number univariate parameter parametric tx sx tx possibly moderate balance flexibility general additive linear view rhs know link disjoint element say identifiable every identifiable group immediate extract second question paper answer mainly various resample scheme prove cross validation approach cluster estimate high spatial estimate matrix utilize link graphical standardize explanatory label explanatory spatial reduction sign dimensional face help dimensionality group information knowledge available spatially dimensional propose criterion grouping try intensive practical grouping drive without sign sign coefficient economic law organize present main notation conclude start concept covariance notation define matrix average eigenvalue thresholding thresholded notice preserve permutation preserve positive uniformity mainly future suppose without introduce view dependency subset fractional cover j w spirit note unless process consider mix related sub coefficient v kolmogorov appear mix process article coefficient notational affect tc state consistency rate get slow level maximize reach offset behind correspondingly slow accord jt retain question ask extent dependence
manner case median svm median smoothed version result consistent predefined I estimation step l call difference svms carry svms package implementation svms loss also result loss think early consistency svms jx reproduce hilbert corresponding assume write f would gx estimate conditional median residual loss function smoothed version smoothed loss estimation estimate one assume unknown smoothed continuous l parametrize logistic purpose introduction newton measurable function say consistent assume predefine median uniquely g quantify r quantify absolute predefine iii plausible nuisance median svms calculation proof integer c measurable complete space borel obviously unique converge sense surely sense ne f existence uniqueness combination risk immediately sub argument combination q q svms svms exist heavy tailed assumption trick shift sense l see function minimize svm e immediately uniqueness consistency difference difference svms immediately uniqueness choose median median conditional quantile quantity conditional well regression svms occur quantile wrong occur prevent kernel view appropriate estimating median absolute residual conditional median conditional need svms type happen approximate g rbf kernel jump happen easy estimate complicated advantage estimation flexibility quantile flexibility type estimate quantile lemma nx h probability follow e apply lemma close separable dense notation define f l nx gx nd g fulfil calculation f negative function reduce definition apply triangular l g r triangular separately four converge superior large part term note define nx boundedness well see easy follow f nx theorem x inequality converge imply therefore guarantee pt ex pt sketch pt ex pt pt ex definition remark definition mathematics department main goal statistical single svms establish estimate like median svms median volatility median absolute value problem conclusion generally location consider two statistic
polynomial one polynomial contribute provide model straightforwardly expert convergence quasi maximum estimator obtain leibler bound generalization pm mn nj total number weight function identifiability permutation unique maximizer remove attain approximation estimation expectation I expert consistency expert specification target belong parameter kullback leibler divergence model number multinomial logistic consider smooth continuous variable satisfy rate assume least finding yet maximum mixture study rate paper long I I case throughout notation h inverse link mixture model glm kx mh kx class density derive consistency restrictive multinomial logistic poisson among function satisfy estimation maximize likelihood necessarily identifiable permutation restriction expert kullback kl divergence minimizer index kullback work consider I straightforward next generating sequence random vector ensure compact np xy almost surely demonstrate identifiability maximizer estimator consistently index I maximum converge also ergodic ergodicity function identifiability distinct every find sufficient identifiability binary adapt specification set adopt identifiability unique maximize taylor around maximize hessian identifiable technical allow flat pg likelihood often maximize expert complete likelihood maximization put q expectation maximize maximize divergence side approximate estimate good approximation follow use kl divergence present shall introduce partition partition fine partition partition abuse notation justify sequence definition bind growth see sub idea approximation control inside fine partition inside cardinality sub geometric employ notation expert expert structure expert actual loss actual expert approximation constant direction derivative polynomial expert result enable important mix many general specification density expert dispersion modify agree assumption therefore sharp deduce model expansion term approximation combine unique maximizer able summarize rate divergence true convergence particular proportional result derive obtain assume identifiable maximizer price pay generality localization process neighborhood favor smooth enough dimension favor simple possibly small big ii iii show quite deviation optimal optimal size situation ii achieve convergence drawback rough polynomial conduct study impulse polynomial increase exponentially preferable remark table compare approximation distinct hold expert know total fixing want table small error h cccc quick exercise conclusion small large compare improvement section mixture expert polynomial link sharp mean rate estimator pm far uniquely specification effect approximation estimation conclude balance inclusion error proof assume include glm ii class instead iii calculate parametric convergence estimator expert sample increase vi polynomial development light acknowledgement author thank expert comment previous version appendix justify drawback work kullback leibler divergence hellinger hellinger distance hellinger density divergence bound hellinger basic inequality relate hellinger leibler stand respect leibler bound square hellinger hellinger rate leibler boundedness support overcome choose hand side large bound estimation convergence I hold inside measure cover moreover universal show task g kx f gx lipschitz cx first inequality take prove likelihood bound rate decay small kx kx ki convexity logarithm application enough combine previous use hellinger maximum likelihood completeness pg estimator let choose sum continuous proceed existence identifiability satisfied assumption boundedness continuity integration limit expansion bound satisfy theorem compact condition remain integrable bound dx kx kx finite assumption kullback ensure dp dp p term choose sc first bind tail e inside eq hellinger inside arrive result rate substituting parallel precisely inside remove allow proof consider two increase minimize decrease minimum iii straightforward kx far first differentiable kx reasoning definite tell let logarithm consider consider unchanged next satisfy metric metric corollary proposition expert give
tend density get hard function favorable easily approximated evaluate real world test use outlier positive house speech contain record extract assign contain word take class dataset write digit consist pixel take level regard assign auc useful systematic auc choice summarize overall small tendency agree bold dataset sort l r diabetes heart speech news news news vs vs vs vs vs vs outlier label py n py test discrepancy input ultimately generalization error py practice covariate setup plain minimization empirical shift q shift tend large importance propose call correspond plain minimization importance empirical give intermediate trade model regular model one drawback hard estimate poorly adaptation cope definition exponential indeed relative importance play weight plain minimization minimization trade consistency importance agree weight unlike exponentially reliably propose adaptation effect behave covariate shift employ ls exponentially change plot show realization ls l give true well figure error l comparable accord next change realization test sample learn l ls test ls tend outperform l effect transfer world transfer human ask perform task hz axis plot stream slide manner slide position reason decide take acceleration step invariant situation want available unlabele obtain train j unlabeled test significance specify bold walk run walk vs activity plain without plain importance different depict three accuracy unlabele sample plain without instability weight useful transfer covariate propose use divergence give direct approximation theoretically parametric show preferable pearson furthermore asymptotic divergence complexity estimator experimentally practical homogeneity outli transfer covariate addition homogeneity test detection transfer machine include multi independence dimensionality cluster probabilistic thus relative homogeneity detection transfer simplicity pearson non pearson pe possibility bound norm rkhs satisfy eq property kernel dense another canonical square derivative term impose rkh spline rkh speak density derive rate book l quantity function large rkhs note rkhs condition pg denote auxiliary sg g pg side entropy show f relation entropy jensen concave similarly show last combine eqs simple b moving complete pf eqs eq also therefore substitute observe complete variance model use follow correctly eq note eq probability likewise sup standard sup asymptotic denote xx regularity linear likewise asymptotic confirm negative increase non negative see negative upper therefore complete yield q expansion complete theorem estimator ratio go denominator apply machine task detection homogeneity always smooth ordinary ratio favorable complexity estimator experiment usefulness pearson outli homogeneity square importance fit fundamental would divergence kl divergence plug know approach unless good approximate without ratio parametric mini max linearity cope divergence pe pe kl thus pe divergence vanish pe estimate give pe computed usefulness various homogeneity dimensionality first non pe govern sup denominator density speed ratio paradigm overcome fundamental comparison quantity call pe q direct q notable unbounded relative density ratio approximation ordinary ratio relative pe divergence pe complex extensive homogeneity pe estimator compare follow pe estimator pe experimentally paper contribution identically distribute dimensional paper let reduce ordinary pe thus regard pe divergence pe relative transpose kernel third expectation obtain easy reduce direct call relative reason refer criterion matlab implementation public acceptance estimator pe divergence pe divergence express pe far similarly particularly simple numerically toy denote denominator contain c c ratio ratio graph relative smooth ratio tend analyze pe divergence estimator specifically section insight mathematical rate appendix analysis density describe space statistical model parametric convergence pe practically dimensional rkh norm represent precise pe tend eq shrink speed technical detail follow pe fp coefficient lead term asymptotic become large preferable term tend infinity pe would advantageous plain pe divergence lead depend accurate large furthermore vanish fast depend another divergence regard tend zero hypothesis approximate value kl pe divergence adopt homogeneity propose experimentally pe two homogeneity contribute hypothesis different correct hypothesis keep hypothesis plain reciprocal adaptive behave two sample homogeneity scenario dataset test plain reciprocal show dataset nan hypothesis plain correctly error control hand acceptance rate toy still reasonably small p critical different want reduce incorrect setup reciprocal setup dataset density reliable smoothed distinguished completely ratio reciprocal plain accurately setup large perform dataset plain reciprocal trade correspond tendency middle overall plain reciprocal work unstable reciprocal tend instability plain reciprocal provide wide assign wide setting work support systematically issue p two correct acceptance specify l r mmd c diabetes heart ratio denominator density positive negative nan acceptance rate comparable face l mmd diabetes heart homogeneity binary dataset pe
straightforward middle always mean unstable reasoning point locally depend one flow dynamic critical exploration finding game action reward h decide pd payoff dominant beneficial nash players ne pd nature guarantee rate contrast observe start regime lack behavior attribute next mp zero player action pure ne ne ne globally position rest term player action decide pure ne ne rate dynamic correspond game furthermore equilibria unstable rate diagram find risk dominant ne game show fig strategy dominant payoff opponent describe game strictly risk show meet critical ne call anti beneficial anti game one behavior game ne mixed ne ne anti game condition anti symmetric ne c xt ct ct unstable consider ne outline parameter value see small single ne game similarly rest critical exploration rate three anti consider additional ne elaborate appearance rest appendix diagram plot diagram contrast payoff different player payoff matrix dominant player payoff mostly make picture preserve light grey hand player role range vertical critical rest behavior grey region parameter region strictly dynamic boltzmann mechanism exploration two action exploration dynamic guarantee reach demonstrate depend game exploration rest structure stable rest positive analytically examine exploration asymptotic behavior ne structure globally observation numerically ref equilibrium fact boltzmann dynamics game dynamic qualitatively game ne exploration rate allow rest ne games single ne rest dynamic rest exploration tend dynamic exploration various mechanism indeed study far limited dynamic insensitive exploration anti fine grain sensitivity rich employ agent manuscript aware reporting use local demonstrate finally analytical diagram possibility rest game ne complementary present thank discussion national foundation grant fa derive rest sake intercept pass intercept exploration happen yield multiple symmetry whenever rest analytical repeat reasoning depict ne learn sake region graphical representation rest fig becomes find require calculation yield xt numerically figure critical solution intercept game converge game nash comprehensive characterization structure game exploration result games ne demonstrate single ne range player tend reinforcement behave trial error single rl extend interact difficulty stationarity learn agent vary agent convergence except well issue perspective dynamical note boltzmann equation additional account similar greedy exploration mechanism schema select high number develop understand behavior categorization learn dynamic game insight body recent boltzmann softmax mechanism decision play competitive indicate decision lee observational human seem reinforcement softmax spectrum behavior important conceptually make prediction outcome analytical technique complete rest necessarily seem previous slow cycle systematically examine exploration asymptotic behavior hand noise inherent aspect human softmax mechanism perturbation show depend one rest point structure vary critical remain rest globally rest describe connection boltzmann non conservative nature asymptotic behavior different game type example conclude remark provide brief connection rl optimally repeat agent choose action receive reinforcement reward agent act cumulative adaptation mechanism call agent parameterize interaction environment agent finite available denote action action need agent greedy selection globally solution one explore less selection temperature tradeoff agent maximum exploitation agent interested continuous scheme game dynamic correspond ne furthermore unstable dynamic generally ne limit insensitive reward mix uniformly behavior exploration rate volume make dimensional dynamical implication specifically dynamical crucial cycle interior rest globally stable nature variable modify read rate phase eqs action games action dynamic eqs introduce point trajectory interior simplex rhs eqs examine interior equilibria interior plot side monotonically increase assume guarantee inspection show type solid curve rhs temperature non increase rest temperature exactly rhs eq alternatively whenever sufficiently branch well separate increase critical branch meet saddle type examine useful interior rest equation graphical easy thus accord consider decrease function state consequently sign side monotonically decrease thus
time select rigorously heuristic currently e subset patch consider involve flat dictionary large heuristic increase perform deviation improvement proximal sparse problem involve sum overlap light connection network proximal norm cast quadratic min algorithm resort certain modularity dictionary come repeatedly loop norm operator cost flow demonstrate apply wide class address formally summarize graph canonical sharing vertex say condition arcs source cost arc cost arc cost every arc conversely arc notice flow arcs flow feasible value arc path node lemma arcs arc flow flow flow equivalent amount flow amount sum flow construction build capacity constraint conversely give flow path decomposition decomposition path correspond arc flow path decomposition arc amount arc flow proximal operator find proximal essentially termination optimality condition primal respectively solution primal problem dual w g g w intuitive flow amount u h old inequality prove convergence correctness also min capacity find cut divide part construction path path arc whereas arc figure arc go cut would arcs inside construction flow go circle draw blue circle thick blue fill place fill red black bend angle auto xshift leave g part xshift u xshift pre dot pre var leave u edge var node right pre leave thick xshift control gr u gr gr v arcs arcs flow cf scalar negative add polynomial operation split process projection flow sufficient call compute min max min j existence jj u equivalent prove correctness correctness use canonical number induction e g since induction canonical algebra suppose ball instance use max scalar u sum optimality condition lead consider case j empty remove notice graph group gr gr optimality derive lemma optimality notice arcs cut equation j cut arcs arcs residual decrease side bit show g disjoint part gr v gr j gr gr gr gr u u want necessarily reason go observe flow nan receive v j gr j contradiction gr u similar summarize g gr u equivalent algorithm give graph perform dual dual g j q introduce g maximization primal imply strong duality respect primal lagrangian sign correctness algorithm show call proposition non empty call canonical induction simple algebra show correct j constraint necessarily inequality follow cut exist set feasible value necessarily algorithm vertex part remark arc go arcs solution induction hypothesis solve replace constraint arc go result equivalent argument exploit show cut section fista duality stop criterion generality look matrix typically compose observation primal place argument duality gap w optimality gap dual algorithm proximal convexity precision duality f k kt k procedure f choose max min compare max experimental namely dataset build basis transform dct respectively compose parametric max flow conduct single ghz follow execution algorithm run statistic correspond sec experiment consistently outperform berkeley california berkeley usa fr de sup paris france consider induce whereas put develop hierarchy proximal polynomial allow proximal splitting address alternative learn tree dictionary video sequence wavelet learn optimization code factorization flow method various supervise combination regularization powerful tool address regardless relationship e spatial temporal hierarchical effort devote designing encode non bioinformatics vision sum norm appropriate variable solution usually difficult part involve nonsmooth strategy proximal ability nesterov handle differentiable paper regularization splitting build upon overlap group hierarchical sum norm solve thereby establish connection literature context fuse lasso maximum flow convex combine method norm gap present proximal regularize demonstrate practical algorithmic introduce sparse build link draw decomposition regularization algorithm different video image natural patch short version publish advance process denoise present method discussion letter pseudo nonzero however sake keep notation I q x r p present work devote proximal gradient algorithm present experiment application paper coefficient zero encode overlap tree structure basically deal computationally concentrate programming method impose priori nonconvex w design induce norm next interested induce generally function fitting structured variable represent coefficient case basis partition e individually know organize explicitly priori regularization improve interpretability penalty set set consider challenge induce penalty see encourage small though structured sparsity overlap purpose union predefine factorization pattern group use rely subgradient scheme approach mention smoothing norm parameter control accelerate depend solve problem parameter norm add smoothed encode g reweighte scheme mention sparse assumption example logistic reference community therein ability nonsmooth nesterov procedure extension gradient objective function minimize p f close constant equivalently attain fast rate proximal reach precision nonsmooth generally define proximal unique generalize projection proximal appeal decomposition operator soft operator u norm soft u onto effect u g norm either include closed form g b q lasso lasso general proximal reformulate propose dual formulation dual proximal problem jj w solution without generality optimality sign sign entry solve flip dual negative magnitude sign know scalar dual variable number primal remove overlap interpret vertex arc capacity arcs arcs flow arc arc incoming flow flow except value flow let group canonical vertex index size simplicity index refer vertex arc arc capacity zero arc capacity flow arc infinite jj canonical figure flow identify non arc cost lead cost minimize graph arcs flow arc arc g equivalent problem canonical simplify edge edge remove capacity simplification quantity compute appendix reduce draw fill minimum fill red mm rectangle thick draw fill bend angle source edge pre xshift mm pre xshift edge pre var u distance node si u leave edge pre node bend auto cut balanced factor practical complexity serve zero optimum duality converse description along computation give z problem derive standard duality efficiently equivalent arcs arc solve amount small flow arc appendix computation dual initial gr uv either sum admit form v ff know svms structured one scope slightly offer regard since operator discussion schmidt choose advantage proximal parameter ability proximal consider variable associate amount lagrangian equivalent r g admm find saddle iterate fix ascent respect summarize respect w influence efficiently cope exploit ni x pi I equivalent problem methodology know usage possibility additional add l compute issue adopt replace quadratic v augment lagrangian proximity w symmetric positive w still demonstrate applicability compare proximal present admm lin admm two sg sg lin run cpu overcomplete dictionaries cosine dct dimensional grid family spatial contiguous sequence case square generate vector w nonzero level sg admm optimize provide low respective sg note step size form choice lead lin interior cast qp program formulation lin except set objective sg second lin lin scale lin admm similar qp lin sparse whereas illustrate single large sum task orthonormal wavelet wavelet orthonormal orthonormal candidate norm sparsity wavelet adapt wavelet wavelet version gaussian good c wavelet run different realization fast dimension proximal approximately take second ghz tool fact subset generalize inverse particularly interpretability expression dataset insight patient decomposition work decomposition I solve lasso problem induce penalty parameter solution submatrix readily identify component bring problem sampling gene expression sequel center run set row involve randomness deviation five initialization conclusion match already partial scheme worth way imagine add low thank latter report deviation initialization dataset background task frame fix camera try segment foreground new pixel w p make account pixel background foreground
without replacement individual sm sequentially new individual combine sample reproduce please always complexitie htb section subset matrix maximum scale matrix correspond fitness scale level several controlling criterion difference typical structure sm probable connect dependency remove multivariate curse dimensionality strongly restrict problem increase dimensional report sm partition group small subset prove sm try dimensional complicated computational sm dimensional sm try perform estimation fortunately control implication sm additional consuming reduce imagine global successfully learn sm outperform controlling section sm flexibility model discuss complexity select representative univariate likelihood study experimental sense include representative gaussian matrix easy implement fair comparison computation implementation comparison within template share flow utility computation building module list select benchmark function shift optima omit find landscape also unimodal separable optima multimodal htb make domain shift shift way also transformation sphere shift nz nf nf nf x z n nf z z bias x nf shifted bias nf b x nf f expand plus see page application fitness evaluation varied besides vice versa people tradeoff balance vary knowledge achieve instead choice every give dimensionality final moreover initialize initialization one individual generation newly sample individual constitute new widely study illustrate population problem algorithm average execute ghz mb ram point calculate pair correlate context experience observe small value result I make may dependency release set constant aim benefit issue value sm preference normal large large vice versa give estimation approximate combination time require user physical implication determine pre preference easy apply later difference good fitness optimum negative small multiple population report fast correspond cpu cpu fig htb marker l e e e e e e e e e e tail tail tail compare benchmark adopt mark bold r separable unimodal facilitate univariate case estimation degradation well good overall significance global shift search cpu size although different cpu good performance cpu grow keep high clearly population performance unimodal problem except bad performance shift performance bad optima shift worse absolute always robust cpu although bad size next sometimes grows explain since apply maximum population keep absolute problem far among global optimum require good approximated subspace dimensional approximate estimation combination well combination experiment experimental fig combination model poor global find significantly scale much time result run show bold nonparametric htb f cost always big superior performance evolutionary curve answer evolutionary curve proceed imply fact converge perform apply therefore large little size perform bad failure population cpu htb f result result include comparison nonparametric gm gm c gm word group fail dimensional quite large optima lead landscape make hard solve estimate although study local optima perform adopt apply shift surprisingly still outperform become intuitively result multimodal perform pose enough group sm common optima present experiment concern characteristic prevent perform characteristic necessarily failure success probably use dependency degree high long failure attribute failure degree setting exactly dependencie consider see essentially include influence indicate algorithm probabilistic matrix replace base four implementation model among summarize table htb nonparametric significance marker marker l dim e e e e e e tail result tail candidate switching dependency help performance promise matter three resource maximal utilize effective long minimal search huge optima increase serious nevertheless although always complicate landscape estimation perform well scale discussion size guess budget fast fast complexity requirement outperform increase also curse optima computationally expensive univariate resource discover efficiency optima separable well three propose begin gaussian usually base cost fail summarize low landscape optima fail case univariate effective success curse dimensionality less control necessary population size rely high traditional high really solve still design investigate several optimization involve whose dependency shape bivariate conditional acceptable code provide dr al variable problem name evolution base framework partition strategy activate although group subset instance code author es several hundred es study g sep es well estimation major es path current usually require es experiment lead sep es deviation derive set result population please refer population keep whether keep pressure test high confidence summarize g es test regard g deviation result result sep es nonparametric marker report leave tailed demonstrate available equivalent significance post hoc sign rank test outperform significance significance es l c e e e e e e tailed result simple separable sep es group separable es covariance separable outperform sep sep es es little well average shift hold much bad shift landscape g group function analyze effectively one behavior huge optima exception explain optimum also partly likelihood estimation high observe perform well curse dimensionality simple univariate handle problem difficulty speak robust optima shift optimum disadvantage compare sep notably consider far successful multivariate significant superiority tune potential even newly test population adopt test htb htb l e unstable current implementation subset dependency eliminate definition become since generally separable good combine empty separable solve non cpu analyze dependency eliminate partition separability scale small separable besides recommend bring speak good impact may cpu cost ask sophisticated partitioning cluster intuitively enough e g available curse dimensionality sm greedy result htb ni remove loop specify subset maximize x xx variable since sm fig sm partition subspace pick outside add variable reach strongly cluster rest outer keep generate subspace manner partition still univariate dependent htb run significance marker marker significant l tail compare algorithm summarize difference test sample dimensional limited population partitioning subspace effective illustrative experiment exclude outperform cluster relatively high contrast partition scale property learn model finding ability characterize advantage recent discrete interaction structure graphic sm also record procedure generation analyze result record strong element evolution plot variable also plot matrix variable row ranging indicate examine even graphic add read omit small purpose evaluation horizontal page length report result effect sm role mutual effect sm separable low grow strong become high interpret sparsity space fix experiment separability grey fitness function value consistent sphere strong correspond plot element show correctly shift dependency chain determine second determine see important structural htbp specific experiment shift condition fig help dark checking table fix among mostly ji ji ji partly speak effect onto hard analyze variable rough figure second negative coefficient third th compare last size etc rough experimental function correctly dark htbp especially test show shift result help explain examine find similar analyze recognize optima look like problem learn htbp shift average plot variable remarkable characterize show although find characterization problem underlie structural remarkable regard valuable aspect however number optima limitation notice implementation try possible univariate gaussian problem property try effort explain problem capability thing address solve run problem sufficient allow aggregate multiple trial role sm interaction sm implement sm compare version role save comparison respective sm experiment population size well sm find magnitude sm weakly dependent simple slightly sm bad cpu sm solution comparable quality cpu time acceptable much speak sm show much robust cost perform slightly sm sm bit partitioning sm fail marker e tail sm htb sm contribute nothing sm sm result strong function global affect eventually partition also modeling become cpu sm characterize find solution sm help model help effectively help properly strategie weakly characterization robust performance sm difficulty due curse population traditional model separable improve reduce adopt identification significantly traditional dimensional optima requirement also besides exhibit characterization solution user variable far valuable black extract design also space available offer may effective appear separable huge local gaussian discussion implementation still restrict test adaptive still issue leave suppose build denote variable dimensional select individual generality g model select select individual estimate complexity new need repeat create cost joint select individual complexity need generate primary multiplication repeat create cost complexity ignore multivariate sampling select individual calculate building identify weakly building building build strongly build building sample sampling time thus grateful yu ray wang comment code support china visit centre computer university uk mail cs ac improve continuous high dimensionality computational scaling continuous necessary explicitly discover useful great benefit box propose novel complexity subspace sm performance population characterization successful model effectively class newly design specifically strength carry kind benchmark compare traditional evolutionary genetic ga neither mutation model promise search space present global statistical well actually model usually evolutionary classified instance originally optimization research domain progress focus decade branch continuous study major branch validate low dimensional rarely ignore continuous difficulty high space rely multi increase computational make impractical name continuous adopt weakly identification sm curse dimensionality reduce overall comparison effectiveness advantage traditional separable optima traditional appropriate motivation advantage discover learn explicitly possible learn simple base ignore deep dependency adopt explicitly control attempt dimensional penalization remainder organize follow traditional present sm model adopt traditional penalization discuss experimental sm investigate sm compare sm advantage partition sm final along typical loop refer evolution p initialize population meet select individual primary adopting form normal define mean matrix base simplicity level solve strong multivariate likelihood gaussian represent normal gaussian graphical factorization introduce computation maximum computational reduce conventional must base performance superiority another however involve normal idea make later poor ability criterion find besides base adopt multimodal hybrid interestingly although accurate true distribution promise nevertheless offer multimodal problem satisfactory explanation phenomenon literature multimodal recently analytical base univariate histogram base flexible consider bin exponentially make multivariate histogram hard together control weakly dependent sm call control covariance covariance variable respectively accord coefficient see covariance evolution multivariate coefficient nearly mean dependency behavior much example coefficient case univariate gaussian significantly hold fact firstly identify
recommend quantity reflect appropriate acknowledgment support award es national institute environmental health sciences view national institute environmental health reference proposition david nc statistical nc years rich variety prior address massive prior normal form cauchy mixture generalize many prior special compete connection develop class variational massive major area rich variety lead interpretation posterior different induce prior induce parameter problem perspective normal choice family penalty appeal strong tail avoid double tail many double exponential near phenomenon new prior widely bayesian framework rely bayesian selection prior mixture mass equal exclude distribution non paradigm appeal accounting learn correspond subset one average subset inclusion predictor include predictor predictor recently may conventional prior attractive bayesian shrinkage mixture gaussian update substantial improvement carlo rapid desire inference many interesting reveal connection bayes approximation truly massive conjugate allow yet advantageous form straightforward sampling intuitively adjust shrinkage control non sparse structure inherent selection brief background primarily possess appeal lasso tail drastically shrink signal heavily interesting representation later form conjugate potentially computational complexity prior multiple hierarchical denote half parameter appropriate represent special name magnitude desirable result tail unbounded create elaborate back cauchy origin explicitly behavior prior trivial exponential ng form yet lack tail user implicit behave see distinct formulation prior able prior formulation furthermore gibb update make flexible class normal appeal name function denote distribution wishart function second q show th gauss propose compound propose denote beta becomes becomes consider behavior equivalence prior observation cauchy mixture half mix parameter place half complete hierarchy help formulate complete hierarchy bring analytical procedure much relatively hyper prior give proposition proposition ab draw dashed line pn normally hierarchical shrinkage common infer hierarchy identical hierarchy huge reasonable exist appropriate reflect underlie coefficient adjustment discuss also error formulate conjugate likelihood sampler conditional b laplace approximation low likelihood may variational posterior posterior distribution iterate reach deterministic make attractive quick mcmc conjugate take straightforward prior regression sophisticated scheme towards flexible normal scale focus many reader sparse hence follow brief posteriori expectation maximization accomplish obtain hierarchy modal term heavy careful hand estimation admissible sparse apparent essential prior density marginal illustrate tail origin drastically jeffreys np ij pi ratio approximately regression variational bayes calculate median model value bootstrappe regularization tune cauchy attain clearly particularly rule choice well due set set randomly component index place due reflect limit sample note adjust global priori whether sampler hour
day train idea rnn fix value keep mind direct layer model seven mixture expert feedforward expert expert expert expert architecture expert mechanism difference expert expert adaptively l py vector bit likelihood bit give denote code logistic online first hold constant kalman dynamic consist observation linearization investigation wu use computational cost seven weight initialize bit jacobian yx table via modify tradeoff compression performance eight achieve bad compression memory achieve compression l change weight layer change initialization refer neural change significant computational cost use manual tuning file difference tune tuned file book book news paper compression result document string string application system compose acoustic component could directly replace recognition prediction string people people slow input device mobile source predict character new create character generate character create file train observational text continuously online character capture syntactic semantic spirit decide soon little whole existence b china name kn name fortunately control example exponential ref show past gauss mu lambda alpha r conclusion prior theory h mean figure long live rnn compare rnns rnns bad rnns difference method continuously beneficial opponent move predict move opponent next ai play reason human human round practical text categorization spam image one sometimes way map several researcher compression usually preprocesse concatenation piece compression compression rate set use near neighbor use achieve competitive rate categorization shape compression classification procedure length approximate minimum neighbor store file class file concatenation file run compression file compress assign compression compression file ff file size minimize fa fa file bt assign class method datum comparison consider time bit get note difference many dataset test file large order magnitude extremely compare compression program access source modify source code essentially copy allow file continue adaptively modify file still parameter default classification experiment performance measure file file fundamentally two file store disk non arithmetic file disk cross entropy compression performance subject neither access source code file categorization assign evaluate news partition evenly category j corpus message retain count os ms windows pc hardware mac window correct classification randomize seem preprocesse methodology percent naive output code train language train split multiclass svm split multinomial naive publish dataset comparative note several dataset evaluation protocol publish result directly example zhang dataset seem dataset shape recognition contour within dataset available dataset binary image five category example orientation resolution ht count seem image text categorization slow object task compression option effective classification al shape contour representation decide percent classification c actual methodology percent nn leave leave leave nn approximated cyclic nn time feature leave nonlinear split convert leave demonstrate convert one first centroid shape project ray centroid point ray around contour measure angle ray convert single rounding run measurement cross protocol parameter surprisingly adjust resulted accuracy adjust classification exhaustive perform exhaustive confusion parameter classification publish show result decrease al image invariant procedure euclidean metric rotation angle euclidean representation try run alternatively invariant small classification achieve compression specifically design image compression perform keep representation perform compression create set learn patch patch scan patch store filter example compress figure compression still create file image visual compress advanced compression exploit human variation high frequency degree sensitivity exploit limitation leave future filter outperform method visual compression hope exposition method area weight try technique include kalman implementation promise technique filtering implement rao filter future combine prediction expert ensemble forest technique gap rnn seem relevance architecture reduce rnns conjunction front remarkable tackle broad metric anomaly detection speech equally remarkable challenge beyond require store well architecture apply univariate seem trivial rnns acknowledgement would understand provide compression would thank discussion available ai run program source compression compression benchmark report detailed show understand improve intuitive module remain hope description increase understanding facilitate transfer recurrent modeling adaptive text playing compression gain great domain unsupervise audio researcher understand intelligence close perhaps machine machine open closely prediction partial compression compression benchmark compression memory record breaking ratio expense memory mb wikipedia specialize web website compression well act concept intelligence number pf book news average file text format reference book crowd principle computer point file topic knowledge format arithmetic code compression format source compression character field compression corpus compression benchmark corpus file appears table test pf implementation compression huge success rarely compare machine core difficulty lack scientific inner good incomplete description source close language extract examine provide explanation learn community lead inspire research develop propagation character character alphabet day make neural contribution demonstrate enable machine contribution novel text classification outperform show achieve develop compression feature problem arithmetic description improvement section conclusion access create consider character alphabet character character character compression two stage first prediction give character encode file scheme arithmetic code arithmetic code preferable arithmetic produce near predict solve compression arithmetic arithmetic two encoding encoder character compress character compress output original character decoder decoder reproduce compression create predict compression generate character case compression character file try character arithmetic need seed encode character arithmetic encoder essentially store process work suppose alphabet character arithmetic character visualize number see assign character arithmetic encoder region expand assign arithmetic measure directly code character character bit code character another common text prediction entropy character suit character redundant contain less individual pixel maximum variant partial compare length ratio text character pixel scan bottom another mapping maximize locality al metric anomaly cluster variety file music li distance string kx kolmogorov string length length program kolmogorov good kolmogorov compression approximate cx cx compress compressed compression close kolmogorov et cx justification use compression without modify fortunately open modification purpose define ex symmetric always e x report al metric perform anomaly well theoretically although use model context match unlike noise long specialize make prediction level sequence make bit detail depend version detect overview architecture general recognize preprocesse modeling ht model predictor pass secondary
discover remove fortunately correct original convex penalty strongly broad mc strongly differentiable hence smooth k bfgs name differentiable lagrange smooth hand tends become respectively recovery impractical fortunately empirically enough recovery theoretical lead mc algorithms fast converge rank positively literature back linearize bregman approximately solve pursuit compressed phenomenon discover converge whose pursuit parameter condition condition equivalent mc rough limitation generic mc exceed matrix mc respectively explicit improve firstly strongly programming rank recovery broad range exist optimization literature secondly prefer objective algorithm give word suitable organized follow section summary exist strongly programming rank recovery explicit conclusion notation define transpose euclidean nonzero equals denote summation singular denote verify important recovered mc problem ambiguity singular positive right vector orthogonal completion let respectively q save notation subgradient nuclear function form subgradient whose shall act letter operator mean symmetric operator subsection exist via give mc impossible unless author decide rank incoherence characterize principle sparsity incoherence obey incoherent eq canonical assumption namely incoherent guarantee exact programming strong assumption easy exist assumption obey incoherent inequality statement mc assume uniformly point convenient zero probability rely bernoulli sampling simplicity ready result completion principle analysis convex let fixed incoherence unique distribute among cardinality write numerical least unique programming enough exactly recover mc problem computable bind result discuss strongly program completion like rank partial show unique dominant exceed explicit sampling unique standard theory assume matrix programming eq multipli strong convexity optimality lagrangian unique sufficient necessary exist shall condition list least obvious satisfie valid dual via assume solution strongly programming theorem high probability verify abc event construction verify practice quantity hard computable bind summarize sampling ratio discuss recover rank fraction fraction programming provide mc present state sufficient unique strongly idea follow feasible perturbation unless arbitrary arbitrary hold eq construction always eq triangle schwarz inequality imply q convexity bind assumption theorem uv sign strongly verify go theorem numerical checking suffice parameter final expression inequality notice inequality choose get low side monotonically easy maximum attain eq ab lemma proof see give upper use q cauchy schwarz matrix convex matrix analog convex numerical recovery principle explicit lower involve complementary result counterpart guarantee observe corrupt least therefore direction completion robust component zhang chinese visit university china multi advance process z completion j pp linearize bregman iteration compress z linearize bregman norm pp linearize j sciences relaxation transaction theory pp plan proceeding li analysis journal uncertainty signal incoherence decomposition journal chen atomic pursuit pp compressed sensing theory pp ph thesis stanford principal physics biology pp recover low ji xu robust video imaging sciences pp ji liu xu denoise pattern liu interior journal pp lin j wu chen fast convex recovery technical lin chen l wu augment multipli exact report min keyword pursuit proceeding conference information collaborative completion cognitive chen bregman iterative mathematical positive program pp linearize compressive sparse pp via minimization w minimization conference decision pp xu minimization mathematic pp simple r nj accelerate gradient least optimization stream international journal vision pp rao robust exact matrix convex optimization journal available w
ii optimal optimizer table take computed design dimension second consider turn point quadratic scalar popular modeling disease cancer efficacy assume setting locally e depend optimal take equivalently bayesian apply locally design et optimal design next description cover model optimality reader design design transform via transformation place sake although red quickly design transform back give interior thank work national foundation dms security state design give formulae optimal prescribe li follow hold respect c lemma far cx subtracting since positive give q last second expression get independence multiplier easily minimize prove claim claim p design li paragraph assume linearly strict li previous apply ix I q complete small eq conversely constant ix vb conjecture section design formulae unknown design single illustrative design give elegant sign sign chernoff chernoff design state dimension involve search sign notation assumption method explicit formulae thus computational design greatly give current sign fast along line without search sign formulae sign practice polynomial regression turning point wu generalize reference recent show design standardize optimize estimation design characterize certain generalize use design clinical carlo efficient scalar normally transpose parameter take independent li li kk satisfy small give nonzero achieve e class admit singular follow elegant origin ray origin design put take
erm unfortunately enough notion online choose sample slightly stability batch however set furthermore know notion review threshold batch online binary randomized uniform show learnable notion loo begin notation definition notion might necessary binary stability notion notion uniform loo learnable problem future open question batch set unknown find minimize define empirical asymptotically regularizer rate whenever mention monotonically I say notion common datum loss risk learnable erm recently much set hold learnable learnable erm turn notion loo stability measure latter general notion shall see commonly notion loo stability measure leave look loo stability remove dataset algorithm distribution index universal average loo loo stable loo loo average stable opposite counter implication direction exception mean matter loo stable loo stable stability notion name slight strong loo simply similar loo need smaller hold hold uniform loo show erm general learnable loo stable learnable erm loo erm nice consequence set attention loo stability loo natural analyze online algorithm batch stability notion notion measure another look mention necessary batch weak another stability except loo stability learnable exist stable addition allow learnable exist stable always potentially learnable convex exist deterministic strongly set observe mention online thought thought supervise classification datum regret definition online say lipschitz convex algorithm regularize erm regularizer functional measure special mirror descent see majority type online stability relate uniform loo weak online difference stability loo change datum point algorithm stability loo weak uniform asymmetric obvious loo online learnable achieve loo case strongly loo loo loo stable well regret loo currently interesting non supervise online mini pick want loo stability sufficient performance unfortunately always exist online loo rate stable online learnable particular I believe technique always long stable able notation regret equation allow online stability rl lr fs I I fs z rl fs fs fs fs j fs fs stable fs I fs fs fs fs fs extra double summation erm easily see fs fs erm regret stable general summation fs fs j fs j I fs z r prove always hard double negligible algorithm eq rl fs combine symmetric loo stable stable loo uniform stable rl fs fs fs fs I I fs fs I fs fs loo stability fs corollary either imply loo erm loo lipschitz rl erm obtain z z hence I l lipschitz loo rate lipschitz regularizers z follow minimize j j rl j z conclude prove alternate regularizer necessarily lipschitz strongly proof ax c cx aa I prove sequence online method method mirror descent mirror low point previously choose rather regularizer previous stability refer surrogate loss minimizer point surrogate loss fs r r broad bound h prove instead stable z exist stable learnable rate apply corollary previous observe dataset randomize weighted generalization distribution online know online learnable learnable randomized adversary problem fall category variant interpret randomized loo stable analysis useful existence randomize formally I functional complexity satisfy might mean tuple gaussian case finite h z ensure algorithm incur hypothesis sample advance definition extend fs average distribution go instantaneous sample probability go g change z randomize introduce play instantaneous apply z loo expert regularizer instantaneous randomized uniform loo finite expert iteration pick satisfie kl regularizer regularization make kl di randomize long point randomize lagrangian choose ji consider choose lead c j tt must loo regularizer expect lipschitz p h regularizer lead lm l c integral statement theorem establish instantaneous learnable uniform loo demonstrate finitely respect online learnable randomized loo restrict symmetric loo stability sufficient give illustrate loo learnable set erm loo deterministic deterministic loo randomize algorithm learnable stable loo loo stable regret hypothesis space loo stable dataset occur show loo stable adversary pick algorithm incur allow allow linearity make hypothesis distribution convex denote learnable uniform uniform loo algorithm stable still uniform loo make universal loo hypothesis change algorithm achieve erm loss odd equal erm pick would thus follow randomize subgradient pick uniform loo fs fs z I loo stable previous plus minus removal achieve whenever pick pick pick pick seek track boundary increase incur loss incur average loss average way learnable learnable learnable show loo stable stability achieve regret stability loo necessary loo strong loo loo stability batch show condition exist learnable learnable set deterministic hypothesis define binary vc finite batch learnable batch learnable erm loo online pick binary learner iteration learner iteration number hypothesis achieve entire sequence even effectively learner iteration learner entire regret regret randomized limit go hence conclude learnable potentially loo argument characterize set learnable know loo learnable potentially loo stable expert play round randomize loo construct expert
km virtue convexity equation concavity factor decomposition gradient matrix eq consider whenever virtue sub histogram computation algorithm computation follow ij simplex cost warm start simple minimize project subgradient linearization subgradient nest loop loop parameterize subdifferential metric subgradient concave part taylor expansion inner loop gradient function increment terminate outer loop realize computed objective far return dc minimize minima link point crucial I pm q qp p z minima good global guess far point interpret r popular initialize minimizer way initial replace discuss q histogram need trick satisfactory seed near phase trick yield explain experimental minimize sum either depend value feasible set unit sum coefficient intractable norm ball pseudo straightforward three condition consider metric algorithm point applicable negative entry solve replace approach dot product handle cone constraint define leave technique section arbitrary table dot candidate table center instance easy table trivially independence approximation tend histogram definite positive call table table briefly explain result write small typical histogram polytope directly tractable large unique minimizer program hessian diagonal compute metric coefficient negative uniformly histogram contrary cone unit ball must order computation arbitrary center independence exhaustive dominant machine bin independently jensen hellinger divergence usually well histogram straightforward euclidean illustrate instance experimental bin enough histogram either co occurrence form prior color bin bin bin distinct support regardless support describe form arguably matrix mahalanobis euclidean root directly information recent review literature follow learn mahalanobis semi criterion candidate model neighbor inspire consider possible learn ground metric little conceptually mahalanobis operator learn operate worth although design vector histogram statistical motivate propose bin parameterized propose naturally operation perturb histogram accordingly describe perturbation semi histogram side key distinction algorithm follow present ground normalize histogram optimization require flow warm gain approximately problem toolbox projection loop objective unconstrained solution positivity optimize slow constrain newton image histogram obtain implementation feature mid computed dimension sum result binary task class form form test amount point hellinger coordinate root public mahalanobis default representation histogram originally euclidean histograms hellinger hellinger mahalanobis build upon euclidean learn mahalanobis significant observe simple confirm classification class class normalize neighborhood directly stepsize experiment fact normalization comparable step inner loop loop progress step tried grid claim select metric table consider machine estimate use train svm bandwidth distance set range fold fold cross select distance general amount regularization minus eigenvalue matrix consider distance recall neighbor quantity average classification point consider select average close neighbor gap compete retrieve task compete svm agree usually near support vector machine hellinger geometry usually intuition far validate mahalanobis perform well histogram show vary use typical table appear impact additional curve point despite overhead table seem table average please figure except hellinger figure simplex explain metric directly progress agree intuition metric neighborhood parameter comparative parameter neighbor classifiers experimental average overlap set classification despite typical independence tune adaptively unique dataset type provide good ground project difference compare compete superior feature histogram ground lot recently argue computation matrix distance metric thresholded attractive compute suggest learn improve accuracy believe distance structural looking minimizer criterion algorithm lie call compute pair warm carry fast implementation could provide computational improvement accept would considerably optimal neighbor candidate learn histogram available u decade machine metric parameterize distance distance choose date practitioner knowledge limit considerably scope lift metric ground follow use descriptor image metric histogram arise frequently language computer vision bioinformatics involve object simplify occurrence frequency feature represent histogram histogram color sift bag word gram follow principle distance propose histogram theoretically popular influential work distance think vision histogram histogram ground machine motivate prior problematic problem knowledge argue universal problem machine ground adaptively organize distance similarity obtain minima subgradient mahalanobis metric inspire much propose unnormalized histogram program sum resp resp element remove sure constraint rank
expansion wavelet expansion truncate whereas thresholded wavelet practical toy realistic inspire study show good moderate wide alternative hypothesis choice publish procedure yield method possibility variety interesting analyse contain measurement galaxy particular event ray sphere event sphere object depend production density universe field interest origin arrive particle interact atom huge cascade secondary secondary detector ground permit measurement direction arrival ray particle energy observe rapidly low numerous composition responsible acceleration least phenomena high energy event decade addition accelerate particle recent extremely recent alternate hypothesis concern origin star extremely yet many origin energy energy extent propagate effect energy cutoff ray physical identify account chemical high energy present knowledge energy order range completely alternate origin alternate production decay big year old publish primary impose limit kind attempt origin statistical signature production likely two rather source second correlation direction arrival nearby attempt plausible production hypothesis probability direction like nearly hypothesis distribution correlate local universe could superposition distinguish understand production year several arrival sometimes difficulty lie source locate ray permit acceleration depend path small small atomic number typical small neither ray factor error direction arrival degree degree heavy long come source distribution analysis sure direction total number event completely constrain meaningful analysis arrival experiment correlation nearby active locate distance km obvious investigate new insight statistically well publish particular work focus question although seem active goodness directional wish discover paper organize ray propagation carlo parameter present supplement devoted investigation tool analyse carlo physical ray emission propagation beyond decide qualitatively simulation although representative toy ray emission one source arrival energy alternate assume uniformly volume radius origin indistinguishable direction arrival identify easily direction address uniformly identical energy draw random energy ray square distance produce study high energy correlation arrival assume coverage exposure compute straightforwardly simple consideration detail exposure display cutoff simply sphere ray subsection model use exposure incoming event direction important medium impact optical range via foreground star align line major star reveal general near align emission infer direct field nearby dense three field direction amplitude measure rotation amplitude amplitude reach local outer galaxy align plane optical scale field disk inside impact energy model physical component arrival induce model center superposition justify central limit theorem sphere irrelevant truncation typical atomic regular atomic alpha propagation length velocity income regular field axis typical field regular part field typical ray come galaxy income ray maximum typical disk distant location universe qualitatively strength less know correlation length large energy although shape region index exact shape spectrum impact various energy field implement ray first ray exact induce refinement easily change practical application high help light particle factor account energie outcome extreme many source source right event different angular capability orientation respect equivalent direction display exposure exposure exposure map assume accept angle incoming period exposure right effect exposure perturbation would measure currently arrive whole subsequent handle choice take acceptance exposure incoming event direction density nan observe direction position physical phenomenon observational need reformulate sphere function directional algorithm small may collect incomplete optimal way pose nonparametric alternative derivation view shall twice old exponent space speak shape neighbourhood nan exclude constant critical alternative essential estimator nonparametric like pixel constant estimate count event sphere could procedure precise contrast nice various context detection uniformity goodness fit test homogeneity uniformity account directional model noise uniform exposure rotation adaptive ideally moment along arrival direction handle formalism sphere clear good especially usual respect sphere consider remarkable concentrate enable traditional basis usual event false second alternative separation optimality eq infimum test collection mean nonlinear analogy homogeneity procedure index respect make implicit know optimally argument concentrate would probably mathematical regularity achieve hypothesis individually define adaptive loss prescribe test interesting procedure plug density property view prove argument suggest minimax exponent estimator approach favorable measure minimax critical radius detection see asymptotically hypothesis respect nonparametric infinite popular minimax technique thresholding post processing constant difference situation especially consider drive density interpretation thresholding quantity one compactly line number actually sphere concentrated replace type finally prescribe deal benchmark er von function high dimension sphere sake comparison run simulation test near point sphere draw take value admit exposure case quantile big exposure uniform quantile point test angular geodesic counterpart evaluate detection typical maximize priori consequently care quantile compare section procedure tune prescribe exposure detector figure carlo replication choose probability scale c pt pt c section power alternative test table online supplement power norm online supplement plotted supplement observation use exposure investigate small choice publication speak plausibility alternative alternative unimodal would uniformly repeat physics inspire rise rich frequency give alternative density put density unimodal observation display family alternative repeat emission source explain source uniformly distribute realistic type matter generalization straightforward source error incidence angle namely exposure detector kind least one ray ray describe source randomly spherical radius assume simulation value namely play incidence assume resp sample resp statistical much energy go detect become simulation scatter source source specifically multiscale alternative understand respect source draw table table alternative percent prescribe level practically band vary j round bold namely case refer supplement curve operate roc procedure along wide procedure curve concave envelope accordingly analyse table represent choice method roc complementary relevant first sensitivity use regular expect unimodal roc supplement procedure figure appear behaviour radius vary case ns illustrate sample produce shape display curve logarithmic highlight comparative performance bad sensitivity case mainly group standard whole near nan distance near sensitive discriminate vary alternative consistently slightly one illustration size low detection supplement less table highlighted multiscale method appropriate future datum set behaviour respect sample soon remain display density alternative power remain roughly separation analogy euclidean space numerical consistent claim base bound density consideration become adaptive pre remain angular sake comparison tune clear give precise close parameter plot alternative figure structure scale observe respect truly efficient parameter level alternative alternative incomplete test exposure column reach point quite taking efficiency test public arrival direction directional event distribution length along perform reference notice early reach minimal interpretable computed value significance isotropic data table datum c c quite sensitive except take theoretical expression statistically monte carlo suggest consider significantly soon turn rule multiple methodology appear evidence realistic alternative
multiclass multiclass website correspond randomly remain problem two version unconstraine constrain classical averaged figure lar classify multi class lar svm learn sequential set fixed note experiment state use stability different value sparsity model c ccc sparsity l show two representative curve summarize sparsity due lar equivalent svm note give linear interpolation compare believe provide ccc corpus un un show sparsity outperform svm true similar result corpus configuration classify comparison figure dataset right good svm less fact solve select feature entire approach propose advantage decision take consideration rank function associate purely redundant inter feature individually induce group tree also although similar filter embed fold describe brief good drawback search perform sequential embed practice classifier remain entity sort black box additionally entire combinatorial resemble similar goal use sensitive similar model optimize quantity decision mechanism inference tie heuristic article classification classify take consider combinatorial inspire reinforcement show problem additionally work easily extend overfitte experimental maintain black universit france technique representation approach whole induce standard classification model classify wise extend inference traditional svm lar class equal performance machine direction modern selection encourage sparsity goal improve prevent overfitte sparsity representation entire choice vary classify infer look difficult underlie balance optimize balance result high preferred second extraction able space dataset onto subspace wise automatically classify use subspace classification iteratively choose classify wise introduce inspire reinforcement contribution propose new sequential obtain term classification wise sparsity e classify classification problem series corpora lar qualitative define explain interest approach approach detail also give qualitative problem supervise want associate category parameter commonly empirical risk solution moreover feature classify risk penalization encourage obtain feature possible classical feature perform feature classify input use classify easy classify one predict label vector convention feature classify category wise advantage feature encourage qualitative explanation extension classifier classify number classify general select classify input differently cc right circle one terminal bold bold red illustrate reward receive reward define equation original decision mdp classify attribute feature classify classify category deterministic currently classify select possible action state set currently correspond stop sequential decision terminal action define q scoring action score quality action follow obtain started follow parameterization reward reward episode obtain see reward initial correspond empty pick classification choose reflect take action state reward feature avoid classify incorrectly choose feature explain goal find parameterization describe let show maximize equivalent wise empirical risk mdp equivalence mdp classifier detail due computed possible state take state action state represent note may representation propose projection restrict select attribute intermediate representation manner action easily distinguish classifier project higher dependent offset amount find optimal parameterization reward wise regularize explore state action policy consist well bellman compose main iteratively choose state policy classical regression obtain generalize estimated action visit iteration parameterize use classify core suffer select regularization classifying training general allow overfitte allow classify input constrain constrain action vector handle ignore type action p cm feature unconstraine un select choose force choose classified
school temperature degree capital city life high temperature cp school list exp aic bic life cp life exp l l correct estimator restrict scenario state aic estimator uncertainty scenario estimator unbounde correctly restrict fold ten correct validation exactly diversity row seven row covariate represent interest specie represents find specie area area km km km original miss full sub aic bic specie area adjacent shrinkage along restrict shrinkage ps error error show thousand represent bic model bic likely notice bic compete uncertainty consequence cause sensitivity restrict estimator outperform estimator predictor capture variability monte conduct positive shrinkage x si si si pi nn initially repeat calculate estimator norm various measure compare amount rmse unity superiority list shrinkage cccc clearly restriction unbounded decay horizontal positive move event small among estimator neighbourhood unbounde fast rate rmse neither suggest maintain superiority estimator wide range shrinkage panel maintain superiority wide range panel suggest shrinkage preferred remain specify statistical correctly one go wrong assume wrong case estimate expression distributional distributional setup much alternative meaningful asymptotic generally simple squared expectation loss write covariance matrix evaluate compare risk prefer say dominate however hold every risk local cumulative eq dispersion dominate asymptotically context propose distributional regularity variate expression form quadratic let dispersion asymptotic distributional present local statistical property estimator restrict gain substantial near review shrinkage estimation multiple bias expression estimator covariate full hand base suitable criterion subset covariate become nuisance step full estimate repeat cross rate restrict superior shrinkage consist respective close misspecification conduct monte characteristic shrinkage size nuisance carlo numerically compute restrict shrinkage study fact outperform estimator unbounde unbounded restrict however approach rmse unity shrinkage perform wide nuisance subset panel either select restrict shrinkage towards subspace space penalize member pls perform estimation simultaneously estimation shrink towards sub space towards change sign practitioner although estimator take care part coefficient eliminate shrink introduction half decade shrinkage around performance shrinkage lasso partially comparative shrinkage absolute find review literature front communication proposition remark positive shrinkage study sm subset covariate contribute restricted covariate may subset combine estimator outperform validate preliminary validity restriction thus outcome preliminary illustrate shrinkage estimation positive shrinkage degree uncertainty carlo keyword phrase estimation rmse branch estimation considerable attention statistical model linear unknown whether parameter obtain insight many situation information contain positively nevertheless advantageous rather parameter datum reliable useful whether accept confirm result mind depend usefulness uncertain may incorporate uncertain prior depend restriction restrict preliminary later estimation stein type take alternative estimator utilize sample prove useful shrinkage attention analytically demonstrate shrinkage preliminary usual maximum study give description large error analytically numerically preliminary dominate stein estimation uncertain various location uncertain available shrinkage certain apply real life purpose affect interpretability sign type stein estimator context preliminary indicator test practical shrinkage incorporate shrinkage life form shrinkage estimator outline estimation utilize model combine sub subspace response prior nuisance subset usual nuisance covariate bic restrict shrinkage prediction shrinkage sub fold cross validation divide roughly size aside term subset use raw correct prediction validation estimate bias random varie
determinant confirm know set maximal determinant extension resolve small resolve order search matrix decompose spectrum maximal determinant hadamard large determinant change sign generality open since hadamard experimental code mapping vice versa confusion thank correspondence convenient determinant split determinant unless subject investigation small three class four order sharp order q factor arise sharp work high gram block see tree order mod open odd order resolve upper use early author infeasible thus computer describe gram order low odd order order determine upper order integer largely take definition design determinant design odd normalize normalize iff iff show odd convert sign regard equivalent row column determinant suggest sign hadamard iff sign permutation matrix gram iff sign order definite furthermore call candidate gram matrix summarize gram gram increment detail admissible gram good step rely follow originally use maximal gram set member diagonal take furthermore minor equal potentially unfortunately multidimensional set expensive restrict diagonal element last entirely latter search done result approximately list candidate complete gram describe involve combinatorial regard row know find row row correspond principle hadamard rely member family gram family gram build column general gram clarity algorithm gram tree application constraint recursive search search implicit subtree root search increment solution simultaneous solution recursively relevant subtree justify sign permutation partial order call level create nonempty contiguous collection frame frame consist width refinement frame level frame consider number entry may weight column output otherwise variable subject follow w zero entry call search recursively subtree elimination always interval variable uniquely non discard preferable basic proportion elimination illustrate candidate elsewhere yet associate vector comment translate together entry entry impose give linear child lead solution outline gram equivalent column relation efficiently know matrix polynomial correspond constraint gram principle new pruning matrix appear block know row matrix depend especially large arithmetic need number case fail backtracking succeed existence decomposition rational far design three corresponding design gram computational maximal determinant describe determinant take hour find nine equivalence class nine twice matrix website g seven candidate matrix decompose run nevertheless attempt replicate involve tree nine matrix characteristic determinant decompose give three hadamard order differ three switch decomposition search decomposition equivalence run program give second similarly second equivalent three design determinant much case give bind previously corresponding equivalence maximal indicate figure correspond maximal factor hadamard least backtrack take hour find website characteristic characteristic polynomial second gram matrix decompose equivalent stop reach explore search decompose find take hour get hadamard order solution expect website upper order order bind attain equal thus last gram cc bind pt pt maximal determinant summary pattern continue case see gram must since attempt construct hill construction base hadamard order fail plausible may resource candidate gram improvement gram take processor candidate hour show decompose construction hadamard appear elsewhere determinant set spectrum include metropolis stein gap gap later result spectra spectra find shorthand computational use find finding run second gram appendix determinant apply minor non diagonal minor block block principal minor submatrix principal determinant q assume consist depend take first thing prove explain introduce replace I expand submatrix operation arise result q leave question empirical search empty induction theorem hold want form subtract expand find suitable element write n r r let satisfy condition true element last column claim determinant claim positive symmetric subtracting
study framework would bias proposal toy aid splitting bivariate shown figure three hasting firstly structure might secondly low separate proposal display log reaction effect improve ability plot create expensive integration separate area low exploration first target rate energy expand parallel recover chain spread target chain accurately mode reach outer mode initial mode hasting exploration successful bottom evaluation scatter normalization sampling scatter adaptive metropolis algorithm half illustrate figure plot bin split bin stay constant show histogram evaluation point show bin bin run uniformity within hence reaction movement paris exploratory stage preliminary stage much attention propose automatic combine upon component adaptive method wang heart interact feature decrease improve explore density parallel adaptive wang interact illustrate modeling lastly overcome encounter spatial contain full pseudo modal remark technology introduce measure device ever grow accordingly linear several decade complex integration growth largely interest distribution arise allow practitioner increase algorithm density invariant current proposal parametrize state accept chain reject state distant chain mode sample outside ensure hasting study sample practice even run convergence accurately approximate reasonable currently available power illustrate highly define high correlation tune manually possible issue monte carlo refer exploratory discuss trait multimodal high connect trait process wang improvement adaptive spatial image distinct goal firstly proposal past sample largely improve already mode might different explore adaptation prevent adequate exploration alternate solution adapt secondly whose goal encourage include energy wang latter distant potentially unknown mode often consume practitioner code merely exploratory believe room exploratory learn particularly would ideally continuous space associate interest user consume model without purpose carlo various idea work aim automatically algorithmic method able much fundamental mode highly multi instead proportional peak idea behind parallel employ chain subsequently dynamically move sequential approach whereby move strategy phase transform considerably across temperature relate partitioning partitioning auxiliary iteratively fundamental wang energy sampler convergence reach temperature metropolis move temperature chain interest distribution use mode move wang partition along reaction energy chain invariant iteration instead chain various wang heart discuss combine energy sequential carlo counterpart central sequence successively reaction coordinate manner largely select monte choose target smc mostly hasting gibbs move increase popularity exploratory adaptively paper dedicate solely namely review result call literature create mcmc principal past sample encourage sampler mcmc consume practitioner save automate exploratory nature complete proposal distribution careful employ encourage exploration exploiting previously explore mode proposal visit prevent reach direction axis additionally combine wang desirable grow recall constitute core improvement task interact encourage end term answer previously generate time admit biased iteration distribution converge infinity restriction coincide restriction multiplicative mechanism improve would analytically invariant validate b straightforward bias univariate biased partition state along left plot partitioning case integral area coincide constant however integral estimate wang generate estimate infinity replace constant wang reaction coordinate choose decrease typically invariant behind increase towards visit tradeoff exploration hasting conjugacy wang use criterion meet criterion meet close last criterion meet control criterion already meet describe generalize wang partition reaction sample sample transition flat ix normalize I explore region decrease frequency useful follow notational simplicity already answer reaction coordinate regardless model reaction coordinate one reaction far wang increase flexibility efficiency sampler know bin partition reaction coordinate depend issue one sample evaluation first wider allow wide initial must decide bin due difficulty select bin bin bin normalize important maintain uniformity within movement realize reaction bin artificial within strongly skewed artificial leave bin bin difficult line within sophisticated outline contrary new create right bin bin start equal specify bin half bin exhibit chain bin bin close split bin bin split check flat meet meet mean easily bin hence keep automatic distribution bin certainly would never reach demonstrate strategy bin allow bin allow bin say reaction wish bin induce tail find new mode become bin include low state take define inner time bin split bin reaction coordinate algorithm idea transition detail alternatively exploratory seed mcmc code wang adaptive mechanism proceed certain choice reaction coordinate inherent example model application could employ quantity interest demonstrate include fourth application walk propose explicitly algorithm preliminary mcmc reaction parameter state interact increase encourage within wherein level variable average measure across calculate consume study convergence towards include describe likelihood employ follow represent select induce explore select noting option would ensure different size emphasize hasting flip high explore ability poorly preliminary value space biased importance use seed traditional mcmc calculate exactly examine number parallel wang aspect examine chain mention target target hasting explore specifically explore much latter run indicate partly seminal paper describe take benchmark smc wang move gaussian mixture weight call follow take invariance likelihood permutation component lead switch label mode replicate emphasize study induce computationally increase parametrization along reaction easier explore refer article choice reaction coordinate default component weight explore multimodal unnormalized weight handle straightforwardly simplex smc parallel detail naive improvement instance plausible reaction propose mcmc chain iteration point draw parameter compute quantile conservative spirit equally divide go twice initially explore instead robust iteration terminal number situation point confirm parameter overall cost acceptance meanwhile adaptive proposal rely evaluation initial number ess consecutive metropolis particle choose induce evaluation depend computational cost mcmc plane restrict plane replicate symbol check visit project dimensional clearly chain modify wang chains final importance bias importance put particle point proportional quantitative admit mode mean run mean highlight context smc confirm explain distribution mode expectation might though high ht method hasting monte smc initial quantity consider hyperparameter simulate high unchanged chain close another likewise particle illustrate degeneracy smc though important exploratory knowledge region parallel mcmc give initial estimation mean parallel hasting smc concentrate average independent algorithmic change number evaluation iteration chain fail mode result huge chain precision suggest unit available l number chain quantity average posterior plane finish identify region identify ice employ likelihood employ posterior similar original neighbourhood interior block mcmc propose flip fail however demonstrate power demonstrate metropolis hasting run preliminary hasting explore divide evenly split time split stop reaction bin conclusion run due flip mixture calculate worst taken example whereas require case wang add slight additional wang encourage alternate explore chain induce mode top leave region bridge central flip metropolis hasting explore absence ice encourage presence pixel tailor overcome explore ise purpose automatic exploration core community literature obstacle implement wang bin speed modern interact algorithm demonstrate density wide discrete mcmc unified practitioner field
bt h conditionally order conditionally remain small conditionally value k event consequently paragraph depend normalise vary hence v view enyi conclusion proposition consider v n calculation imply entail v survival mention eq collect n conclude let sake proposition application conclude let remark quantile eq condition asymptotic obtain heavy tailed explicit simplify lead separately since asymptotic impose whereas impose entail eq collecting imply finally mm st team france fr address estimation heavy tail functional covariate quantile range near boundary depend extreme introduce investigate quantile extreme value dedicate extreme quantile quantile tend propose heavy tailed adapt recorded period covariate consider fully model observation similarly spline estimator fit penalize likelihood covariate establish besides covariate curve come illustration quantile covariate hand smooth functional deal move window hand extreme comprehensive methodology framework make tailed parametric amount survival decrease polynomial three situation zero enough quantile quantile locate boundary outside define finite infinite denote cumulative distribution function cumulative quantile speak quantile rate vary characterize tailed account regular distribution satisfy given precisely want focus end point tend go infinity estimator use move window response ball role design concentrate go covariate belong quantile mean estimator adapt highlight unconditional examine de consider summary result slow wise conditional quantile interpolation slice c thus rely extreme extreme quantile tail estimator paragraph note adaptation achieve magnitude estimate tail index heavy give notation condition establish quantile control respect variable ii asymptotic distribution satisfy eventually situation arise result tackle satisfy situation eventually small appear sense application next paragraph family tail assumption simplify paragraph heavy tailed family introduce base log large integrate extreme quantile function let situation arise lead follow hold weight hill weight detail simple example heavy decrease case vanish quantile example proportional fr extreme value theory furthermore distribution extreme quantile european express around physical water answer dataset transfer co proportion transfer see spectra function discretized dataset co support machine regression see overview proportion perturb perturbation eq q identically fr furthermore value estimate conditional quantile end distance derivative denote base b chapter discretize depend index select thank estimator extreme two function associate denote
hierarchical wishart use link analytic technique variance make agglomerative particularly balanced agglomerative criterion produce cluster criterion dissimilarity thereby obtain partition partition p c variance criterion l vc c dissimilarity class note singleton early efficient hierarchical respectively cite early make mid present briefly exactly computationally way construction neighbor mutual chain rnn consist nn follow necessarily pair reciprocal rnn dissimilaritie tb nn irrespective rnns soon arrive use traditional store dissimilarity store algorithm criterion previous ambient particular something normally tb q dissimilarity linkage centroid either account des select grow pair rnn cluster update rnns step rnns return day find sum square latter term monotonic variation criterion sequence unweighte general dissimilarity also method optimally pack address include construction whenever reciprocal near meet distribute implementation parallel dissimilarity method et chemical database reciprocal near hierarchical assess al scientific social science cite linkage pass make create subset linkage analysis comprehensive agglomerative property agglomerative method induce hierarchy identical observation cluster important software package quite handle limitation impose adjacent near neighbor feasibility take self k et application hierarchical self review include hierarchical map term follow grow discuss local hierarchical wang model analysis cluster et identifiability next hierarchical gaussian cluster analogous cluster identifiability criterion criterion refer agglomerative feasible cut concerned chapter closely eigenvector sense interest innovation development eigenvector reduction algebraic give min frequently somewhat direction cluster focus xu classified select base density split split information dense region dense step create structure partition cell sort density center traversal neighbor cell category wang spatial rectangular cell represent hierarchical child level space difficult cell child grid cluster database noise partition recursion multidimensional grid cut minimal dense half use grid turn clustered topological search algorithm main step calculation sort e traversal block cluster define uniform dimensional datum represent datum set grey scale treat create grid cell image great deal transform segmentation grid lee et base algorithm dense low therefore arbitrarily distribute important advantage noise important discover arbitrarily shape since find et lee et integrating apply cluster cluster large spatial database et distribute cluster arbitrarily deal divide consider cube near neighbor neighbor inner capable find spherical computational detailed presentation wang see study integer expansion advantage system system case string common us metric ultrametric example bound order index place split stre leaf cell assign successfully retrieval root domain hierarchical long agglomerative development cell algorithmic computational domain area back decade decade early mr york analyse la paris rd ed ms cluster behavioral application york massive collection thesis hierarchical distance liu xu th international conference database dc pp agglomerative hierarchical journal classification des des structure hierarchical self hierarchical map journal signal processing xu discover database nd international conference discovery pp wu algorithms journal ad classification review statistical international conference parallel electrical engineering dc computer p history span tree history la agglomerative automatic document journal da base discover cluster noise proceed international conference discovery mining york towards curse dimensional international conference basis surveys mf relational cluster scientific de par en les de analyse des self rd hierarchical journal mathematical imaging ic analyse paris le b correspondence von survey map connection journal art f multidimensional cluster self bayesian segmentation image vision correspondence haar dendrogram journal institute massive ultrametric embed journal scientific nh lee clustering stream record semantic multidimensional behavioral sciences york cluster span computer journal xu density hierarchical th international conference zhang database journal optimally single computer numerical hierarchical constructing manifold principled transaction pattern intelligence van nd ed wang wang proceed conference mining rd international conference basis pp principal subspace visualization transaction visualization ed science technology hierarchical cluster visualization j de ia na red de ia agglomerative processor journal mode analysis reduce effect ed xu survey algorithm transaction xu computer xu international conference dc computer pp xu j fast large database mine spatial international agglomerative algorithm discuss software environment look mixture focus hierarchical finally develop grid agglomerative dominant embed scheme aim reader attention practical method point view effective view helpful target attribute numerical vector space formulate form rectangular inconsistent relation cover cluster interactive user storage retrieval recognition survey coverage include xu review include role retrieval make van et various mathematical view look normalization historical remark motivation agglomerative formulation wide theoretic reciprocal near neighbor near hierarchical overview self survey development grid particularly us consideration consideration relate comprise group decide group similarity pair symmetry iff triangular triangular space distance family positive integer chebyshev distances special cosine retrieval match vector query query close cosine dot product mahalanobis review metric distance distance map datum determine widely optimization proximity many salient answer dendrogram agglomerative hierarchical algorithm hierarchical group linkage method unweighted stage clustering hierarchical member
entry cone equivalent positive cone cone configuration jj ii ij configuration true dd property dd condition cone relax monotonic cone relax positive condition huge cone unknown positive cone configuration simple prove lemma rank n dd size matrix express lemma read dd k k recursively dd dd dd dd dd dd remark keyword true le university universit universit des des pr universit sciences ef le pt cm university sciences et france mail fr pt ref pt ptc david j er yu wang compress paper square absolute hyperparameter lagrange multiplier great profile tradeoff penalize condition optimizer increase also generalize norm take propose level signal condition homotopy lar norm dominant compressed solution property cone decade compress signal nu common compressed sensing theory elementary residual formulate operator concern apply straightforward evolution typical colored entry find theoretical view profile tradeoff help l example correspond plane pareto curvature tradeoff discovery algorithm homotopy angle lar develop advantage piece reconstruct whole hyperparameter interpolation colored evolution homotopy lar solution obvious homotopy lar usually start decrease necessary homotopy maintain update entry zero contrary previous work strictly yield iteration homotopy reduce word homotopy homotopy yield condition sparsity paper sufficient signal organize sec exist total variation denoise sec extend total variation call row row meet notation n k size transpose k n dd principal minor nm subdifferential system piecewise piece constant permutation nonzero entry omit sake brevity multiply since n increase piece continuous straightforward absolute decrease zero remain exist sufficient obvious hadamard monte satisfy whose find utilize high low column random enough recovered optimization intrinsic bernoulli dd within distribution step give kk correct homotopy property instance obey mutual homotopy run stop solution
bind improved previous follow theorem proof small restrict bound real see arbitrarily large entry triangular big consist row diagonal observe tm achieve margin dataset nevertheless margin row guarantee bottom derive imply doubly technique previous admit quantity technical corollary together bad price pay next end completeness norm decomposition lemma quick every lie om light optimal loss although bad dependence optimal need key decomposition margin finite conclude invoke similar dependence dependence modify second technique om omit long conjecture om rate converge state conjecture feature therefore c w bt induction iteration rewrite term product second since initial round need suppose non negative constant induction base hold thus side u equation separately quantity matrix ai ji proof first suffice without column k kk kn orthogonal act subspace recall linear algebra without loss independent submatrix form finish suffice k follow k vector pick coordinate set suppose point coordinate contradiction reduce solve om bf addition slack standard everywhere else know norm slack yield strict inequality entry segment therefore arbitrarily next matrix decomposition multiple possible nf positive banach f equality item mi mi bound adaboost combine weak slightly strong converge exponential previous minimizer exponential adaboost norm round bound depend round achieve within descent perform slightly adaboost understand focus simplify precede easy functional gradient iteratively example descent move adaboost direction adaboost choose add updating value exist asymptotically converge loss minimizer address state polynomial hypothesis q round rather weak typically low without additional assumption minimization aware bind convergence adaboost namely within adaboost achieve situation constant may within proof call margin loss far loss class prove determine proof consistency impact adaboost converge practitioner wish exponential well violate appear finite depend measure minimizer case variant also hold generally point compact space improvement exponential exponential loss round adaboost converge descent section conjecture associate section provide hypothesis compute hypothesis choose hypothesis specify adaboost pt weak hypothesis tx td hypothesis refer combination hypothesis adaboost define contains write compactly coordinate maximally decrease coordinate coordinate elsewhere recursion see term also adaboost hypothesis round divide side exponential drop depend ti ti weak rate criterion weak hypothesis absolute line long drop rate hypothesis however choose enjoy say small continue hold leave explicit simplify adaboost attain parameter norm serve reference bind thereby pose show section follow rate adaboost loss high behind large round indicate large guarantee adaboost grow proportional either way progress idea concrete lemma notation measure logarithm exponential combination achieve interested r notice loss round trivially also attention boost edge polynomially compare adaboost edge round monotonicity non negative great l otherwise next put loss negativity adaboost create summation rewrite target last combine complete assume lemma prove grow fall growth large prove firstly rearrange complete prove b proof show suffice chain tr tr complete provide desire believe tight slack decrease improve directly never decrease monotonicity holds obtain absolute constant fast rate convergence except round scale back largely hypothesis j ts great decrease exponential ts instance distribution e modification adaboost loss round continue exploit adaboost improved back effect weight becoming show write summation add true gs denote minima throughout minima finish round together never occur adaboost identical rescale imply adaboost would hope show achieve argument tailor adaboost coordinate decrease connect solution require round show wide achieve dataset example get wrong correct round row boundedness hypothesis predict magnitude lie contradiction loss round thus assign mt mt statement directly weak entry feature property achieve fix every row example row triangular diagonal row complement loss lemma round reach least picture construct loss loss might achieve add therefore least margin combination satisfy imply attain exactly therefore hypothesis confidence rate arbitrary arbitrarily construction requirement integral norm achievable achieve norm two therefore margin least fourth triangle round boost cause cause rate increase drastically fact arbitrarily absolutely rate weak arbitrary confidence purely suffer highlight rate solution important convergence dependence round parameter solution achieve target dataset realize want round decompose part solution loss however introduce section depend additionally show necessary converge previous although respect polynomial upper adaboost reach depend technique upon adaboost mainly particular hold achieve adaboost combination optimal round situation fail yet optimal technical hold solution make progress progress always sufficient next result quantify nature use empty c z exist example finite deferred immediately denote c achieve zero finite margin thereby name l proceed finite illustrate wrong solution serve round adaboost proof appropriate subscript exp example sl combination end round edge mean loss unnormalize affect immediate decomposition recall put weight get x round matching bound henceforth loss margin example use lemma together show good progress use progress make solution early appear depend generality subspace nan nan necessary matrix along rescale bound derivative suffer I let schwarz may put quantity inside everything adaboost round solution lemma pick entire c edge last inequality step k mc complete boost loss dataset k claim apply sequence achieve round suffice throughout otherwise state boost simple rigorously define loss admissible combination weak zero subset loss complement margin build final subsequence empty pick example ever away subsequence attain begin subsequence repeat subsequence sequence converge call loss finite margin show extract combination item suppose combination
estimation observation equivalent alignment fact claim derive work space criterion consistent definition proper eqn f restrict result fisher rao metric f fisher rao fisher compare probability rigorously focus family metric nonparametric find important curve attribute preserve simplify calculation modify introduce interval go qx cc study velocity include parameterization absolutely integrable vertical obtain invertible understand reader metric smoothly tangent fisher rao riemannian dealing function cumulative classical fisher form rao sign metric fundamental riemannian metric invariant play important role information geometry geodesic geodesic geodesic connect simplification motivate represent elastic rao compute simply negativity fisher rao distance find compute involve calculation represent elastic step elastic represent set metric use induce elastic q domain belong distance proper satisfies negativity pseudo distance quick relationship rao metric item fisher rao dt w dt f geodesic numerical straight sf use framework align improve match peak across cross function align template derive template alignment element define individual function elastic geodesic minimizer function name become manifold rao understand geometry represent derivative transformation dt root geometry sphere simply length great connect rao define mean fr algorithm fr function initialize compute mean square rather dynamic programming align although prove global th iteration optimal q decrease iteratively converge template align function condition identity cross prove existence depict condition definition minimize fr fr fr apply setup utilize algorithm finding template align set note simulate transformation simulate give f second panel align function panel panel deviation ccccc std std tight alignment peak effect remove remain mean band alignment simulate function variability show leave peak peak mixture panel function well next estimate panel original practically cccc simulate gaussian phase variability shift tight alignment function leave remain align compact ccccc std std family multimodal phase show mean show huge apparent amplitude align phase variability perfect alignment ccccc std panel plot occur different discover bottom panel top row mean deviation alignment function mean fact average similar perform curve fig consist original growth deviation tight peak suggest several std std std std signature datum effectively elastic handwritten signature acceleration signature time study variability alignment show signature analysis panel align suggest align function much peak alignment due cccc cccc original std std spike sequence sequence train language focus convert spike train domain st dirac train spike smoothed spike train ft kt panel spike train perform path movement deviation neuron peak mean standard growth signature increase decrease deviation observe std align std example temporal microarray time cycle measure period minute fully cluster related phase fig gene expression many goal focus subproblem deviation alignment panel right panel align strong functional publish conceptual utilize discuss section criterion obvious criterion alignment three provide comprehensive denote function cross validate total variance align original correlation pairwise pearson well alignment least time least criterion variance align original alternative synchronization well achieve compare rao curve principal expectation package self method present match simulated result alignment datum even easy perform alignment visual alignment sometimes three fail evaluate alignment performance shape signal shape wave job evaluation number involve choose challenge scenario original signature method table representative complexity c r matlab matlab r matlab sec sec sec sec sec automate basic fisher define elastic distance template select identity align distance separation variability achieve framework template include development amplitude class functional corresponding idea relatively limit apply mention albeit context proof ft qx directional directional derivative tangent w dt rhs eqn proof dt corollary lemma dt q important last q strictly lemma minimizer minimize everywhere ft ds ds qr dr fr fr mathematics university north geometric separate frequently study framework rao metric convenient square transform rao simplifying distance align estimator scale translation idea real data berkeley growth handwritten curve spike train signal superior several publish alignment application nearly range processing easily align problem principle hilbert space compute distance cross function serious challenge arise inherent variability variability possibility tool increase elastic keep mind allow value function differ peak location term variability extract correspond function right function observed variability parsimonious consider height berkeley growth growth subject highlight growth growth rate discover broad pattern would technique show gene train relatively proper align across g require manual natural comprehensive alignment unsupervise fashion principled align elastic important unified fundamental idea treat process different scale observation vertical translation seek use mention introduce brief summary
solver author address fit approach search web contiguous dataset dimension common english may require dictionary available approximately mean rarely thereby unlikely occur even empirical study achieve binary fit memory extremely however often load training svm much severe arise memory situation publicly available dataset gb disk input format exceed document small dataset bit sparse text hashing use issue matrix solid foundation nonlinear effectively sketch hash definite linear svm bottleneck partition block repeatedly update however bottleneck memory loading iteration number disk approach work hash either widely apply permutation repeat time hash prohibitive bit problem store low bit convenience th bit low bit approximate formula numerical comparison probability permutation apply hash theoretical property hash foundation bit hashing behind construction learning call pd follow pd whose th entry ij nm ij te b hashing pd dim pd pd bit pd expand expansion dimension exactly svm popular representative package include sgd solve regularize logistic important demonstrate effectiveness bit hashing simply achievable approach permutation feature store store merely bit expand originally binary digit expand vector fed solver inspire bit hash pd theorem note total datum exactly work closely conduct public randomly sample demonstrate effectiveness gb ram system format fit bit hashing substantially advantageous testing merely mb try train result hash second accuracy match test therefore benefit provide hashing demonstrate able tuning parameter logistic conduct extensive experiment std deviation test achieve randomize experiment report mean illustrate produce averaged repetition hash solid accuracy use dash red computed repetition become extremely e second training take minute load take course study collection store multiple near neighbor hashing testing see merely include loading efficiency processing hashing may often color available accuracy bit hashing figure standard deviation train second second summary effective reduce testing svm training interestingly bit curve represent repetition red color original method projection although dimensional related min dim estimate inner multiply sample er v eq include projection provide small satisfie equal I paper refer correct version sketch hash vector element task estimate product suggest generate remove bias variance difficult mention major heavy pre multiply original vector equal correspond apply multiplication hashing e er follow see interestingly choice vector element wise paper test bit substantially size time example bit small projection storage hash fold bit bit substantial make I size zero preserve mean vector number exceed easy zero zero focus reduction introduction relatively often zero develop useful excellent tool indexing due property extremely indexing reduction reduce original data technique achieve huge expand point binary like vector use substantially especially expand actually generate binary hash size may provide insight straightforward formula b kt datum vector generate bit hashing counting exactly denote estimate unbiased complete reduce trade figure verify intuition bit bit quite accuracy additional bring accuracy bit bit practice understand practice use simulate permutation require stage
solve numerical linear solver essentially result incorporate improve several provide rank notably underlie amazon random moreover hide hadamard projection projection run approximate leverage dense brief review relevant main algorithm contain main conclude approximation main environment column column ax square square alternatively finally identity let thin general singular finally orthonormal subspace range orthogonal onto frobenius norm provide call l set transform projection probability matrix draw let accuracy strong transform vector project orthogonality quickly specifically find let randomize hadamard use efficient construction recall unnormalize hadamard transform hadamard equal simplicity generality numerical property apply energy need access transform vector distribution formalism denote imply property generate summarize combination use arbitrary fix orthogonal probability additive approximation leverage start high recall goal orthogonal span fold second multiplication application projection bottleneck computing eqn compute approximate matrix preserve idea uniformly sample fail return meaningful matrix identical row find row preserve rank row see formally recall let example approximate approximated take efficient bottleneck need euclidean norm thus reduce specifically vector sketch essentially dot different randomize approximate leverage matrix output idea improve sketch let svd let note implementation approximate approximation specify basis choose leave approximately compute norm next state matrix lemma say eqn compute qr rather multiply triangle inequality obtain lemma dot product score approximation within first follow pairwise product achieve compute use improve leverage result theorem algorithm subroutine heavy provide sketch proof theorem proof subsection point lemma event estimate improve run since orthogonal prove preserve alone need inner relate actual cross efficiently product preserve condition event additive approximation rescale I e full rotation unitary invariance eqn vector preserve let vector j expand square algebra multiply throughout u u j homogeneity together return return squared row norm implementation r nd nr q simplify product among denote pair schwarz call norm heavy first pair notation heavy cauchy schwarz sort initialize initialize stop otherwise increment first heavy none pair heavy occur add loop pair next decrease otherwise occur whenever pair norm heavy norm heavy number operation pair heavy whether heavy return heavy pair apply n tu nu tu eqn j f conclude return satisfie extension computation leverage score case specifie arise compute capture interested low majority capture parameter score normalize k ill degenerate leverage well moreover leverage leverage trivial degeneracy help singular singular example play role solve distinguish singular th get leverage score leverage score leverage might spectral useful feature develop algorithms care score notion approximation give approximation rank instead eqn ill pose hand instead normalize leverage good seek number leverage remove leverage popular spectral frobenius approximate leverage score input namely approximation draw I statistical input close write essentially prove computational detail appear see conference remainder technical report purpose section detail sketch gaussian matrix k consider negative x algorithm approximately leverage score assumption span hold clearly normalize leverage leverage rank summarize normalize rank parameter approximate rank output namely rank rank draw compute orthonormal compute vector leave q return worth ta ta lemma close member constant unlike spectral provide closed formula return approximation normalize gaussian I eqn standard linear prove rearrange take square side row matrix orthogonal follow eqn leverage normalize matrix take computation discussion give compute leverage conclude broad related constrain variant streaming environment statistical leverage proceed tw argue leverage obtain nontrivial provable compute truncation negative perform order final return algorithm truncation leverage score maintain positivity matrix notable approximate separate compute dot positivity estimator manner thresholding provable albeit weak guarantee bad direction considerable evaluate empirically leverage score score quite column well time given argue compute use order make statistical leverage proportional leverage main result failure solution least square eqn probability satisfy constant right vector q diagonal strictly positive full apply singular value hence ready theorem opt u tv derivation drop term change fact simplify imply opt compute computing sum satisfy eqn rt p opt remark first constant clear leverage approximate leverage hadamard multiply hadamard analyze follow line interesting evaluate experimentally score relate statistic way stream stream pass space depend integer size stream compute qr decomposition compute outline effectively row euclidean norm equal find row idea bit demonstrate concern sampling vector treat matrix along copy row set linear matrix ta norm multiplication serve magnitude streaming set additive reduce entropy bit pass compute effectively bit natural obtain leverage importance sampling sampling score compute finally procedure proportional bit pass identity row definition proposition corollary david square top singular popular problem completion nystr om low statistical develop precise I relative several practically important cross extension idea streaming vector correlate basis usefulness recently randomize relate popular nystr om base approximation leverage compute singular leverage statistical leverage outli score bad amenable implementation useful domain detail I well dominant value qr projection span randomize compute qualitatively arbitrary algorithm assumption precise opposed na I corollary coherence addition practically underlying definition denote singular th
differentiable w scale norm bregman divergence compact ergodic mirror iterative maintain gradient specifically stochastic receive select dependent project descent since continuity bit fx denote measurable element field sample norm consequence expect though still function definition essential presentation distance hellinger factor different supremum square hellinger fact hellinger metric hellinger denote mixing probabilistic exist time weak version assumption mix field markov uniformly ergodic chain space indeed assumption mixing randomness process stochastically mixing allow wide process assumption begin expectation section sharp numerical factor sample algorithm assumption arbitrary assumption hold eq assumption bound immediate jensen hold satisfie method similar least theorem obtain additional arise care couple nonetheless corollary clear uniformly hold mix assumption make broad turn slight bound intuition attain process stationary process uniform geometric variation mix hellinger condition assume assumption update stepsize integral simplify stepsize multipli parameter corollary generally stepsize argument choose shorthand choose descent mirror generalization attain sharp conclusion hold high replace step al ergodic multipli penalty scale bad classical approximation choose nonetheless average yield see class ergodic remark homogeneity reasonably work similar assume conclude process geometric simply present slowly process concern optimality result informally oracle query return oracle th receive distribution xt xt xt random set set mix return stochastic assumption definition minimax norm minimax oracle satisfy minimax complexity implication match discussion al dependence bound optimal brief definition proximal theorem constant attain collect consequence statement begin concrete abstract principle complete mix ergodic previously example previously incremental descent al ram et come optimization scheme processor computer function goal minimize procedure work pass processor token indicate hold iteration update token move processor draw distribution algorithm update token evolve doubly stationary case true th doubly stochastic q denote spectral addition recall significant consequently follow corollary evolve evolve doubly consequence theorems corollary mix walk hellinger distance rate somewhat original incremental gradient update euclidean geometry essential obtaining method mixing return sharp finally ram rate et algorithm convergence rate never weak strong random walk bind uniform web receive click impose result ranking order lead total order impose certainly challenging sample rapidly develop permutation partial transition order show chain follow mix consistent objective evolve update multipli turn example broad guarantee generalize markov gradient random communication consider autoregressive require expect total distance assumption appendix procedure token processor transition token doubly token et al notably k take apply define example applicability agent communication link algorithm failure suitable obtain doubly logarithmic roughly increase suffer bound suffer example statistical standard chain apply space simulation phenomena autoregressive em assumption markov ergodicity difficult require essentially allow difficulty focus moving model identify linear area paper particular condition assumption obtain sharp universal corollary contrast ram apply property ram et exist strongly connect example motivate corollary fast rate brief discussion process establish ergodic assumption markov exhibit rate hasting sampler stationary assume simplicity hasting sampler markov denote condition density transition accept hasting mcmc generates associate away metropolis hasting uniform mix main update distribution give fix statement statement multipli apply along enter minimize derivative inspection slowly mix incorrect set incorrect substantially slow mis suitably choose demonstrate provably yield convergence slowly mix require mix incorrectly note mis ergodic suitably quickly nonetheless corollary stepsize hellinger total regardless notational infimum term decrease yield probability borel argue occur suffice ft present experiment algorithm guarantee essentially interesting natural understand benefit mirror descent problem natural identification surface pair bi variance system q minimax ar study alternative generate sample case call replication specify repeat one sample analysis limit infinite step system generate accord autoregressive process replication sgd sgd begin use sgd proximal yield analogue descent n fx taking computing yield figure simulation obtain sample figure replication approach theory still convergence stepsize enough multiple performance gradient plot figure convergence plot sequentially rather draw efficient cc task robustness modification stepsize second numerical experiment study motivation instantaneous distribute incremental mirror draw sample sampling vector flip sign perfectly uniform analogue offline use objective effectiveness non euclidean robustness stepsize nearly denote analysis al yield I norm whose corollary simulation distribute left plot connect cycle cycle mirror circle denote dot plot th optimization set predict theory mis take spaced plot right figure show optimality mis certainly degradation capture behavior analyze necessary subsection proof expect prove give collecting consequence make proof represent subgradient expectation fx fx make substantially easy proof essentially early care establish relevant optimization setup understand impact equality may take expectation essential idea allow nearly parameter lemma stochastic unbiased formalize whose proof hold stability show value apart let non eq hold nee apply precede lemma use proof turn lemma sum require control final either lipschitz obtain complete statement holds begin point expansion show small stationary mind appropriately behave approximately martingale hold provide appendix combination proof start obtain remain probabilistic hold guarantee lemma guarantee accuracy probability inequality guarantee consequence state hold description intuition returning employ online identical minimization packing hypercube classical fact cardinality sequential pair otherwise construct uniform sequence step tu yx denote inspection al enough sampling set different block mutual entropy sub within block apply al lemma construction see coin oracle lemma subgradient descent extend elegant desire difficulty reasonably fast ergodic stochastic generate show strength analysis believe version carlo sampler clean derive convergence full g provide low numerical constant special nice property thank question thank anonymous suggestion begin control make result subgradient take
smoothed matrix follow static select number static evolutionary heuristic many cluster number cluster accomplish review modularity heuristic cluster employ choose affect face challenge cluster matching cluster cluster weight common object general multiple merge splitting multiple beyond scope reader address problem affect framework tracking measure criterion seek rand agreement ground rand rand spectral propose evolutionary experiment track gaussian distribution dimensional first step size new membership run cluster cluster incorporate mse plot estimate oracle calculate cluster membership application fig sep alpha oracle increase nearly walk oracle appear oracle alpha rather steady estimate continue objective close fig draw identity mixture proportion draw initially initially proportion cluster unchanged sample tp time use mse experiment vary membership choice mse significantly exclude oracle cluster membership move perform stationary experiment plot cluster compare simulate two filter short memory rd order memory fig good rand poorly begin overlap overlap slow filter index rand place historical membership change rand affect tp plot iteration notice visible outperform move membership change finally cluster perform appear converge low oracle effect natural phenomenon phenomenon propose object govern try centroid try keep move experiment initially regular interval time path note behavior simulate change change membership one tp tp rand static iteration run experiment linkage previous true computed membership various display memory experiment rand various list see cluster simply move toward position rather switch cluster affect modularity equip run membership maximize rand tp estimate iteration tp tp rand method two rand separate summary list outperform good drop notice iteration rand perform unlike interesting observation affect cluster contribute rand leave step experiment mit reality phone activity student mit phone record medium mac address nearby device device proximity student proximity divide week partial truth namely dominant could proximity business school likely school cccc school iteration cross experiment real simulate instead fold believe substitute rand entire school year list surprisingly static cross validate leave step estimate similarity membership contrary object leave six important mit day class notice estimate drop physical change student physical similarity time begin break estimate matrix notice physical particularly change similarity fall estimate factor appropriate evolve set namely price daily stock exchange operational construct stock coordinate price day vector subtracting divide deviation day period cluster stock ground label stock list stock evolutionary rand affect standard error five leave cluster stock list rand c ccc capital stock non service stock finance health stock stock rand index static iteration iteration estimate drop mit mining happen market occur suggest tp evaluate scalability affect varying object cluster stock compare affect evolutionary algorithm ten intel processor fig computation affect run affect consist iterate static cluster notice affect stock cluster iterative nature fast increase decrease fast great iteration affect propose track cluster accurately track recursive control order square allow evolutionary selecting unlike exist framework synthetic good outperform evolutionary future factor converge improve finally plan proximity optimize perhaps certain dot sample mixture calculate oracle factor section drop simplify arbitrary independence eq calculation involve expression variance confirm indeed possess assume structure acknowledgement thank anonymous suggestion improve national science office grant nf xu partially award science engineering application evolve evolutionary typically static propose often smoothness tracking follow present evolutionary adaptively parameter shrinkage improve I additional include evolutionary synthetic indicate evolutionary algorithm scenario obtain tracking find time vary stock market mining machine processing object change short term variation na I datum extremely sensitive produce cluster unstable inconsistent long short variation several evolutionary cost static penalty prevent cluster evolutionary commonly agglomerative penalty manner paper evolutionary treat track static dissimilarity mean view involve past perform static state estimator improve raw estimate past call formula optimal evolutionary factor adaptively adjust dynamic propose evolutionary tracking extend dissimilarity handle cluster accommodate object leave time demonstrate affect three namely cluster spectral synthetic set outperform recently evolutionary cluster algorithm adaptive evolutionary cluster framework several advantage evolutionary static enable extend proximity input evolutionary algorithm outperform static cluster exist evolutionary increase static iteration extension evolutionary propose static insight effectiveness experiment commonly use algorithm affect term notation assign cluster datum store dimensional th feature vector create proximity object dissimilarity represent adjacency denote edge vertex edge agglomerative cluster dendrogram dendrogram certain obtain flat variant agglomerative hierarchical general vary dissimilarity common dissimilarity object cluster dissimilarity complete linkage dissimilarity low dissimilarity dendrogram attempt minimize centroid object cluster close simply square euclidean object close rewrite dot product algorithm dot product integer calculate centroid compute centroid cluster similarity positive definite similarity similarity laplacian spectral cluster association aa np solve relaxed relaxed variant optimal consist eigenvalue similarly solution consist optimal relaxed typically normalize tp li unit algorithm normalize ratio cut eigenvector instead ignore row average association large ignore step contribution area constrain evolutionary incremental type cluster type stream focus stream object target evolutionary cluster incremental incremental could apply type consider focus incremental low computational expense quality incremental often bad static already introduction evolutionary concern capable constrain optimize fit objective use hand use preference evolutionary evolutionary historical result evolve object step unlikely cluster divide segment differ significantly one remove unobserved matrix reflect mutually filter correspond filter impractical model secondly covariance enough evolutionary simple recursive define expect static rather proximity objective static estimate also lead cluster result I static use disadvantage unstable inconsistent adjacent incorporate potentially step allow rate allow amount smoothing smooth unbiased combination bias representative term statistical estimating similar variance convex suitably notice shrinkage smoothed proximity proximity shrinkage intensity manner shrinkage estimation frobenius proximity smoothed optimize try estimate know shall henceforth refer mutually independent risk eq note conditional derivative set rearrange minimize risk lead minimize risk knowledge proximity trying suggest replace w tw tn belong cluster membership propose proximity use object two object two distinct belong structure proximity fig short inside possesse assume ti gmm gmm parameterize mean manner sample determine object draw gaussian change component membership stay show dot similarity observation structure although rather information shape block model order framework beyond scope area work model assumption block obtain variance know cluster membership around membership membership logical choice membership static substitute estimate substitute static clustering result result refine improve quality clustering find rarely
birth death general previous focused birth death often nonlinear birth death rate particle provide mean sophisticated realistic biological example population sometimes growth capacity genetic allele depend allele rate researcher assess death progress birth death researcher longitudinal observation observation likelihood partially since write close challenge researcher progress none development likelihood general birth death major likelihood expression expectation miss relevant expectation em researcher derive unfortunately expectation pose challenge joint distribution generate able expectation certain rate depend particle development promise technique sophisticated realistic review little applied researcher parameter apparent provide derive estimate laplace transform fraction continuously em describe expectation expectation transition laplace transform expectation technique costly integration previous provide example em fast compete validate open question evolutionary particle time happen happen instantaneous rate homogeneous efficient birth death rate complete likelihood transition ordinary parameterization possible transition term orthogonal polynomial special find even rarely outline additional specific insight specification advantageous essential maintain probability compute expectation necessary laplace transform equation laplace transform rearrange recurrence relation exact laplace write compactly numerator denominator rational function fraction combine although transform transition close fraction fraction representation fast series transition probability fraction control generate stable probability unobserved conditional expectation discrete wish find maximum parameter birth death state realization spend let step realization denote total particle count amount demonstrate concept figure likelihood maximize em complete maximize step surrogate clarity omit parameter assume uniqueness expectation death unobserve state path adopt combine author derive resort approximate relevant expectation simulate path recognize conditional expectation statistic conditional expectation formula appear markov chain whose integration product prohibitive however integrable laplace transform easy laplace property inverse laplace formula offer substantial integral inversion involve inverse amenable acceleration method transform method death representative example complete analytic maximization simple unknown solving update estimator continuously probabilitie expectation algorithm sometimes population enter arise model point dna sequence suppose new arise process rate regardless many exist represent become derivative concave maximize take preserved maximize respect expression weight proportion state illustrate specification evident state examine typical population often incorporate limitation environment model growth stochastic previously roughly deterministic growth support growth beyond balance restrict growth suppose parent decay death interpret capacity roughly appear q preserve place epidemic infect become infect currently infect sis disease specify birth death number rate new infected population allow assessment novel way associate process birth death rate linear possibility covariate ease arrive surrogate maximize step surrogate differential process note become ill condition difficult newton option perform carry avoid ascent check newton present application evolution outline subsequent algorithm usually simple evaluate fortunately replace summation eq expectation greater slow near appropriate exploit acceleration introduce implementation give figure likelihood accelerate newton guarantee achieve ascent advantageous efficiently formulae analytic expression analytic generally termination em also outline purely technique hessian formulae achieve exist expectation various time table employ rejection reject method I convolution use compute domain outline implementation effort share code sis list table fast method term time stand achieve logistic finding construct markov method finite process result aware affect simulation hand enough path upper however condition fail sis model issue quite largely computational rejection condition quantity convolution logistic kt sis several draw start integer trajectory record glm simple parameterization table simulated observation regardless find factor generate path start finite size partially stochastic point se linear sis glm short character dna molecular responsible repeat researcher analyze evolution however many depend nucleotide addition rate fit composition generalize examine evolution evolution common repeat size nucleotide find human control extended inference evolution draw introduce novel modeling deduce implement insight must acknowledge evolutionary human human evolutionary separate human mild reversible assume stationarity stand justify evolutionary regard vice evolutionary human equilibrium line kolmogorov therefore observe human evolutionary separate scale unity evolutionary evident small number necessary mutation interpret justification avoid g past researcher grow suggest equilibrium distribution length assume addition depend present birth equilibrium close form nonzero addition always evolutionary likelihood equivalent distribution also incorporate achieve impose barrier iteratively barrier composition heterogeneity covariate define control model together surrogate become eq report infer size evolutionary great least unique largely consistent justification nonzero distribution formal statistical repeat mutation cccc covariate se birth correspond nucleotide composition control difference birth birth death death compare mutation statistic flexible realistic become observe establish require birth death rate exact ingredient hessian numerically previous completion typically numerical simulation rejection condition comparison convolution conditioning markov chain sis parameter laplace offer chain nucleotide substitution availability yield parameter update model mle close majority return update available one numerical section likelihood slow surrogate lie far newton require inversion surrogate condition speedup laplace convolution acceleration slowly toward na
label thm lemma de universit de universit b f france criterion recursive adapt deal allow arrive converge surely particular attention average automatic descent step average compare term speed estimation partitioning means finally cluster profile every hour dimensional recursive approximation fast domain biology computer vast literature recent argue computation procedure large slow focus deal fix non sequential sequential require find local element belong cluster drawback base consequently may represent fraction get develop consider norm spatial proved certainly partition around local consequence reduce subsampling computation distance accuracy scenario outlier distance cluster center simulation study sort bad execution idea estimator dimensional propose recursive advantage one recursive nature automatic store descent value empirical step sequential order run present third proof rely technique sequential technique small number initialization point competitive moderate small major search element consequently adapt deal temporal indicate appendix copy take mean aim finding follow non take present new notation recursive equal set point recursive start arbitrary group allocate allocate recursive fast rely exhibit denote check partial stochastic suppose measurable denote gx gx classical descent choice estimation set point realization cluster minimizing follow risk classical descent step adopt asymptotic could suitable practice attain parametric algorithm experimental define sensitive carefully simulation particular show inaccurate always even choose prove geometric corresponding classical averaging around prefer slow provide center start one value converge surely absolutely lebesgue measure n absolutely almost h fulfil stationary towards converge surely converge randomly absolutely moreover get decompose nan probability scope establish convergent provide continuous version estimator relate property fulfil satisfied datum deal fulfil infinity simulation recursive function default make average recursive parameter far consist approximate recursive justification automatic tuning work well important beyond scope however intuitive certainly group would consider value present highlight strength drawback recursive realization bivariate random vector cc cc contamination sample dimension also perform vary multivariate covariance structure autocorrelation play draw dimension directly cluster evaluate comparison term small contamination equals estimate even contamination level secondly nearly even perform replication equal present figure performance well look solution among sample explore enough look provide error around attain consider multiply factor mean presence stable term empirical recursive value drive average chosen drive summarize center mean evaluate formula run average classification measure partition indicator place define equal perfectly design detect automatically replication size contamination term drive fraction outlier figure clearly affect performance even well performance strongly large contamination median unchanged meaning fraction small one term effective recover code call average fast get mean allocate computation drive recursive sample size large ccc critical recursive mean take second second average
auxiliary lemma admit z let star initial admit unique z moreover z since yield second yield h z ph sup des paris france france sup dr award general dr interest communication theory wireless gray rgb pt corollary conjecture remark remark interference wireless digital line multi division access decode channel multiple parallel channel orthogonal successive interference cumulative signal ratio response reinforcement differential tucker ordinary paris may analyze non game channel available channel possess unique distribute evolutionary theory stochastically game unique equilibrium employ stochastic still theoretical novel nonlinear dynamic nash potential view decentralize wireless distribute assume protocol goal resource etc question whether policie deviation equilibrium reach require accordingly attract wireless communication concern allocation viewpoint subject optimal reach boundary rate assume knowledge central hand allocation capacity achieve power unstable decentralize consist operate channel focus answer point ac analyze despite apparent relevant channel allocation throughput control analysis focus scheme separately treat incoming signal instead low decode overhead latter consequence decode suffer property rate theory channel iterative converge special conditional fail static thus capacity region understand attempt carry allocation admit nash equilibria potential equilibrium uniqueness fail uniqueness surprisingly turn though static nash uniqueness characterize system behavior get regard consider pricing interference similarly show local channel overall subject mild interference water equilibrium game result enhance drop modify water water present base often player require solve water study extensively game property understand nonetheless unique open fast adjust learn payoff achievable ergodic game theoretic game admit star convex game equilibrium fast convergence establish notational span canonical measure finally indice player one simply setup consider wireless receiver orthogonal typically may subset denote channel subject represent user channel power analogously pre k k denote clearly achievable rate allocation gain much duration power extreme coherence short correspond interesting intermediate block instantaneous game channel efficiency user bandwidth gain draw transmission spectral efficiency user high lead player player scale simplex allocation vector space strategy player payoff game nash player finite even payoff multilinear hand possesse examine channel remain transmission know gain la evolve evolve channel regime evolve transmission relevance power ergodic counterpart payoff admit whose depend calculation crucial numerical begin notion profile resp resp admit potential existence equilibrium already practical view equilibria crucial evaluation etc static potential profile strictly might singleton admit detail fact quantity call scenario receiver access almost always strict convexity promise determine admit construct channel coefficient admit equilibrium sketch proof whose vertex kp kp kp impose acyclic assertion version potential game accordingly realization admit equilibrium potential give nash equilibrium theorem clear whether decentralized environment distribute cognitive present attract point view player assume policy optimally unfortunately monitor large large decentralize limitation static water channel overall interference point interference except number rely player know possibly hard algorithm discrete adapt action space one start aspect dynamic suffer drawback algorithm game q lead term ensure utility behave nash equilibria stationary instead seek player alone water user kp update scheme clearly unlike water close develop power instantaneous iteratively little overlap coincide popular total unit calculate base appear shall set equilibrium power spectral configuration coincide black rest converse vertex necessarily nash nonetheless game dynamic interior solution dynamic kullback leibler divergence nash equilibrium rate see dynamic review kullback divergence channel thus power quickly fitness specie phenotype fitness equilibrium see extend dynamic really water water contraction contraction fail equilibrium state least sake equilibria equilibria power equilibrium q k k grow follow deterministic evolve stochastically dynamic p kn kn knn concentrate channel uncorrelated case expectation gradient mean asymptotic follow represent notion block channel satisfy uncorrelated nash equilibrium interpret either discount interpretation purpose limit device evolve convergence note ergodic fast instantaneous information update power little static user learn obtain equilibrium ergodic exponential equilibrium mean static fig equilibrium channel fig channel theoretical begin rate achievable aggregate capacity game equilibrium similarly equilibrium static channel equilibrium unity attribute equilibrium alone fair simulation much channel rate equilibrium reach begin channel realization time channel user within ergodic block plot system user system ergodic equilibrium slow follow version dynamical deterministic space consideration possess long stationary simulated well user km fig run plot power evolve instantaneous channel quantify evolve remarkably dynamic evolve equilibrium within ms ms km ms number channel previous tracking channel time present
open loop turn another formula request ucb policy good item ucb step go sequence uniformly report good good ucb discover request adapt high missing mass reference therein heuristic principle reinforcement prefer ucb mass time step make request expert high start brief good strategy rely new interesting item make expert issue address effort subsection version discrete element element mass item occur instance error eq denote modify applying get moreover conclude confidence bandit good arm tx tx tx tx tx order ucb tune cn tp ucb design general iii explicitly outcome request request bound validate ucb behave meet focus show number know every information make choice expert item note expert index n k k n nk find expert every step horizon close strategy loop strategy request high interesting step alternatively request expert step expert successively shall remain expert besides require collect time go infinity together converge go deterministic compute strictly decrease mapping define q mapping appendix go ii infinity sequence infinity go nf nf lemma proof omit expression oracle policy allow consequence sufficient upper bound request proportion item yet satisfy I line least good optimistic make request request time collect good ucb round close loop policy theorem proceed step expert obviously missing denote expert distinguish either belong easily request expert draw twice otherwise nu il u il absolute th request expert place I u il u u kn il q il n accord surely sufficient show converge loop policy respective number request appear limit recall n f proportion interesting find request go almost proportion allocation define technique eq lagrangian denote get first conversely item open loop policy q ki performance ucb balanced side ucb algorithm proportion arithmetic unbalance simulation behaviour ucb illustrate satisfying assumption proportion display item ucb solid balanced simply alternate dotted representative run averaging remove variability obvious moderate oracle take rather course improve ucb probabilistic geometrically ucb sampling remain ucb oracle former sampling entire discovery process choose tight estimator well suggest proposition value discovery expert simulation extend first analyze behaviour less restrictive optimality good ucb remove fairly oracle policy clear remove though assume ucb good ucb complicated explicit mass whether convergence whether sense another interesting item infinity possible good estimator contribute experiment decrease comparison show eq hence converge write nf nf il nf reciprocal sequel suffice expert almost infinity elementary geometric distribution eq fourth develop markov permit converge infinity accord eq proof proposition assumption engineering nj usa universit ac paris france fr analysis probabilistic expert address base optimistic paradigm estimator strategy uniformly attain probabilistic provide suggest behavior still weak optimistic algorithm request expert request latter draw discover request expert arise real security power often amount credible may security perhaps e g country power hour
noise level weighted group fuse identifie point asymptotically without position scale unweighted group fuse first point tend outside fuse lasso find first tend e robustness method fluctuation position point classical multidimensional video likely rule profile come genomic tumor gene although level change point lie change function result focus detection however effort conjecture fuse analyze check point single point kkt correct change strictly point single case weighting estimation change independently may although multiple change confirm indeed correctly multidimensional select segment multidimensional find would expect square criterion successive value error approximation appear intensive impossible subset programming strategy ease notation physical reality implement slope derivative real processor gb lasso behavior profile length dimension run fused lar record run descent slope exponent fuse linearity curve initially sub extremely practical limit technology critical second fuse perhaps hard surprisingly increase eventually deterministic fuse lar suggest say version later slow run lar relatively converge final lasso increase complexity cubic already slow lar fix axis test empirically fuse multidimensional profile jump position add theorem trial fuse identify unweighted case occur across theorem variety resp unweighte fused weighted vary change location weight fuse remain fluctuation extend share length consider profile draw center profile run one implementation weight record define percentage detect resp group fuse outperform lar unweighted although lar reliable lasso experiment exact may worth burden accurate demonstrate change point number profile nine center either length possible application frequent gain among copy comparative genomic technology purpose adequate joint check segmentation profile remove fused lar subset post processing piecewise share calculate green c segment red large segment believe gain analogous loss obviously statistical carry frequent segment joint loss hmm outperform several benchmark jointly help gain constant interval point segment figure segment red positive believe loss segment profile profile exhibit region performance roc curve auc follow run default auc second weight fuse lar take auc method time cancer cell originally cell weight fuse lar fuse lar number change programming total second point hmm give author hmm look top panel stochastic code case common gain suppose advantage hmm predict gain fairly genomic weighted lar hmm consider publicly tumor profile give relative quantity dna patient fuse lar take second take minute profile version result smoothed version tumor profile fuse selection transform score correspond hmm vertical boundary comprehensive common genomic table hmm frequently arm q frequently lose region loss h figure select difficult verify frequently arm useful precise figure clearly gain loss hmm find perhaps account h weight fuse propose extend multidimensional approximately theoretically empirically group fuse change likely several theoretically empirically profile share change encourage accumulation genomic profile share try union present profile reduction detect point profile post fuse modify g tv fuse constrain signal frequently finally view solve proximal constraint multidimensional tv biology indeed theoretically help commonly conjecture hold technical successive beneficial oppose implementation follow lasso fast lasso would nice exist criterion change strategy result carry implementation claim define group lasso center zero mean compute cumulative formula get namely compute I u complexity significant compute step satisfy u j j ni n st conclude multiplication memory q easily check admit convention define row expressed similarly recover translate point locate reach row euclidean compute hand dirac jointly covariance particular similar computation equivalently tends union select union select converge lemma tend asymptotically check deduce whether select tend always result index th u happen denote eq convenience notation rewrite let event clear event q probability derive union bind lemma shift independence tend let support correspond possible weight unweighted move towards symmetry also otherwise lack generality treat suffice never e france paris france paris present detection share occur signal constraint multidimensional piecewise approximately consistency evidence simulate array comparative genomic place dimensional change way question several field common want find multidimensional processing detect computer economic time analysis another situation dimensional believe genomic profile patient increasingly biology variation genome microarray biological share individual cancer genome measure segmentation multidimensional signal dimension profile signal fix technology individual increase patient statistical view develop identify signal increase number exist vast point segment multidimensional approximate piecewise constant quadratic know segment dynamic prohibitive technology alternative global point segment change detect point signal piecewise signal global minimum benefit detect base multidimensional multidimensional signal approximation multidimensional increment reformulate yet problem lar design point length correspond signal genome design benefit though within signal towards consistently able correctly identify change notation section lasso empirical comparison copy number preliminary version publish two integer frobenius case vector indice b ji pp profile length store profile th column profile signal noise location tend share profile detect benefit possibly profile change profile function quadratic solve eq dirac otherwise jump solve although impractical current computer reach million genomic profile combinatorial relax replace jump variation tv recent solve fast like find change add pi tucker kkt expect strategy brief maintain remove kkt condition result optimization line check outside active fulfil group active kkt therefore need convergence group line must several time compute takes bind correctly would times kkt active provide coordinate strategy center inactive c computationally intensive interest implement lar regularization lar approximate solution path affine intend solve straight add line justification original lar require storage design benefit lar descent step need equation overall change line memory change center initialize point wu next theoretically extent estimator recover
proof monotonicity actually independent prox increase language operator objective associate whose infimum attain differentiable pointwise infimum index derivative objective develop obviously strict negative progress ensure convergence eq derivative whereby subtract follow cauchy flip sign apply conclude bound gx fx k x sufficiently large help crucial eq eq note scalar plug side since inequality obviously true scalar limit work allow reduce case applicable claim say norm write prevent enter region fall behavior pay strong assumption vanish variant apply scale composite objective function sequel decomposable advantageous differentiable analyze backpropagation incremental aware generalize fail composite function disadvantage even incremental proximal splitting induction combine shorthand inequality complete constant whereby requirement incremental allow invoke aim illustrative application nonconvex nmf nonnegative add allow nonsmooth regularizer study covered stochastic like method analysis vanish deterministic rewrite form amenable eliminate attain subroutine ii nmf rank factorization google column remove dataset objective nmf art toolbox dense optimize fair unlike equally matrix fig expect time line speedup numerical compare stochastic start well stepsize tuning good stepsize tuning substantially return sparsity htbp cc cc bar plot high well factor plot bad objective nonconvex composite permit error practically specialize incremental large scale factorization indicate state art numerous algorithm example incremental theoretically open analysis remark nonsmooth nonconvex objective includes extensively study convex nonconvex introduce powerful framework avoid assumption error within new incremental splitting even application scale nonsmooth factorization form continuously differentiable possibly nonconvex semi nonsmooth compact make written reach machine learning formulation extremely familiar blind deconvolution neural primary contribution algorithmic specifically new split smooth nonsmooth proximal splitting notable allow capability critical scalable incremental knowledge incremental nonconvex splitting distinction lie notably require realistic inherent error make simplify perhaps remarkable nonconvex choice general allow nonsmooth idea proximal prove projection nonsmooth proximity operator attractive important implementation associate split similarly nonconvex subgradient method fails exploit structure iterate batch recently briefly discuss nonconvex monotonic introduce generic nonconvex nonsmooth gradient algorithm exploit composite despite benefit splitting raise especially allow scalable descent base simplify presentation replace formulation primary via approach define component proximity function nonlinear introduce orthogonal notably algorithmic minimize alternate forward proximal formally suitable tie tackle nonconvex
regardless datum dot box represents learn permutation close learn clearly relative associated parametric outperform test permutation conditioning sample provide adequate probability structure alarm network size number capture arc likely relationship would data conclusion test goodness network new model preferable shrinkage permutation cover network shrinkage fit bic positive network learn size contingency therefore value soon test large shrinkage coefficient mutual classic increasingly shrinkage sample shrinkage systematic behaviour produce shrinkage many lead parametric affect bayesian discrete shrinkage test goodness network parametric permutation close dependence sciences ac uk network structure always heuristic algorithm dependency base algorithm cover overview develop shrinkage article whose publish communication taylor communications available online www graph node refer direct arc connect stochastic connect either marginally independent conditionally specify principle choice function aim random univariate random variable usually task network artificial intelligence consist ideally coincide correct terminology classical application probability optimisation despite terminology fit combine approach deal global distribution structure extensive study heuristic technique influence strategy independence associate I threshold equivalent sample pose serious conclusion decision heuristic therefore conditional present behaviour tune appropriately investigate behaviour permutation test discrete datum class test base pearson information mutual consider global multinomial latter independence frequency xy level condition classic contingency information proportional ratio differ relate pearson contingency table respective require permutation statistic norm considerable know curse cause exponential issue maximum multivariate discover investigate increase overall maximum likelihood n usually call closed mean derive multinomial improve definition shrinkage likelihood compute classic reason classic behaviour shrinkage test observe gain additional shrinkage freedom independence shrinkage test indicator goodness fit score prior bic learn well score dependence indicator estimate true distribution alarm network alarm arcs network hybrid use conditional investigation combine hill since brevity asymptotic test pearson mutual bic score
follow readily conclusion acknowledgement grateful improve early greatly careful reading discussion subject thank de france universit et paris france construct asymptotic divergence bootstrap include particular choice several propose subject phrase consistency weighted bootstrap parametric estimator empirical bootstrap set modeling flexible provide powerful variety therein good source reference research limit divergence crucially propose general sophisticated confidence student useful tool inaccurate attractive one procedure datum population draw inference confidence receive study topic main summarize follow estimator objective identically bootstrap draw mention alternatively exchangeable weighting resample propose extensively name vector order consider empirical subsequently index example nonparametric regression statistical procedure purpose paper survey formulation present drawback namely observation difficulty formulation weight bootstrap computationally many distribution smooth statistic stein bootstrap usefulness survey bootstrap refer organized procedure base divergence property section censor propose flow presentation mathematical development recently introduce sign absolutely continuous divergence kullback leibler respectively le respectively divergence sign correspond real notice whole strictly example chapter mention real kl well survey consider unknown basis shall satisfies whenever class divergence divergence assumption duality divergence represent optimization procedure accord function fix put eq display reach naturally divergence instrumental divergence estimator suitable robust maximum mle robustness influence treat numerous model keep definition bootstrappe measurable weight rewrite bootstrap exchangeable sequel transpose vector shall condition exchangeable ni ni nonparametric satisfied nonparametric condition list condition restrictive nonparametric double two source quantity e rigorously divergence space order relevant canonical randomness throughout independent order outer reader cite follow open contain assume hold q definition refer function brownian bridge brownian covariance separate optimum well separated define class possess randomization multiplier randomize admissible page normality q class eq limit may state precisely consequently take capture variance weight refer follow quantile bootstrap stand dirac take show nh shift contamination position global divergence associate maximum mle affect subject dual refer mind reference example divergence divergence associate detail kullback leibler x x divergence estimate example define parametric closed exponential gamma pareto unfortunately also ml practical use algorithm powerful since objective concave estimate nh remark consider power divergence calculus lead maximum estimate kl n simple calculus exponential eq equality consider divergence gamma gamma imply power divergence routine divergence pareto eq simple calculus give last equality n lead maximum upon choice divergence criterion divergence value indeed leibl hellinger divergence infinite negligible corresponding finite also classical therefore automatically contamination effect misspecification parametric reasonably property progress automatic mention minimum estimator involve solve multiple root consistent estimate necessary survival survival f ng censor observe pair whether censor q time death risk generally weight cf integrable sequel follow write censor situation replace estimate recall formula censor power divergence x give define infer lead independently censor open conduct examine provide coverage implement indicate divergence divergence hellinger consider simulation take estimate limit indeed coincide mle detail subject refer mention weight table mse various normal expect produce close look simulation hellinger mle term produce ht table estimator notice inferential value nominal ht subsection simulation censor censor censor take censor robustness ht
countable trivial one probability person information know extra agent information extra agent first new get inequality agent x prove finite countable consider set disjoint exist finitely one conjecture minimal assume define whenever disjoint j n h h ij px k concept neighbor efficiently site minimal efficiently neighbor neighbor coincide neighbor positivity every row equally positivity neighbor cm cm mm hypothesis section categorical university road bc neighbor field field site site belief site condition give neighbor positivity keyword field categorical spatial process definition field show joint various site neighbor positivity hold uninformative set field site belief agent site change conditional information site sense site question less event suppose change belief might well conjecture wrong probability take shorthand sure denote equality extra hence well general positivity imply positivity mean categorical field
viewpoint involve class discrimination framework drawback map discrimination prove function popular boost pose weight act study involve framework address allow discrimination label method pose tradeoff estimate fashion avoid act classification dictionary incorporate avoid involve technique modular guarantee module incorporate organize follow map section methodology validate framework digit art benchmark give probabilistic representation optimization formulate setup seek maximize couple prior parameter dimensional combination atom type need nature e explicit identity dictionary belong sharing simplify sample encode class membership linear classifier classifier vs setup classifier completely scalar quantify assign label functional section seek linear model denote formalize consider note determine keep represent code dictionary matrix combine classification unify allow consist dominant peak determine pp simplify representation prior behind encourage training representation encourage respectively classify sum prior respect encode representation enforce representation obtaining lead regularizer regularizer select effort estimate context code however representation dictionary develop specialized parameter tune require dependent descent alternate learn fix procedure estimate r weight term form exponential optimize adaboost optimize incorporation additive e pursuit omp refer require represent sample one vs vector element update facilitate define use however easy apply project newton derivative around p pz z p z pz z well outlier overcome cost positively correlate moderate cost correlated cost outperform htb l logistic z c logistic hinge fix update avoid inversion require efficient svd iteratively generate atom case iteration reader ji sample overall scheme experiment randomly scheme produce similar dictionary scheme compute use scheme summarize stop initialize classifier test infer seek maximize belong respect dominant maximize p inner exactly label testing account dense g identity compute independent speedup computationally expensive algorithm omp exploit parallelism suitable dramatically speedup training number initialize previous moreover framework sample modification update representation handwritten digit face digit standard benchmark mnist acquire write technique error handwritten rotation need shift version face vision sparse traditional treat estimate winner outperform method achieve significantly outperform baseline training dataset comprise pixel baseline four type plot indicate yield minimum reporting square low clearly comparable adapt label yield suit increase overall performance reliable square performance classifier variation predict value mm cc mm figure learn form observe change notice close indicate reliable reconstruction take contain significant corresponding linear visualization interestingly atom particular proportional variation require representation comprise improvement comparable comprise image implication discriminative much significant loss cccc c digit recognition exp log hinge exp hinge c address discriminative linear sequence study art thm address linear probabilistic framework single discriminative dictionary learn jointly oppose capable classification cost avoid technique update code validate apply digit face benchmark signal popular processing learn sparse column dictionary significant progress predefine recent result face recognition solve np hard empirically underlie improve upon art denoise refer refer
b entry dimension time summarize dimension three sparsity element fix artificial generate unit however suffer point generate much ccc cc structure solve regularize nontrivial log propose three time solver method expression consist thousand measure experimental interesting model consistently consider correlation consistent module prominent assertion proposition definition remark matrix tractable concentration rank matrix represent marginalization bregman mild fast artificial gene correlation observe explain latent class covariance dimensional arise draw poorly behave often essential robustness gain popularity recently covariance matrix covariance covariance concentration precision encode variable conditionally sparsity equivalent independent estimation covariance estimation formulate tr denote positive explicit definite method concentration efficiently however real provide instance prominent movie recommender influence latent social densely correlate marginalization therefore sparsity alone may gaussian simply submatrix concentration component specify latent marginalization hide variable impose underlie maximum latent formulate decompose denote hide whose parameter et show grow observe strictly high due appear penalty positive problem use purpose log program solver order involve produce believe efficient fast large derive adapt close subproblem bregman method expression find latent explain well correlation gene explain latent aspect analyze follow split bregman also consist utility expression datum split bregman include method admm multiplier increasingly become method recovery proceed bregman reformulate emphasize although bregman solve firstly bregman fitting adopt secondly appear provide formula divide symmetric update subproblem term couple make amenable bregman detailed although bregman demonstrate multiplier admm presentation split augment lagrangian define lagrangian augment term lagrangian dual apply gradient ascent multiplier ascent efficiency iterative algorithm largely solve note augment lagrangian solve minimization run time equation split bregman alternate iteration rough eq convergence convergence bregman quite mild irrelevant involved check square root symmetric root definite whose square first eigenvalue correspond adopt routine divide eigenvalue decomposition routine fa sl use symmetric eigenvector thus together diag easy routine divide full eigenvalue together latent graphical initialize kl u k bregman trial implement bit art semidefinite need problem demonstrate generalization ability datum guarantee theorem would affect involve artificial
interesting multivariate vector show adapt theorem need hold inclusion ii finally sign mean divide part show complement union give fact bind line last focus c spectral great inside line stationary jensen conclusion follow far tx line projection us line inequality line tell lasso choose zero ol know write tell adaptive biased equivalent converge optimal converge already cr cr device central limit z treat increase theorem remark asymptotic adaptive lasso regression weakly asymptotic distribution ol present study method access become main area traditionally test theoretic first fit later adapt general sequentially unnecessary number regressor method spurious huge assigning want distinct choose quickly greedy another face candidate model feasible identifiable shrinkage successfully matter leave shrinkage method several eq parameter matrix entire path handle relevant modification property modification lead adaptive estimate candidate large weakly enter coefficient enter enter effort understand lasso advance little series economic autoregressive choose variable weakly dependent framework integration paper suffer candidate small size large technique framework lasso factor forecast goal order predictor forecast possibly explanatory extend regression converge distinct rate parameter consistent estimator impose already integration number number sample size candidate possibly condition impose error candidate finite panel understand price portfolio know object include interesting evolution time series predictor lag autoregressive lag present show estimate final remark delay index meaning process weakly weakly stationary process hold mix rate mix jt jt z assumption common model particular derive multiple integrate invariance decide directly relate candidate coefficient nonzero stack matter case letter parameter assume without know result interested space problem concave minimization satisfy tucker condition paper include lead ic condition presence examine violate comprehensive ic section converge oracle assumption satisfied perturb standard perturb adaptive minimizing minimize estimate issue lasso iterate consist weight precisely initial ridge regularization stable provide penalize prove adapt consistency conclusion adapt implement cross projection adaptive dynamically fix small affect report study want accuracy four specification performance mean large effect except variable consider moderate effect moderate among highly error tail select correct select coefficient resample estimate mean correct coefficient zero coefficient c nz nz nz table adaptive frequently select correct set zero coefficient candidate variable effect moderate affected structure proportion select particularly model correlate error correlate large correctly zero accuracy parameter accuracy square estimate standard use resample table mse affect large oracle square quickly oracle mse expect mse moderate observation decrease indicate really negligible observation ols oracle ols ol ols ol c ols oracle ols ols ol oracle ols oracle c ols oracle oracle ols table estimate calculate resampling assume knowledge reasonable interested formula data
h take toward tend specify thereby enable prior toward toward little information inference correspond confidence credible presence physical parameter inference inference intermediate extent ambiguity utility discrimination level extent difference pointed process science quite bayesian suited try progress optimistic modeling frequentist frequentist bayesian try frequentist aim acceptable conclusion stand minimax attribute attribute idea motivate place facilitate make inference definition formalize face toward mathematically economic identify distinct type agent toward ambiguity vice toward ambiguity differ ambiguity balance statistical distinguished belief latter act bad consistent motivate frequentist technical literature conservative interval coverage rate term assign operational definition ambiguity ball color ambiguity action accord display gain accord utility display toward ambiguity action ii would concept require extreme minimax red utility nature red action iv ambiguity generalization conditional utility posterior posterior posterior insufficient spread physical variability ambiguity study decision approach minimax frequentist sense strategy minimize maximize member traditionally special utility take action maximize minimize review extend posterior physical cg reduce action complete similar spirit call cg strategy drawback prevent cg impose parameter cg cg necessarily frequentist cg procedure reduce procedure complete notation theoretic identify limitation cg demonstrate generality note call exchange brief discussion frequentist preliminary vector model realization p inference simple possibility nan posterior understand every triple probability call knowledge yield inference plausible posterior corresponding prior prevent confusion system would purely plausible posterior posterior admit inference inference p denote surely eq random posterior scalar borel encode nan device confidence probability probability give observed loss confidence posterior coherent associate denote set represent posterior necessarily rather derive confidence posterior law probability recent simple distribution benchmark respect moderate inference absolutely continuous many literature kullback information statistical gain replace plausible physical distribution specifically gain call plausible oppose bayesian posterior high follow make inference define bad inferential minimize maximum bad gain moderate work solve let write define decision take loss consideration q extreme decision work posterior minimization lead framework fact q finding moderate imply thus thereby reduce immediate convex moderate eq one posterior work bayesian nonempty confidence close posterior p nonempty unless nonempty whenever posterior lack example involve use bayes depend confidence posterior verify p xx quantify prediction cg dp imply entail substitute unique drop parametric plausible prior quadratic contrast moderate involve typical hypothesis selection parameter hypothesis equivalently term work correspond work let side versus sided example ix posterior nan equal widely applicable bind nan hypothesis restriction eq bind hypothesis great two similarly stand apply ip I I omit brevity posterior recover interestingly case simplify evident unique say equation uniqueness moderate close work p ix work plausible balance bayesian encounter various bayesian posterior improve lead frequentist illustrate unclear would lead work benchmark attractive whenever corollary word inference irrespective posterior posterior plausible posterior specify member many involve member derive impose procedure average member explain posterior simplicity version posterior moderate three applicable version summarize generalize beyond version inference upon
ik note resemble specification stick stick break product fix reason specification construction different second concentration marginal aspect follow section sake clarity discuss specification independent thank result marginally precision measure hold true precision component base process h assume two process marginal use along precision process cluster observation cc h h row purpose analyze let correlation component q fig stick break right graph white gray area parametrization interval stick assumption eq q parameter bivariate beta breaking measure worth mixture globally exchangeable distinction limit h independent obtain consider block exchangeable case assume independent marginally take I ig give proof appendix recall process parameter take section dependent marginal random proposition extension slice shall sampling beta subsection propose extend product poisson stick distribution start introduce variable finite latent indicate finite crucially introduce allocation take value slice already conditionally vector ij hyperparameter law assume hyperparameter v exploration monte carlo gibbs sampler iteratively multidimensional trivial generation allocation multivariate stick break efficient conditional let q shall k subsection conditional u conditionally conditional strategy full depend kernel joint metropolis walk acceptance k employ independent conditionally important remark slice full almost conditional precisely small new dependent dirichlet normal apply simulate production united states european h synthetic base normal bivariate parametric section shall describe sake omit indicate full hyperparameter order sample n conditional inverse respectively previous parameter break obtain gamma respectively v alternative independent simulate accept turn inefficient rate th previous simulate walk proposal conditional proposal parameter acceptance rate simulate independent component alternatively model component normal mix cycle even feature regime cluster growth could expect regime cycle dependent multivariate dirichlet process literature usually consider day adjusted index regime py forecast shift shift intercept volatility dirichlet prior product inverse gamma var consider improper normal gibbs cc frequency period logarithmic solid line dash empirical note non oppose capture non linearity business cycle distribution row histogram allow conclude cycle result coherent result switch consider growth finding economic inspection posterior mean see economic business phase substantially pi coherent suggest regime nevertheless finding matter research four consequence solid fig tail dash ty predictive density conditionally evaluate effect correspondence realize similarly expansion phase atom fig approximated period expansion posterior distribution expansion fig second column one volatility concentrate expansion order identify dp mixture posterior associated country originally successfully allocation mcmc one cc st us marginal l il minimize sum square q specifically show cycle column vertical dark gray band correspond growth true gray column fig sample every mcmc iteration two make allocation comparable dependent estimate probability one cycle another cycle belong white light gray share atom allow find cycle fig identification expansion group atom interpret strong phase cycle phase coherent business exception present growth dependent apply dependent context non mcmc computation particularly cluster joint analysis business cycle parametric able highlight issue gibbs describe principle sample infinite proceed number check summarize pt algorithm step initialize suppose involve come sample metropolis gibbs old eq direct calculation obtain hence measure h sake simplicity place rs addition set eq give conditional definition proposition remark context may across motivated non modeling dirichlet multivariate process stick break weight effect provide efficient illustrate simulation united european index keyword paper concern statistic rely probability measure measure process stick break stochastically weight break construction beta dp paper generalization dp dirichlet take generalization stick break construction dp univariate bayesian non univariate dp incorporate prior provide extremely specifically model kernel availability simple use field aim process naturally generalization model divide different shall observation exchangeable exchangeable recently dependent dirichlet specification incorporate dependence atom process covariate exist dependence structure dependent random upon hierarchical structure stick g probability normalize vector measure employ bivariate dp stick weight tractable inference procedure multivariate poisson show appeal dp new dp posterior model data augmentation extend slice conditional series non time increase employ independent dirichlet process stochastic flexible stick breaking capture extend bayesian parametric model allow accounting clustering parametric usually introduce breaking process property introduce dirichlet process modelling propose chain mcmc dp simulated united european conclude value divide may specify assess information introduce ic iy dependent stick break stick break propose vector atom hypothesis stochastically probability determine via break stochastically independent dependent measure vector dependence measure affect underlying density
simple picture reduce dynamical half exploit helpful system organize provide background model reduction extend balance rkh set determine nonlinear control control input approach observe lot require small energy determine coordinate state reach significantly coordinate generalize approach nonlinear control advance suitably balanced need lyapunov equation balance impulse truncate balancing approach assume study circuit unbalanced approximation operator rkh provide laplacian build trajectory reduce system principal carry control balance control observable two generate input column span coincide many reduction lyapunov several idea behind balance align find f balanced look gap singular responsible output relationship assume negligible unstable integral however exist lyapunov balance find transformation positive balance unstable balancing method exist computing cholesky problem energy lyapunov development around observable asymptotically stable stable equilibrium origin eq origin asymptotically equilibrium hand normal output system coordinate theorem neighborhood transformation value case sort sort remove important balance systematic expansion statistical white gaussian balance transformation differential topology combine drive analytic approach balance xt linearization origin origin treat estimating operator reproduce rkh primary contain original pca extend generalize development lift reproduce brief overview reproduce theory heavily let hilbert functions rkhs kx kx call property reproducing reproduce unique kx kx kernel kx definite reproduce satisfy every sequence strongly pointwise convergence positive kernel hilbert feature feature fx kx use iv feature rkhs positive covariance matrix essential go existence reproduce rkh important define neither computable eq play minimize empirical eq solve eq hence minimize possibly infinite uniformly sample input rotation however signal xt see column observation response describe fix measure response q coordinate separate convention lead clarity understanding briefly review relevant background pca generalize pca dimensional work product center accord subspace show zero leading sort magnitude q similarly sort eigenvector feature principal space kx kx essence collect perform simultaneous component estimate rkh space worth transformation favorable practice center consider additional note view scale similarly collection q equation kernel simplicity choose terminology kernel root matrix system behave linearly assertion prove immediately proceed center order consider distinct convention center separately let similarly build feature representation c mn feature quantity c center immediately remainder drop quantity balance dimensionality state truncation calculation original behave order reduce discard project balanced reduction balance accomplish empirical feature map center space let mx space counterpart similarly since kernel balance rkhs last transformation onto eq map recall form top root system linearity yx yx c zero eigenvalue nonlinear close original accuracy nonlinear construct correspond reduce space refer appropriate inverse difficulty approximation get difficulty approximated element rkhs state assume series last might direct account reproduce case ability capture nn taylor approximation avoid state ix u th characterize typical behavior even trajectory seek ik z nk true regularize square notational convenience far ix j broad characterize separable estimate component solution order taylor approximate update dynamic desire expansion length eq ones calculation certain choice exploit fact derivative perhaps polynomial polynomial degree dx particular polynomial recall dx yx invertible would potentially empty reasonable convex pre image formal noting trajectory reduce counterpart virtue way formulation reduce recognize representation final theorem derivative kernel approximation need close dynamical jacobian notation jacobian see give close nonlinear control solely reduce p reduce exist expect input leave approximation appear future dynamic possibility approximate familiar ix ik hx rkhs different coordinate note map compare define define close simple linearization preserve first paper empirical datum rkh system brief reduce linearization system linearization observable reduce approach linearization follow control let example x rearrange write u truncate affect output response space time degree kernel retain retain sake eigenvector system follow control choose hz square wave peak sample map problem kernel output variable bias account offset use fast remark rd rkhs fact reduce instead simple square wave peak cycle zero demonstrate square wave input summarie taylor
algorithmic implication problem constant might solution entry minimizer one support minimizer follow minimizer local minimizer contradiction concern minimizer theorem minimizer trivial local minimizer illustrate theorem minimization datum part minimizer minimizer low minimizer easy vector satisfy eq order minimizer vector satisfy claim give desirable regularization problem tractable efficient unfortunately strong however minimizer basic solution exactly stationary difference follow hardness facilitate np np hard hard prove useful technical minimize optimal unique optimize maximum first partition optimal must partition present polynomial reduction strongly partition subset sum consider minimization remain strongly np smoothed problem form suffice hardness change eq ij n equality come hold z q one know argue reveal hard original complexity global minimizer hope problem enough desire nonzero theorem choose bridge estimator sample tend nonzero bridge design matrix typically standardize small covariate constant find sparsity minimizer estimator become meet efficiency bridge model eq q hence minimizer likely could choice consistency estimator unconstraine theorem example wang unconstraine minimization minimizer extensively least square minimizer concavity minimization np carefully certain desire sparsity regularize keyword nonsmooth nonconvex variable reconstruction bridge minimization least dimensional datum original entry subset regularization regularize deal derive continuity boundedness nice property oracle theoretical distinguish entry approximation bridge advantage minimization nonconvex analyze approach develop globally convergent guarantee computational open problem attempt draw hardness hard np polynomially bound admit time np objective admit polynomial unless penalty precisely find hard smoothed hardness side sufficient desire minimizer global local estimator encourage global organize follow meet requirement minimizer minimization hard smoothed version even though lipschitz gain advantage term finally literature choose regularization
intervention control passive joint signal processing community concept probabilistic causality several causality causal kernel representation causal relationship distribution w acyclic dag correspond causal system naturally think diagram overview paper motivate system framework motivate intervention sequential recursive whereby natural induce factorization accord correspondence kernel correspondence direct generalization quantify strength compare distribution theoretic interpretation back conditional analog statistical diagram communication without figure message map deterministic intuitively clear message cause message joint eq q factor statistically dependent indeed inherent channel decoder message causal message opposite break symmetry end assume suitable sequential happen make assignment look input assume mapping message assignment hard assignment one replace decode map map effect assignment q clearly random action conclusion decode absence causal dynamical multiple feedback loop influence system couple equation eq specification sound sense equation dynamical system loop use seminal system generic dynamical feedback loop arbitrary dependency causality limit system order system causality markovian sort factorization direct vertex direct dag graphical heavily sequel associate dag let direct eq go impossible markovian simple study causal effect variable examine assignment main start call intervention let distribution add value assign claim effect upon would intuitively intervention intervention original equation original intervention define intervention reasoning follow intervention consider intervention intervention eq intervention intervention intervention distribution induce follow write outside following suppose observe system evolve accord belief graphical markovian offer visual way correspond dag let couple consider joint intervention word upon diagram eq depict channel sequence feedback intervention represent turn construction distribution capacity tuple order course subsequent depend equivalent conversely way q step view mapping tuple define channel specification intervention system originally direct complete encoder decoder controller contrast causal various causality oppose markovian system let point sx coincide strength intervention divergence expectation w marginal induce obtain distribution recognize information turn context communication channel information causality conditional reliably ordinary must along direct path dag pointing toward definition arbitrary markovian markov factorization product dag marginal accord edge dag correspond numerator denominator exploit independence relation encode dag point status causality involve disjoint define conditioning eq sensible structure let direct quantify flow markovian dynamical contribution ordinary happen direct information give disjoint q brevity equal similarly eq fundamental discover influence observational datum reduce canonical causal structure chain direct relation structure direct cause moreover direct go since direction reverse active study causality concern causal effect markovian dynamical whenever causal feasible become index express disjoint dag distribution
however neighborhood portion propose allow understanding first generation markov monte draw correspond present sample procedure equivalent drawing try draw measure initialize graph g lr von fisher distribution edge size converge graph topology think employ assign yet approximated employ process uniform unobserved small graph spherical basic lk example generate lr g adopt study fit statistic simulate network fit simulate either degenerate secondly edge graph neighbor divide compute census proportion set lastly minimum median network node outline graph random network show capture statistic generate resemble observed geodesic indicate cccc indicate line effect high starting depict model generate sign misspecification cccc summary indicate summarize benchmark whether degeneracy phenomenon network summaries degeneracy severe mle specification correspond original generate cause degeneracy feature vector relative convex polytope issue spherical belong boundary interior outline social degeneracy sampling spherical g von fisher individual geometric mode spherical realistic author thank v c nsf award generating degeneracy locally graph contain relationship bioinformatic interaction researcher area develop network network long history generalize social nice suffer issue degeneracy instability place distribution empty result purpose degeneracy discover cause graph hull become mode second insight embed surface sphere convex vector belong hull could serve avoid degeneracy issue approach embed surface sphere distribution spherical possible thus degeneracy additional start present generate realistic exponential synthetic realistic conclude future similar self loop undirecte adjacency zeros diagonal label fairly random network feature explicitly define used triangle star subgraph pattern feature show edge triangle star model use vector partition much feature view mass refer weight q exponential family polytope statistic graph p interior hull possible essence preserve observe extremely fact approximate e pseudo likelihood instead often suffer degeneracy type degeneracy first mle exist mle type happen reliably significant graph degeneracy consider viewpoint undesirable justification place mass vector place little undirected graph feature vector graph display diameter topology cell count accomplish diameter give mapping edge table sized space graph pair count cell feature distribution hull subset black estimate different pmf triangle estimate indicate indicate mention degeneracy unless otherwise degeneracy mode focus achieve attribute give structural edge triangle graph gender however degeneracy know minimize degeneracy task knowledge accurately reasonable make geometrically geometrically dyadic share specification avoid dependent specification degeneracy entirely surprising mode investigate degeneracy suppose set boundary hull hull observe log observe include boundary maximize attention illustration node specify solid black number boundary correspond mode unique tt vector mode outside cone maximizing explain many place large graph degeneracy change geometrically weight geometrically distribution arise fact match concentrate mass grid uniformly spaced model evident exhibit degeneracy issue describe cc right plot indicate indicate provide ii degeneracy sensitive geometry geometry place little mass place e onto belong define section spherical sampling technique graph ng von hyper feature spherical new model generate approach propose locally embedding spherical short execute outline algorithm graph possible otherwise walk step graph away result embed mapping spherical approach dissimilarity distance euclidean outer beneficial preserve map locally small graph spherical rest neighborhood preserve estimate spherical position unknown radius sphere x u n kk von first member unlike distribution mode determine concentrate see detail wish single also treat control concentrated obtain
possibility index exploitation epoch player cause mistake exploitation epoch part lower decentralize armed bandit unknown multiple player markovian play evolve accord passive player arm suffer loss reward decentralize arm decentralize construct achieve arbitrary nontrivial logarithmic find financial engineering mab independent arm arm play choose select large reward select selection case grow certain lead constant regret growth good result accommodate multiple simultaneous reward evolve play classic mab refer ucb achieve time ucb extend liu formulate study decentralize classic I arm reward player choose information player player arm one receive desire decentralized mab player fair share logarithmic centralized player share eliminate centralized perfect scheduling bernoulli decentralize mab player policy player decentralize markovian unknown markovian play accord address ucb achieve reward know play well epoch epoch length balance optimal markovian reward model latter long know strict decentralized model former arm action player state evolve accord arbitrary play extend logarithmic stronger know couple single player liu adopt policy achieve weak system available use structure propose outside epoch learn strict mab specifically arm govern stochastically identical policy arbitrarily logarithmic positive l decentralize mab arm player choose activate offer amount reward arm arm time arm arm markovian irreducible arm arm occur play obtain reward reward arm permutation specify history th play notice random share obtain arm play mention sec performance evaluate arm regret follow random induce minimize growth growth rate slot epoch slot epoch arm decentralize r l exploration exploitation index index epoch exploration exploitation epoch epoch decentralize base divide disjoint epoch epoch illustration epoch player index arm play obtain exploration epoch exploitation epoch sufficiently accurate fig beginning select high index arm exploitation epoch player play epoch detail obtain exploration player epoch play epoch decentralize exploitation epoch epoch information phase observation exploitation epoch dynamic epoch e point epoch depend epoch htb exploration exploitation epoch begin exploitation exploration play exploitation epoch play epoch every spend exploration epoch go exploitation epoch index current sample accord time high index arm divide player exploitation epoch step divide epoch arm play agreement different subsection agreement eliminate logarithmic regret player join global synchronization epoch agreement arm consider player play arm elimination achieve join system schedule regret order markovian irreducible reversible markov chain state I satisfy condition epoch get decentralize epoch epoch agreement decentralize also appendix dl achieve arbitrarily formally arm finite irreducible reversible markov chain increase conclusion hold model appendix decentralize multi armed bandit player aim term
part recover cm cc section structure prior e document goal probabilistic representation semantic collection latent dirichlet document represent predefine number vocabulary usually vocabulary e representation corpus fundamentally argue lda multinomial cast problem formulation present call representation vocabulary e component fix constrain entry nonnegative decomposition interpret ensure describe structure regularization light decomposition document also share force conversely deep dictionary typically non extra inclusion acyclic dag select induce norm present section eq variable direct acyclic introduce norm associate application find estimation consider resp active predictor aspect add step efficiently collection able appropriate possible term potentially doubly blue dag review structure convex norm norm readily norm introduce dimensional unsupervised interpretability increase predictive european project grant nsf award anonymous whose comment greatly method aim parsimonious combinatorial admit relaxation norm structured deal structure dictionary concept scientific take form selection two interpretable admit look sparse predict variable tool benefit reference therein develop theory selection individually regardless exist relationship structure hierarchical merely improve form functional voxel discriminate state localize area face show plain norm fail encode background order come bioinformatics diagnosis profile genomic feed profile characterize profile specific genome discriminative feature contiguous improvement accuracy find present nucleotide snp gene target share gene hierarchy information design induce scheme capable encoding sparsity mention available pattern nonzero support induce induce norm structure way e convex optimization theoretical analysis easily traditional norm popular select overlap group different type introduce application namely section variable vector letter bold one j nj w row extensively norm review induce norm learn predict typically observation usually regularize suited indeed pair follow sparsity typically nonsmooth non reader description regression norm pursuit lasso take pursuit write correspond limiting span counterpart correspondence notation notation describe variable represent correspond notation express usually representation consider regularize solution depend ny belong normal cone sufficiently ensure solution ball favor correspond shape pattern double pyramid exhibit point align subspace tangent sparse ball norm section norm cm axis align origin norm induce norm display priori reflect ht cm sparsity arguably priori disjoint block ignore form positive zero square regularization interpretability model block share task learn group size depend desire discussion seem reality collection encode quite structured key present construction overlap complementary section overview encode relevant setting group show sparsity remain entire reflect extreme point unit pattern obvious overlap interesting tie belong group solution obtain form mean take intersection group moreover mild possible hoc solution belong norm adapt figure select diagnosis profile since genome constrain support grid set relatively half may pattern sparsity inducing build upon group performance background wavelet denoise fourth hierarchy variable assign forest select hierarchical exactly instance wavelet hierarchical model model order bioinformatic exploit network task scale mining fmri cognitive contour tree represent group set zero gray remove root subtree obey ii select possible set aforementione example complicate topology consider discretized discretized slice application direct encode developed overlap theoretically addition group interval weight globally structure model support scalar group generalization norm group solve norm equivalent interpret expand careful zero keep zero component lead select convex relaxation encourage knowledge take graph connect prior co gene favor neighbor select variable formulation point lie align circle correspond group hull ball ball leave without rely fa therein sparsity induce convex envelope convex ball structure far submodular thus lead different type level mainly convex define viewpoint context focus without theoretic non convex community statistic name fuse lasso w signal recover piecewise function sparsity penalty select exclusive inside mention grouping priori penalty term encourage thereby form review solve regularize technique adapt constitute take advantage simplicity present proximal least gradient logistic assume process numerous reference therein therein chen provide strong convexity assumption function take q stay close solve iterative constitute name proximal technique split iterative furthermore rate function scheme extend accelerate case proximal converge optimum rate rate order non overview core define rewrite regularizers proximal close make turn norm compute operator efficiently group operator close g variable computation proximal composition g overlap proximal expensive technique norm proximal implement open software generalize structured proximal done traditionally analyze load denote solution traditional consider model e x possible criterion consistency assumption generate loading especially characterize hardness refer oracle regime consistency regularization validation large try collection along usual model ol hybrid without validation phenomenon characterize compatible additional encode assumption sometimes relaxed interpretation global formulation orthogonality dictionary signal also impose terminology observation coefficient product coefficient correspond factorization formulate constrain pseudo norm say typically sparsity matrix assume convex versa convex
motivate em provide mean start list lead experiment cluster yield list method continue guess c c c c mse runtime sec ir cr na na km na na motivate follow mse th singular fall formulation utilize use next parameter start form individual think include set mn tt estimate list minimize mse separately fit least square regression index column corresponding response predictor z consist least find similar th let individual estimate span whole estimate individual km perform joint regression perform explicit various along ce discuss runtime dataset compress sense projection replace dimensional sketch consist split dataset relevance ce dataset calculate ce fraction vice versa method base respective matlab implementation intel core ghz machine number km choose parameter mse mse ce summarize km requirement cr perform individual regression run km method km report mse cr improvement ir far see great km runtime performance ce associate algorithm appear arbitrary never know eigenvalue unity suggest cluster propose show substantial ir method cr query take cluster sensitivity match remark shall snr regime figure cluster size method linearly super growth near mse ce runtime km sensitive compare estimation compare method runtime km scale bad complexity section order ir th operation multiply transpose operation multiply matrix cr operation complete em e iteration operation vector compute operation km need well explain cluster experiment group close regression km multiply thresholding operation thus operation individual regression compute need similarity operation individual operation ir per see cubic growth consistent seen figure growth km growth unlike real km algorithm much compare em ir cr growth shown mse load also understand introduce end vector trial ir cr noise noise level low find cr less ir hard neighbor pick cr phenomenon clear close level total filter pooling low balance well level since em iteration converge optimum trial something close similar optimal snr use ml regression point following denote fisher last choose I normality estimator tell normal mse empirical even db predictor introduce various experiment yahoo challenge near optimal performance relatively insensitive associated mathematical understand prediction surface hope present deep future manuscript general non appear th experiment notation begin kk p represent eq find I maximize em pick pr mn z ny mn mn thus denote r eq maximize derivative complete em column mse constraint relevance whose th block g dm obtain stack one problem usa experiment cluster experiment relationship experiment cluster relationship predictor exhibit regression focus several applicability yahoo associate study diverse adaptation maximization inspire mean value thresholding adaptation collaborative filter dataset simulate datum good near good reasonable method light choice regression relationship response variable predictor rich employ field often collect reason share relationship example yahoo challenge query multiple relevance score relevance score many query cluster consider problem cluster already unknown study arise outline main level find mse factor collective give notation matrix bold vector transpose
datum histogram point heat leave white fall common string figure look find equal rd digit digit us plot figure leave take value share digit digit thus highly correlate share digit share digit digit digit plot digit figure share plot share value share part match third digit show find level precision word example value least percentage share five retrieve fall build figure depict frequency gaussian surface three dimensional top panel first concentrated precision uniformly gradually go precision peak digit distribution peak discriminate peak top digit peak traditional ultrametric ultrametric useful case digit relevance share digits ultrametric common digits distance show store advantageous digit instance distance basis chemical comparable due semantic tw email house safe tw ex uk induce ultrametric quadratic agglomerative apply distance digital survey costly neighbor keyword ultrametric agglomerative axiom triplet point dx dx dy dy consider ultrametric triangular ultrametric dx max dx dy addition positivity triple metric map onto ultrametric dissimilarity often metric ultrametric agglomerative hierarchical pairwise merge place cardinality closeness distance closeness singleton singleton inter cluster dissimilarity define wish step agglomerative hierarchical pairwise dissimilarity show reciprocal near neighbor quadratic time present carry linear theory bad node term dendrogram set embed totally strong subset hierarchy ultrametric show embed constructive vector involve series pairwise propertie iii h I ultrametric euclidean use minimize change dissimilarity agglomerative constructive construct ultrametric search ultrametric inherent agglomerative furthermore help greatly find inherently ultrametric view coding seek string could determine compression partially express infinite common close distance value digit exponent notation range digit integer order order place take example position distance ultrametric dealing number hierarchy dendrogram dendrogram general non integer ultrametric present rank geometry shift focus impose encoding summarize aim rather agglomerative quadratic individual seek read number observation p read scan dataset interested encoding define digit bin digit bin digit level neither deep rise suffice bin require operation value constant read cardinality digital produce position million galaxy al
candidate gene time disease respectively promise ability global disease curve begin ht top top disease select set unlabele complete pathway table disease reasonable disease cancer diabetes breast successively disease size p gene list top list precision unknown list training namely see cancer limit large list test c c disease cancer diabetes gene disease know disease across still retrieve disease validate gene list database report ten cancer cancer diabetes list cancer tp diabetes col c col disease order cancer diabetes cancer breast cancer publication find relevant disease gene extract pathway tool discussion introduce gene unify able information rank causal disease disease gene share real database outperform art method disease dirac prevent disease generic disease relevance enhance dirac influence final believe much room improvement phenotype could integrate induce question recall top choose ready addition think disease known situation gene already disease intuitively need information disease dependency global bring local number check light illustrate plot standard disease least known surprising difference disease criterion suggest practice interested rank list top soon go gene novel learn score fitting augment disease identification unlabele bioinformatic good pathway cancer patient high protein protein protein us disease gene aim gene typically human genome disease include expression annotation assume source product gene think accord representation kernel gene collection phenotype know phenotype gene disease complement far disease available short candidate gene disease disease retrieve gene disease gene disease near list single disease source list must candidate gene exist quantify disease learn motivation behind scoring initially gene represent retrieve scoring usually try gene could dash letter green negative area represent solid consequently positive density practice assign binary discriminate label assign point logistic training implement positive unlabeled proxy building set binary lead biased bag equal practice bias weight fact represent confidence negative namely add bag contaminated speed scalability take specify successive aggregating positive element decrease detail observe dramatically affect default iteration multiple datum source gene expression formally encode result source first kernel gene similarity widely often heterogeneous integrated score method building non optimize weighting differently mkl formulation discard give gene straightforward mkl instead svm train disease disease treat disease know characteristic similar disease often gene suggest instead treat disease beneficial disease across disease disease attempt disease causal property disease obviously jointly disease problem adapt gene candidate disease scoring order instead positive gene disease disease kernel disease share precisely consider variant various dirac two disease disease treat disease strategy treat disease disease define allow basic sharing disease causal dirac disease extent disease gene disease share low information disease exploit knowledge capture disease share disease many work prior phenotype cause gene across principled disease practice disease plug disease mining measure measure gene kernel slight variant phenotype add dirac eq motivation disease phenotype incomplete characterize disease disease similar addition allow distinguish disease give importance associate phenotype result four gene score variant summarize c l name kernel share disease across disease share share share identity differ share disease third column disease variant apart disease variant follow gene easily translate gene mean kernel variant restrict simple average leave dataset association extract disease train score positive pair training list implicitly know pair remove candidate rank sample receiver operate curve formula therefore gene method cdf association rank list function gene first mkl outperforms method mkl gene kernel mkl website compare design disease function query gene disease similar gene similarity disease label score smoothly description v multiple status leave gene expression est functional database activity text consist data product define create ensure phenotype measure measure automatic text mesh medical subject similarity disease mesh disease finally collect association disease gene association involve gene acknowledgment grateful yu make grant centre f france paris france paris france france basis goal molecular linkage analysis modern throughput long list hundred candidate time consume candidate propose disease implement strategy allow integrate various share disease search disease datum show outperform human decade considerable genetic rare disease diagnosis towards biological disease discover disease identify contain linkage population identify hundred gene gene consume sequence whose activity disease sample genome scale identify list candidate gene among agent important promising gene likely information biological pattern expression promise candidate availability genome set variety heterogeneous store various obtain candidate expert perform task via approach previous work attempt identify promise without gene annotation phenotype investigation successful disease gene share know phenotype phenotype perform use art integrate heterogeneous information candidate decrease use label protein protein interaction able know disease gene disease web disease association already new bring main first paradigm score like gene exploit know disease gene candidate disease gene machine paradigm positive disease play disease phenotype disease disease allow rely disease relate disease machine implement share disease disease heterogeneous include scientific literature datum integration differ approach limited scoring database disease disease gene gene share across disease assess ability retrieve disease disease share across disease extract association nine source expression profile functional annotation interaction activity encode pair accord way source mkl compare differ unlabeled mkl comparable pair sign integration though begin picture behave come refer train panel global curve panel begin visualize scatter directly rank small rank rank list many mean dirac phenotype kernel phenotype concentrated side indicate favor kernel generic although plot different rank share disease beneficial across disease beneficial restrict associated share disease one training procedure leave disease association show cdf share information retrieval behavior depend look globally mkl rank systematic difference pair sign summarize picture indicate significantly confirm variant mkl cdf likely serious investigation cdf curve mkl increase curve mean yield truly additionally proportion identify top list show
infinite neither convex function set requirement banach sign bound continuous value function vector call pre banach everywhere satisfied pre let q condition unique another consequence hence banach functional possess regularizer q point minimizer obtain borel subset exist index countable norm put transpose former result minimal interpolation minimizer form hold ready satisfie hold lemma x bf recall x inequality lemma interpolation minimizer q exist constant bridge reproducing introduce bilinear say regularize square literature compact measure definite reproduce open ball radius cover q consist eigenfunction eigenvalue cx minimizer problem hypothesis regularization choose enable discard immediately ef z ef proof regularization error get since turn decompose z bound law surely suppose lemma reach rate cx eq discuss lemma maximum achieve substitute higher impose symmetry might strategy thm thm thm thm zhang regularize bind error reproduce linear bring discard automatically reproducing banach reproduce banach space least square reproduce banach linear theorem recently purpose rate regularize square sequence row reproduce network extensively rate respectively programming attract much attention mainly sense able yield sparse result reproduce estimate regularize least explain machine learn fundamental instance measure minimize eq conditional unknown competitive precise intermediate banach nonnegative decompose quantity hypothesis carefully reproduce space definite regularizer regularization need
algebraic bad dimension believe obtain number k artificial present extraction employ artificial method approximate randomize oppose expensive fact comparable space suffice create via run dataset formally would obtain dimensional run state notation obtain running mean low space eqn projection drop without frobenius equation eqn provide also k optimality svd first eqn fact appear argument proof eqn rescale accordingly preliminary extraction present work prominent feature mac core processor ram empirical finding exhaustive extraction satisfactory rather far believe constant analysis cluster select construct execute feature compare feature method summarize follow modification matrix calculate naive multiplication first contain singular vector fix particular matlab execute evaluate allow use practice employ experiment namely every mention mean initialization repetition ht whereas right ht whereas right column generate center uniformly dimensional center point center center include center provide optimal use handwritten uci repository point normalize take object image image grey pixel ten take homogeneous position tolerance object database person pose illumination dataset pose expression dimension identical datum cardinality correspond normalize mis base label example correct finally report important report run dimensionality propose five ht dataset whereas present relative dataset mean demonstrate approach respect mean effective apply separate method world dimension result decrease increase dimensionality compare laplacian dataset laplacian superior notice naive poorly notice necessarily increase take evaluation run nearly separation gaussian thorough evaluation would far finding quite encouraging topic consist approach explain dimensionality approach dimensionality reduction wants explain connection text modern reduction mean test practice encourage fast dimensional provably novel extraction interesting research design provably efficient relative mean follow proof rr k drop without change spectral ii side singular value inequality assumption proof multiplication involve sampling subsampling row notice observe inequality norm bind apply together bind failure rest assume algebraic variable matrix rescale four wise independence construction chebyshev side conclude definition transform row eq failure notation rescale matrix set appropriate applying index j multiplication claim say row contain entry random independence bind drop limited repeat rescale consider respectively crucial bounding failure last k apply k happen follow event let event k obtain bind add spectral sub individually bound lemma k last eqn k markov theorem axiom theorem false green yahoo new york ny computational department two select apply construct feature construct despite cluster heuristic address provably provably cluster random decomposition towards understand cluster provably extraction method upon feature extraction fast svd factor objective cluster engineering web perhaps know attempt address euclidean point positive integer distance point center minimize cluster application modern massive considerable challenge cluster dimensional datum computationally inefficient existence feature allow practitioner address selection subset construct rigorous selection mean mathematical study point belong collection jj ks jk approximate cluster feature construct actual dimensionality extraction construct artificial original formally optimum partition reduction construct compute project plug optimal notice mean study clustering technique helpful observe favorable rp svd rp svd despite address provably accurate feature extraction method know describe projection imply row immediate mainly due prove pairwise distance multiplicative factor pairwise correspond within reduce artificial clustering briefly section corollary suffice provably accurate selection algorithm present k achieve theorem compute present feature working approximate approximation approach compute synthesis paradigm paradigm extraction probability result show small obtain approximate outline rely paradigm paradigm describe extraction employ decomposition construct dimensionality almost show svd switch algebraic mean notation eq represent belong non zero belong cluster slightly indicator identity hold identity I jj pick scale cluster formulation minimize quality evaluate cluster framework new reduction along give denote number indicator denote formalize approximation correspond run cluster run run dimension considerably fast algorithm though although admit well employ algorithm section dimensionality compare similar run arise one dimension key subset column algorithms I new combination extraction algebraic perspective main analysis approximate rank reduction cluster procedure orthogonal short proof completeness completeness value rescale procedure close orthonormal r prove leverage rd conclude rescale column randomize close intuitively subsample affect frobenius define tt j r linearity definition worth note set equation rescale rank orthonormal sampling correspond compute rank two another rank factorization theorem proof switch bound inequality almost state point project multiplicative precisely show orthonormal research transformation gaussian random recently fast transform construction prove rescale computed utilize summarize interest defer proof matrix norm square scale comparable analog fix effect sign singular rescale construct simultaneously w analog analog lemma rescale matrix random worth note random require dimension close ok r c c construct approximate right singular technique section one replace algorithm exact present corresponding proof replicate replace working considerably algorithm kk r ok suffice roughly point theorem formally introduce obtain partition fact run k first drop equation short svd q eqn eqn failure triangle inequality drop eqn drop
forecast pi successfully several forecasting kind input yield concern combination sequentially sequentially remove motivated flexibility deep exploration initialize previous concern winner test winner combine alternative average proportional error present eight forecasting show respective forecasting variant forecasting loo variant forecasting autocorrelation criterion forecast avg strategy value weight testing assessment compare strategy forecast measure general comparative two stage compare post perform high significance rank average tie hypothesis nan degree show improve statistic nan accord degree hypothesis difference least hoc among eq asymptotically normally compare comparison possibly pairwise incorrectly reject procedure adjust value well adopt correction strategy statistically sort post competition selection procedure obtain phase equal quite good competition forecast series illustrate forecast htp deduce pre mostly competition phase overall weight comparable improve big perhaps version strategy consistently consistent reason put burden model forecast pattern input beneficial mixed whether perform construct concern combination winner winner finding forecast combination persistent finding base competition true competition phase true finding concern selection persistence validation select good strategy good pre datum intelligence competition competition hard forecast horizon uncertainty comparative review ahead apply forecasting competition comparison promising finding output impact persistent selecting testing method could make rgb rgb ahead forecast deal literature extensive missing aim fill gap strategy ahead forecast theoretical term attain strategy large experimental forecasting forecast combination ahead forecasting follow finding consistently support multiple strategy perform forecast long forecasting forecasting learning forecasting playing role field science engineering economic finance step forecast ahead accumulation increase forecasting influence however increasingly adapt many useful model bilinear autoregressive autoregressive bootstrapping assumption distribution nonlinear forecasting development last decade machine attention serious box drive historical stochastic find artificial neural conduct conclude appear neighbor create area mining forecasting forecast attention forecasting forecast critical topic however ahead forecast point good knowledge ahead forecast ahead forecast fed input contrast strategy return value combination strategy behind direct previous introduce preserve value characterize time series multi multi scalar single strategy call preserve direct strategy flexibility present sometimes thorough unified review well comparative ahead fact collective outcome regard forecast little use favor strategy fact strategy strategy show strategy give direct recursive strategy theoretically author theoretical experimental evidence favor concern strategy et al provide bad forecasting recursive forecasting strategy dependency forecast previous forecast finding find truth relative paper experimental different ahead forecasting competition pose usually multi forecasting existence series dynamic experimental comparison configuration regard comparison recommendation make forecasting conduct rather show forecasting influence forecast adopted successfully forecast task also forecast competition goal assess compare intelligence competition potential present strategy describe forecasting methodology apply discusse finally give summary conclude ahead also series task next historical observation forecasting horizon give presentation comparative strategy common notation dependency refer dimension series predict intuitive forecasting perform step forecast apply subsequently use one ahead continue forecast give depend time forecasting suffer ahead especially true forecasting observation reason potential recursive strategy accumulation horizon intermediate forecast propagate forward forecast subsequent limitation recursive strategy forecast many serie like neural consist horizon series learn model compute independently induce independence affect forecast complex dependency forecast break forecast large model use multi ahead forecast near neighbor combine principle strategy like input forecast like learn forecast outperform direct series set research need far single see introduction strategy existence dependency affect learn model return model preserve characterize time series conditional independence accumulation successfully real ahead forecasting preserve structure constraint flexibility forecasting appeal direct strategy forecast horizon block fashion thus ahead decompose size task direct value number equal intermediate value allow dimensionality dependency maximal provide preserve large dependency predictor strategy strategy clarity series forecast learn successfully forecast nn five ahead forecast task strategy forecast relationship forecasting strategy link combination direct recursive combination present multi ahead forecasting ahead forecast task series horizon forecast n forecasting obtain forecast give task number computational recursive allow strategy computation hand need equal follow conclude five forecasting cm computational series accumulation accumulation trade recursive reduce flexibility trade forecast forecasting forecasting estimate scalar value represent dependency paper multi ahead strategy forecast require setup instance local since series forecasting present learn section type forecast future value past series value parametric describe value analytical whole nonlinear regression assumption divide combine divide two different module cover space base inference radial basis tree modular intermediate scale identification basis global excellent complete rather predict also call point local locally weight combine proportional distance predict nearby weight distance well know intelligence give rank competition rank use work algorithm learn query contrary local parameter extensively model technique defer forecast number neighbor next local discard repeat query reduce consideration motivate multi step ahead context definition namely ref adequate configuration paper number selection essentially control bias strategy length second future consecutive section series pair output different select associate leave loo loo disadvantage repeat effort fortunately powerful sum calculate set aside loo training sum perform index replace already calculate method loo error neighbor loo error prediction return query sort increasingly neighbor look multiple response quantity criterion compare model different multiple output strategy sort respect index th closest j e look loo loo lk extension loo cross capability model loo aggregation error note output happen single e direct horizon suppose large time forecast quality forecast measure autocorrelation consider autocorrelation query symbol concatenation represent pearson discrepancy two part use note partial part discrepancy autocorrelation concatenation training series sequence autocorrelation evaluate minimize discrepancy training series number preserve property sequence return q sort increasingly vector k equation consider prediction testing consideration prediction query exist mainly
subject control failure hypothesis tight calculated parametrize define ball provide later short length shortest embed relate replace lagrange multipli ml constraint controlling cost restrict hypothesis f cover calculate derive later cost handle accord regularize beta formulae norm decrease decrease generalization broadly fact derive constraint specifically intersection operational equal bb bb x b regularize incomplete beta influence operational able specify something operational guarantee able operational result use volume spherical space cover uniform improvement lead arbitrary totally result cover number cover volume theorem cover convergence lf e relate cover number covering follow l first lipschitz cover fx l supremum empirical involve number cover metric follow norm relate loss view function function mn substituting use relation spherical b complete several decision operational framework power grid abstract necessarily explore application hypothesis operational quadratic constraint arise current type lead intersection currently shoot decision decision epoch mdp improvement present call framework advantage quantifying size generalization ml operational potentially school management technology usa operational exploratory theory knowledge implication reliability cost failure estimate also present framework essential influence explore range reasonable operational smoothed different distance belief cost minimize operational combine step sequential process operational term operational simultaneous process thus simultaneous simultaneous process organization simultaneous require subproblem solve solve vary full simultaneous know vary range cost operational belief operating cost unlabele instance sense close connection reality regularizer company give answer policy rely accurate abstract deal calculate deal us department grid document engineering inefficient future need without operational substantial capital decade utility implement inspection whereas new york city electrical go half time separate include inspection electrical service program extensive replace possibility serious company scheduling inspection project characteristic history past failure serious serious event repeat failure failure rare probability fail framework make fix assume terminology indicate time encode electrical previous whether within specific feature unlabele substantially carry physical permutation failure estimate form b logistic lf e logistic send change long operational cost repeat failure visit come make first make failure process discretize approximated bernoulli cost application application fail convenience begin return differently mean first visit visit definition equivalently start visit start start proportional failure visit visit occur within unit failure determine random thus distribution I failure visit visit trading visit failure l give become visit relax assume vanish failure node would expect failure visit node return term indexing cause failure visit cost reflect failure govern occur probability ip proportional similarly influence visit early incur reach schedule visit later node high chance visit total failure present failure probability cost derive cost quantity visit visit failure failure summation cost mean exhaustive define operational minimum problem cost proportional node operational proceed operational property operational true operational unless implicitly major objective program md md k different goal minimize traversal case need visit goal visit np total view item customer stop customer remove goal customer start program later flow flow drop yet visit variable flow edge integer restrict self loop form ensure one going come start constraint quantifie relate define estimate change sum failure form integer choice model version intuitive minimum particular individual serve purpose depend operational necessarily map nonetheless subproblem completely subproblem alone discuss solver simultaneous generic solver solution bind relate exact np hard cycle hence unweighted mixed integer formulation c show property motivate large accuracy triangle point end figure color black also plot color simultaneous failure cost model label possible lot choose enough training process cost unit illustration predictive operational cost exist feature triangle unlabele right indicate number edge indicate highlight company set give problem show simultaneous second experiment find predictive operational third issue experiment predict failure course year failure arbitrary interval failure hour fine scale develop secondary electrical design predict represent encode type electrical past source encode source prediction period test operational equip relatively distance obtain google driving note distance coordinate drive especially city limited inspection probability spend outlier simultaneous generally prevent many logistic misclassification size prediction roc auc increase tend cost fix range know could see substantial reduction varied away sequential process auc solver dramatically failure cost probability increase unit eight operational yield failure probability estimate due uncertainty cost nm marginally horizontal auc obtain simultaneous increase increase horizontal represent regularization failure horizontal horizontal regression different first provide I probability sequential failure depict I view I solution sequential solve electrical grid simultaneous substantially na I little understand truly believe belief combine justify varied simultaneous expect small operational cost simultaneous showing operational helpful regularization term much effect sequential similarly observe experiment seven problem hold decision pick whose size solve simultaneous involve sequential pick coefficient subproblem simultaneous achieve value good encode right per per type figure auc hold varied different training decision infer test small size simultaneous knowledge operational cost simultaneous sequential perform nonparametric versus auc median auc
present boost objective converge even issue demonstrate experimentally model standard good result widely detailed select direction error previously study kind primarily overcome projection final rest operation within hilbert discuss quantify term exist discuss descent recently natural match representation vector convenient present hilbert boost restricted descent input class lebesgue measure fx hilbert natural quantity expect perform need compute gradient functional say space iff functional functional wise pointwise differentiable respect outline boost replace descent necessary optimize space computationally generalize restriction directly primarily first descent second quantify explanation direction minimize generalization al special multiplication directly projection operation way direction restrict projection operation schedule subgradient onto hypothesis nearest f restricted quantify relative restriction refer boost weak convergence generalized notion equivalent norm search direction later restriction project gradient either h projection though hypothesis interpretation specific function traditionally weak classification project equivalent classification problem norm gradient output notion quantifies performance advantage hypothesis equivalent notion refer improvement result multiclass weak learner modify statement edge satisfie extend version analyze descent descent direction boost meet approach rf analyze extend space smooth functional result number work rely smoothness learner performance tailor pac weak learner learner train two unconstraine optimization theorem strongly restriction iteration size smoothness requirement upper guarantee progress select gain require iteration adaboost tight reasonably convergence result applicable wide derive benefit broad weak learner perform problem equally albeit efficiently require much approach quickly case objective objective two always give final space find f second optimizing take keep residual leave force past possibility functional residual functional restriction ff restrict like extra come serve mechanism error projection analysis norm residual derive observe increase decrease projection requirement residual present complete repeat weak algorithm weak learner evaluate preliminary task classification problem plan policy learn feature policy smooth loss minimize cost demonstrate adapt boost naive algorithm know three planning domain weak learner tb second experimental microsoft learn specifically web use learner learner final algorithms uci repository letter dataset experiment multiclass hinge multiclass learner interest naive fail letter rate strong case repeatedly learner optimization acknowledgement helpful feedback conduct laboratory technology adaboost weak classifier statement equivalent non negative achieve error inner formulation examine break incorrect weight side inner finally adaboost converse give edge requirement proof show imply increase relation adaboost style notion boost framework strong training multiclass multiclass classifier output baseline requirement extension classify simply weak output label convert weak learner output otherwise adaboost style cost reward requirement fine modification edge performance multiclass framework multiclass learner sum sum max multiply show existence implication broad strong goal formulation norm use n n sum adaboost requirement inner strongly definition strong examine objective subtract apply f rearrange conclude norm strongly functional restriction tt potential descent sum q q give final repeat convex restriction g ff f start convexity bind final theorem potential augment step multiply get restriction let ff f similar theorem boost powerful learner boost framework analyze descent boost introduce performance generalize
maximal matlab duality glasso ghz intel processor create p entry entry kp table set component take large give precision computation block machine improvement report operate block loop across sec speedup graph glasso glasso glasso glasso glasso glasso screening reflect list component thresholded give fail table improvement propose strategy glasso glasso write demonstrate help improve baseline screening glasso large parameter problem observe glasso probably associate strategy make quite attractive time thresholded connect microarray around obtain problem range micro observe vary thresholded split non connect size machine budget graphical micro heavy regularization solution analyze array process filter expression stanford patient body gene expression gene third gene signature breast cancer expression thresholded covariance arrive entry grid thresholded range value b color bar gradually decompose size become isolated differ great size appear appear cb ct cb cb encourage speed screen glasso appear glasso run independently choose emphasize screen c speedup partition glasso glasso range value leave column maximal average decrease thresholded negligible example size beyond scope glasso solution average glasso grid sort absolute diagonal thresholded glasso partition novel characterize solution lasso surprising pattern concentration thresholded seem insight solve large around thousand notational kkt order notational convenience already thresholde graph q represent block eq v kkt note thresholded satisfy kkt entry construction kkt condition block problem solve problem ij ij partition fine partition component proof direct establish estimate precision graph thresholded thresholded nest within connect component vertex thresholded graph inside component thresholded sample covariance conclude component definition department stanford stanford covariance regularization form decompose component component thresholded induce connect solve use graphical lasso proposal infeasible scalability synthetic life microarray comprise realization positive ie inverse know achieve sample variable use parametric criterion penalize develop efficient active field convex machine appear special property ignore paper surprising equivalence induce thresholded focus aforementioned observation screening insight behavior simple screening arguably challenge convex use sparsity separation connect introduce notation terminology entry denote also theory undirected order tuple set undirected edge symmetric matrix convention adjacency zero maximal connect v g relation graph k partition note label throughout refer component admit say vertex connect graph along relate algorithmic numerical conclude pattern edge skeleton eq v namely edge suppose imply one perform covariance admit construction operation perform arguably message connect obtain path solution understand screen within edge concentration graph numerical non monotonicity edge remark detect screen operate isolate covariance author show isolated solve paper concentration admit connect isolated treat connect unit learn also discover screen glasso improvement exist glasso glasso screen immediate block update glasso exploit glasso operate fashion zero iff pointed screen notable node screening glasso glasso check go implementation go optimize regularize fairly challenging proposal depend b subproblem form handle compute thresholded graph size graphical simulation observe
portion project call would piece ad hoc manner help especially adaptation specification use technique near neighbor computation retrieval contrast retrieval expansion expand module maximize eq permit mapping satisfying goal compute ic ic quick instance discovery retrieve currently compute possible graph combine match hypothesis structural consistency still event become computation score approach retrieval implementation cognitive science cite influential analogy computing provide mapping domain previously information give meaningful structural pick analogy sum score possible discovery concept label probable relation domain tag relation operate fall make mapping retrieved use match expand possibility stage technique retrieve would domain modify update retain newly retrieve case base instance cite even apparent argue limit analogous resource scheduling fact category cluster international resolution structural similarity international incorporate international desirable social international field effort international project maintain international research agreement third party together management aim classify approach lack knowledge basis contain detailed information enable fall arguably limited category technique future large fitness structural make suitable modification graph account ability support base feature highly advantageous law reference important practical life diverse domain among law resolution international management form situation component meaningful effort ai development base generation mention improve concentrate ai perspective lastly framework domain fellowship grant de next thank constructive reasoning une school mathematic international conference reasoning method reasoning reasoning allow satisfy requirement expect utilize domain transform issue utilize reasoning employ flexibility circumstance contrast specialized incorporate range international reach agreement fail field law e resort include type explore solution experience draw similar handle instead within field intelligence study develop together recent al theoretical computational mention tackle variety approach stem incorporate matching component recall past different extent innovation retrieval stage score structural new knowledge mapping retrieve component stage reason present idea illustrate discussion end conclusion solve ai utilize past experience view type make generalization derive domain model decision tree lack generalization particular inherent base keep decision valuable explanation support structure base describe form cycle module detail cognitive onto newly encounter similarity play role aspect cognitive capacity memory collection inherent nevertheless almost restrict indexing matching develop implementation capable retrieval domain especially adapt past target research international b dash edge reasoning correspondence concept pair domain base exist domain link dash correspond successfully address reasoning module expansion concept relation concept analogy possibly semantic relation base target want embed division goal allocation reasoning ability maintain regardless semantic fundamental capable create view retrieval successful adapt current utilize middle retrieve goal description implementation together flow agent newly acquire base solution holds successfully fully describe denote respectively associate goal solution even subgraph concept relation embed list indicate concept relation correspond indexing purpose possibility
entry clarity satisfie consist apply forecaster matrix round achieve round adversary choose different round imply trace rademacher thm round apply require trivial explicit bad rademacher variable recall rademacher shift replace convert outcome set shift one respect outcome apply rademacher author support ec fp online di universit di microsoft research usa online receive recent attractive lack informally speak prediction choose learner goal attain excess predictor precise standard batch convert method effective batch distribution instance needs reveal fix already compute online analogue instance need probably technique extend use prediction static expert forecaster minimax outcome expensive easily randomization case binary convex lipschitz random randomize round unseen way depend result trading regret polynomial online interestingly focus link concept forecaster use method subroutine forecaster determine comparison class measure statistical setting explore recently idea imply extend design learn trace constrain matrix straightforward application learning mirror match know setting analyze inferior rate pose whether forecaster rate wide loss efficient online thesis emphasize forecaster minimizer prohibitive whenever erm efficiently claim practical nevertheless seem tool online often work technique appear technique employ derive online context formally prediction play forecaster adversary outcome forecaster reveal forecaster predict almost expert irrespective outcome sequence horizon horizon advance simplify forecaster derive problem quantity one explicit tf ty erm scope outline optimal prediction work calculate round etc remarkably crucial rademacher probability vanish round explicit complicated recursive little class variable simply cumulative adversary forecaster predict receive suffer guarantee summarize static expert forecaster forecaster erm round variable ty randomized forecasting least statement statement statement expectation algorithmic easy infimum sequence oppose infimum different outcome minimax forecaster outcome limit applicability outcome trivial deal convex loss function outcome propose essentially forecaster subroutine carefully lie interval round randomized round forecaster describe version subgradient prediction forecaster present upper tp ti ty z z tp tp ty tr convex lipschitz condition drawing observation convexity use z tt martingale difference step relate z might whose influence method q hoeffde probability finally equal forecaster prediction round outcome thm rademacher straightforward rewrite simplify theorem empirical approximation reflect precision improve runtime trade note algorithm single forecaster constant know readily generic initially new actual equal enjoy regret moderate multiplicative factor union know game end horizon mild function forecaster suppose loss assume supremum easily find infimum contradiction suppose adversary adversary adversary aforementione compute round adversary regret round least hand end play prediction forecast prediction expert far permutation unchanged irrespective adversary consider sequence example round must provide reveal learner incur learner tp fx main class vc dimension exist empirical whose minimization imply learn hard boolean major pose rate learning batch thesis also non prediction vc dimension online forecaster achieve respect rademacher computationally risk minimization forecaster unlabele example reduce remark mapping function expert remark care forecaster thm use thm online dimension dimension vc turn present online collaborative regularization matrix well consider weighted stochastic observe pick free fail practice stationarity online distributional assumption require
md md mse top panel md sampler ratio md sampler md demonstrate effectiveness dag md sampler mse time md huge probability domain estimate domain representation bn moderately complicated local mode fast md investigate keep mode term comparable domain confirm indeed major burn period cell properly basis response flow construction towards various treatment disease use network pathway natural activation cause b network imply change reaction refer chemical physical modification exist dag protein flow flow throughput technique cell cell cell huge al protein network naive cd cell perturbation pathway intervention infer perturbation measurement discretize level low naive continuously extensive annotate among causal edge report flow md burn predict probability edge predict tp md predict annotate unnormalized dag md md algorithm table term mode md reflect high standard deviation md sampler jump show network true threshold notice mean threshold identical fact close size edge predict md sampler demonstrate md underlie compare advanced md md sd tp fp fp tp fp tp fp dr run result probability evaluate likelihood p g dr annotate predictive expected mean low domain literature superposition please give boltzmann superposition approximate function often physical respect sum domain employ md carlo sampling estimation mode superposition md sampler carlo deterministic method li carlo liu metropolis use find minimum improper space point mode promise md utilize recorded proposal md far enhance future construct sample md domain collection candidate direction current appendix appendix establish convergence md conduct doubly sampler adjust equip generate nonempty denote write measure j q notation target distribution accept scenario involve I e doubly adaptive doubly fix routine place norm ad working theory apply md remark update translation md establish main mathematical always mh jump either multivariate multinomial c ia jk c aa c imply liu routine decrease almost become deterministic drift projection give p irreducible condition imply verify component straightforward calculation verify global lyapunov u algebra I ensure interior c assumption immediate outline drift exist c assumption al essentially kronecker infinite desired convert proposition corollary routine distribution unconditional mean decompose space mode domain probability expectation informative unconditional expectation construct domain representation multimodal network throughput model landscape accuracy multimodal protein wang unknown observe posterior belong distribution chain monte carlo bayesian include metropolis hasting mh metropolis sampler review recent monte computation chen expectation approximate sample however unconditional expectation offer summary multimodal mean locate conventional multimodal major mode calculate statistic neighborhood achieve partition domain unimodal parsimonious minimize domain use rigorously later illustration partition trajectory domain various expectation domain summarie unconditional expectation dr practice sufficient efficient multimodal construct representation arbitrary multimodal sample utilize wang far generalize however accurate reason partition accord partitioning method complete nontrivial job detect mostly rely simple move coherent move dynamic partition sample domain devise utilize iteration enable fast transition feature multiple domain md md particularly problem field mainly concern structural causal network monte carlo zhang acyclic dag md construct landscape insight study protein cell highly pathway pathway constitute network critical disease recent advance allow measure collection rich statistical inference md build network discover pathway reveal distribution approach domain representation md ergodicity test example study human article conclude discussion relate future differential mild basic descent maximize index eq local start k set zero default article integrable write domain representation array dr provide expectation decompose contribution summary multimodal distribution every domain reason mode index define partition md k geometric posterior nf leibler regard may manifold potential index drive form domain representation necessary identify differentiable application gd local space structure naive quite carlo belong partition domain identify domain simple drawback efficiency design monte carlo version generate major mode inaccurate addition information overcome drawback md multimodal representation computational wish majority density neighboring often exponentially mh sampler across multiple allow region domain consideration partition md sampler mode include mode give density partition domain empty empty nonempty indicator dd immediate representation commonly strategy multimodal stay g substantial jump domain representation local move multimodal step size overall estimate respective domain design jump one one proposal md burn initialize mp lk regard visit decrease visit choice follow design employ sequence et al originally wang adaptive modify wang md initialize give c sampler visit since update default set normalize sum see covariance use mode mean accurately gd proposal proposal domain give work mixture multimodal efficient global well approximated identify mode proposal covariance single cause big domain covariance incorporate utilize guide summarize partition wang sampling gd complexity step move jump mode covariance domain move verification regularity furthermore iteration roughly converge acceptable iteration jump stay may divergence small md local mode update dynamically burn algorithm scheme construct representation real application evenly identify iteration mode routine new k tm tw tw routine record record mode replace old density adjust interval update routine help explore part kl kl tw j input note record update routine routine burn search density powerful finding mode md scale cover around low dimensional default propose mixed effect mode later representation liu however post burn carry convergence md document unnormalized please supplementary discussion section effectiveness md statistical estimation especially step example routine burn md algorithm update modify w iteration consequently work effectively q gd domain estimate highlight effect md sampler md basis evaluate mode positive combination five mode respectively mode permutation sign md independently iteration million burn numerical integration error mse estimate conditional mean layer expectation save cccc cccc mse md md md md md naive md show md especially domain sampler estimate partition contrary md domain improvement domain jump md md md default fold convergence md sampler without mixed jump slow reflect five fold iteration mix jump move md sampler joint construct code bn connect variable via joint area equation randomization model experimental intervention parent conversely fix intervention affect perturbation take state take state parent bn give parameterized infer type fix intervention infer causality intervention extensively context calculate mix observational intervention collection parent structure specify
give expectation forecast loss straightforward use law complicate rademacher iid rademacher still way iid introduction sample expectation expectation rademacher lead dependent quite rademacher differently online depth dependent follow introduce recursively start discuss q rademacher introduce z sequence path adversarial thorough treatment z z structure structure choose leave child order probability take selector become scenario past bound example aid three aid independence case simply note datum difference control hoeffding trying predict sure bind thus entirely result side independent imply essentially datum define everything iid hoeffding give decrease probability bad iid control predict future order handle rademacher complexity expectation need adversary unnecessary clean form datum rely application certainly case choose surely loss undesirable requirement rely infinite depend amount theorem department pa edu department mark edu generalization wherein outcome result iid concentration concentration behave version much machine prediction iid ergodicity may predict precisely control error sort behavior situation predictor obvious past observe ease remainder possible empirical risk result datum optimistic erm sample restrict model rather function class observe characterize include dependent derive give standard affect conclude idea future section stationarity flexibility throughout measurable measurable limit go field input notion stationarity variable stationarity measure complexity predictive model rademacher seem fit draw joint eq q independent everything expectation term inside supremum large empirical minimization fit outcome sample ergodic overall absolutely nothing predict past iid developing derive random behavior priori knowledge
requirement lack dimensional scenario still setting nlp coordinate motivate descent advantage give quite useful optimizer simplex pair density weak corollary strong take advantage euclidean geometry nonetheless term dominate rate high gaussian problem clear geometry compact smoothing remark corollary algorithm return fixed essentially corollary benefit satisfy condition corollary set achieve instantaneous dominate grow exploit assume oracle expectation output supremum similar hold set query bound corollary corollary dominant term issue query similarly strategy logarithmic application experimental illustrate result parallel computation imagine assume query imply consequently optimization error briefly simple master worker master maintain access uncorrelated smoothing master sample aggregation speedup stochastic exploit improvement convergence serial implementation centralize adaptation improvement second interest calculate proximal expensive update may beneficial several proximal concrete statistical semidefinite cone form include covariance completion therein problem metric belong desirable discover guarantee illustration involve cost reduce appropriate project simplex benefit randomize technique gradient cost stochastic sample update unit randomize stochastic experiment section describe experimental explore benefit study solve describe essential whether smoothing device ccc b number achieve use result b scale average algorithm use uniform dominant dominant invert small theory predict assess accuracy prediction robust identification denote experiment index dimension setting similar behavior decrease number allow roughly increase batch linearly decrease qualitative complexity c achieve measure learn sec mean induce pseudo consequently solve stochastic give choose pair uniformly give experimental optimality gap performance even require clear give predict improvement objective suggest indistinguishable achieve simple descent reasonable smoothing necessary point simple dual averaging answer though experiment suggest smoothing experiment iteration optimal query oracle averaging accelerate randomize average plot stepsize favorable indistinguishable complexity grow corollary proof corollary full lemma follow recognize smoothed fx close non apply rate approximately appropriately complete corollary relatively tight smoothing convolution lemma fy moreover lipschitz couple proof identical appendix lipschitz fx fx eq universal f vector sub lemma claim recall strongly remain assumption obtain specifically q proof auxiliary result show provide sigma assumption prove appendix constant ensure involve corollary smoothness add challenge essentially idea fx xt modification partially auxiliary sub collect ts gaussian consequently sub parameter recall jensen replacing complete linearize indicator adopt shorthand triple regularizer bind substitute simplify yield write tx z side combine inequality bind eq divide equality since consequence successively xx recall note f f x eq use z sub define step return establish later bind book yield claim involve invert bind prove zero recall see thus invert return intermediate claim replace x eq property smooth sufficiently x density already somewhat smoother differentiable finer f continuous notation proceed almost everywhere differentiable abuse inside integral expectation everywhere differentiable measure abuse probability end uniform lipschitz continuously tight iv tight factor satisfied provide smoothing respect b u lipschitz moreover continuous iii let f uniform ball b lipschitz lipschitz exist simultaneously lastly lipschitz respect use normal smoothing de consider function norm purpose carefully quantify norm continuously lipschitz continuous iv illustrate bound show distribution smooth f continuity begin lemma replace lead term factor also corollary build accord norm density symmetric attain shift huber loss turn inequality fx fx since use jensen estimate f du directly imply f symmetric calculation complete auxiliary lemma end let u volume eq see claim proposition theorem triangle follow dimensional trivially fix show minimum one index zero logarithm clear minimize derivative decrease increase increase decrease though distribute uniformly jensen fx jensen inequality early part consequence follow consider function early throughout give f verify f continuous apply lemma control order technique make inside invariant variable exploit combine early inequality constant consider f c symmetry imply dependent remain normal modify lemma convexity difficult constant z z q take arbitrary complete see follow equality symmetric inspection purpose book moment n know satisfy let exponential rely complete density completeness note side complete hand side hand integrate similarly multiply side grows follow gaussian result union chernoff van behavior hoeffding adapt lemma martingale establish sum realization quantity control martingale martingale value defines bound martingale since conditionally sub slightly define exponential zero variable equivalent characterization expansion variable whose square jensen recall take combine bind em j berkeley edu berkeley department electrical computer sciences department statistics california berkeley berkeley rates procedure gradient method smooth provide decentralized yield distribute procedure problem value close potentially throughout paper domain satisfy convex almost include regularizer procedure throughout mostly function mainly reason many evaluate usually throughout work access consequently focus stochastic address difficulty several researcher deterministic regularization method proximal method convex objective smoothing reference difficulty require impractical except algorithm solve subgradient et gradient stochastic stochastic relevance instead return unbiased exploit work rate oracle optimizer box update oracle subgradient optimizer issue iteration technique non smooth stochastic start note particular perturbation smooth lebesgue smoothed whenever measure analyze procedure smooth density main contribution issue theorem tail consequence iteration achieve domain constant implication statistical area discuss remainder organize achieve randomize several appendix outline smoothing approach main specify fx denote subdifferential shorthand notation ff fx bregman divergence value denote large modulus transpose draw motivate state convergence behavior descent though omit reader assume sequence mild guarantee instance strongly moreover generate update refer reader paper aspect subgradient obtain thereby result stochastically accelerate base improvement accelerate stochastically objective stochastically perturb iteration slowly decrease perturbation algorithm quantity rather oracle point query draw step stochastic compute throughout distribution procedure ensure scheme accelerate strong objective version series point query receive sampling evolve average scalar accelerate scheme vary smoothing scheme number iteration extra
site loo avoid know site tf site bind representative loo calculate base tf loo cv representative repeat loo sample repeat loo compare rank loo procedure obtain loo loo cv mark loo cv experiment mark expect rank second cv comparable first perform statistical rank evaluate collect loo cv compare include probability scoring centroid compare include nucleotide pair centroid non content loo framework rank consistent marked horizontal bar bad centroid fig tf perform tf centroid well significance observation test centroid outperform outperform outperform value cut fig relation median content method utilize low tf loo however similarity tf nucleotide incorporate bind rank tf rank tf among tf method except tf centroid combination measure optimal tf reasonable similarity tf achieve cv content relationship rank median datum centroid aside intercept median relationship scatter straight represent relationship median simple content view measure site tf bind site predict reveal perform fig centroid centroid content pair tf divide term three factor median ic length bind site comparison tf I tf median ic tf significantly great ic tf perform interpretation centroid perform centroid tf ic short bind site comparison centroid bar suggest tf centroid statistically column display centroid centroid report four improvement centroid always significant centroid identification centroid outperform centroid confirm hypothesis tf median ic employ site rr rr rr centroid centroid centroid centroid centroid centroid ic well whether use nucleotide nucleotide nucleotide tf support test value relationship denote centroid second fig content position display information content improve scatter site bind site respective scatter bind bind site respective centroid bind site bind separable non result fold improvement scatter plot bind site respective centroid identify scatter plot bind bind site centroid solid nucleotide abuse moment become nucleotide factor since therefore nucleotide pair score look first nucleotide pair transform variable pair find correspondence also implication vector obtain tf signature bind site centroid display method imply difference gave give average lie weight letter position similarly investigate example utilize centroid compare state art rely loo bind representative set loo single good search method measure tf method tf cv experiment nevertheless pair centroid tf tf low median identification centroid effective bind site couple conduct factor tf genome claim centroid site search database show matrix vice properly space bind aim extend method handle bind length seek sequence alignment consume seek similarity investigate support national foundation edu lee factor bind site identification decade utilize discovery bind site search understand role bind limited centroid propose benefit negative centroid scoring method perform method argue tool translation protein determine certain gene cell crucial tf considerable past decade bind double dna bind genome bind find gene locate bind site factor bind site identification discovery search may contain algorithm predict pattern discover discovery algorithm latter assume learn supervise know devoted evaluation search evaluation article therein search dna bind site tf bind tf bind site score bind site tf encode score letter count position observe position extend nucleotide pair information loo cross validation conduct totally bind improvement use representation mutual matrix scoring call background method bind site loo tf bind site superior example vast amount bind specificity site may site bind previous study role improve review negative benchmark positive benchmark rely recently wang fold vector require human artificial tf tf investigate novel extension centroid sequence employ incorporate centroid rely upon centroid performance assess cv experiment factor discuss method accurately bind bind advantage couple organize study novel leave cross establish conclude conduct datum first bind site tf set one contain bind tf genome release tf bind length crp ns crp introduce centroid similarity two sequence letter indicator set content position probability observe letter share account similarity indicator set content letter respectively apart embed dot vector convert gs illustrate l transform measure preserve sequence embed embed bold e bind tf centroid score l centroid bind site non tf centroid centroid bind centroid employ centroid bind site bind illustrate interpret bind measure similarity bind site length equals angle form illustration computation equivalent computation equivalent onto centroid centroid rise describe bind bind tf exceed constraint non stay flexibility threshold introduce distinguish non difference two equivalently minimize scoring background length pair distinguish take background nucleotide nucleotide observe separate sequence chain observe position transition background consider background markov show experiment section conduct loo cv section comparison step briefly loo cv tf region construct negative sequence bind comprise loo scoring negative test sequence forward reverse produce
evaluate however reproduce could forget model eq four exhibit dependency product model mass true carry large analyse importance dependency particle already naturally particle strongly concentrated want keep diversity system spread entire figure set explain sequential product logistic acceptance product rapidly half contrast logistic particle diversity product kernel decrease stage regression carlo algorithm acceptance rate odd rate step jump ahead however see marginal posterior easy reproduce distribution move decide fail ever logistic simple completely approach unfortunately incorporate monte alternative strategy fail provide suitable section parametric monte function interaction additive fact allow formulae ever author like much cite unfortunately impractical coefficient remove term eq exponential simple recursive factorization propose approximately compute approximation logistic fit logistic fit conditional binary family parameter option repeatedly literature first function unit moment adjust feasible replace drawback latent variable wise probability model carlo context potential random attempt binary vector construct precisely frank family sampling binary unfortunately multivariate non therefore yield sequential mention sequential monte carlo recent approach evolutionary difficulty modal thorough advanced need consideration high particularly modal build polynomial carlo present tend fail multi yield something nothing include logistic supplementary posterior distribution monotonic run smc particle sequential monte need evaluation problem markov distribution observe significant change update monte recommendation adapt generate package run ghz scientific source numerical evidence along publication source work provide project format result run window obtain marginal linear visualize variation plot monte varied run figure white box contain carlo box add bar smallest obtain otherwise indicate monte precisely run chain day monte extremely run data set algorithm major chain markov monte carlo standard chain adaptive chain outlier box start probability completely wrong outlier black chain confirm intuition make robust sampling constrain difficult key performance respective less running markov term computational chain set average complementary figure standard acceptance rate smc I I key complementary mc mcmc computational acceptance move concrete smc concrete strength set key indicator complementary mc mcmc min acceptance chain I concrete smc I concrete I concrete compressive strength restriction indicator complementary sequential adaptive min evaluation protein smc protein average complementary mc min min length move I smc mcmc effect complementary mc min acknowledgement n support thank two valuable acknowledge uci machine repository provide corollary definition example sequential sequential variable say past define present high binary practical motivation variable bayesian version raw version sequential monte easily vector satisfactory key provide carlo life instance sequential technique carlo algorithm adaptive sampling besides sequential carlo adaptive carlo aspect adaptive need parametric distribution property family performance quickly reasonable past space clearly practical analogue family binary space allow exhaustive enumeration whole order construction suitable discrete continuous counterpart major regression model encode linear explicitly constant approximate like expect value rich find monte approach variable view problem firstly grow global track initially spread robust chain carlo latter largely illustrate gain year multi central graphical computer regression motivate section briefly principal monte carlo commonly integrate ingredient algorithm family target construct rich core discuss completeness construct example variable problem fail reliable successfully write vector index sub model explain include drop linear choose conjugate prior bayesian q follow flat interval variance quickly eq normalization minimize criterion schwarz criterion basically coincide structure integration approach least absolute shrinkage optimization computation large compete essential bayesian model feasible alternative practical reason markov monte rapidly go carlo approach advance parallel combine elaborate move evolutionary ess thorough comparison scope metropolis hasting stationary distribution estimate expect markov chain discard give chain towards stationary stationary trajectory theorem apply chain work move metropolis hasting copy shall kind transition trajectory stationary modal ik iy parameter case chain iy refer acceptance likely interested chain explore mutation single drawing uniformly permutation kernel sampler proceed wise involve sequel gibb sequentially marginal correspond acceptance mutation copy state approximate predictor matrix ensure vector row trajectory update mutation gibbs largely computationally evaluation adaptive already produce adaptive proxy evaluate comparison propose current state component mutation probability modify move average gibbs component simultaneously suggest word value propose far mix property problem numerical benefit update sequential proposal dy markov chain kernel refer kernel current practically coincide obviously work choose close high acceptance average reliable shall sequential carlo general transition particle tailor work reader familiar sequential rather look correspond implementation ingredient sequential carlo smooth purely instrumental support smoothly theoretically initial pilot recommend simplicity reliability sequel associate convenient natural geometric alternatively sequence procedure sequence produce kk k nu weighting step reach instrumental idea control degeneracy move track particle degeneracy sample eq effective particle bridge size merely degeneracy length step effective value solve bi see procedure stable newton involve solution advance fix tuning parameter always speed sequential n draw nc n resample resample unweighte multiply size demand improve rao scope repeat weight resample rapidly particle reservoir reduce different particle totally inaccurate key decay move provide unweighted contain multiple copy particle idea monte carlo draw kernel measure particle change target almost move kernel particle within therefore locally review practically impossible hasting kernel proposal sufficiently acceptance particle convenient sequential come determine move system health particle distinct particle diversity quality analogue recommend keep system diversity uniform particle target reach beyond particle lot would identical change firstly aggregation multiply sample state aggregated size presence particle concentrate seem distribution find secondly keep sort way repeat move break particle copy move separately split aggregating might copy summarize complete method sequence index counter procedure n recommend store unnecessary latter systematically particle copy particle multivariate parametric need construct metropolis use want expensive binary moment want wise entry already generate rapidly evaluate metropolis hasting ratio metropolis satisfactory rate explain stem multi contingency table interaction theory completeness brief binary model reason sequential provide family meet requirement difficult important part paragraph mention unfortunately closed fit conditional suggest regression conditional valid regression
learn continuous space say acceptable minimizer important property pairwise element every acceptable lying span lie equivalent ensure minimal notice function nc xx nx direction q linear theorem suffice interpolation minimizer show k form particular cardinality finitely minimal interpolation f x g f ingredient preserve topology compact metric universal input admissible introduction requirement admissible construction requirement easy present requirement demand kernel arise brownian bridge bridge admissible space direct xx tx nk n requirement requirement function suppose hence clearly function side equal take side equation distributional sense infinitely continuously lebesgue measure nonzero sufficiently rate appropriately constant linear relaxed loss regularization satisfy characterization rise admissible x may x k allow equal vector singleton conversely satisfied shall discuss paragraph x n x completes regularize xt constant kernel exactly demand lebesgue bound regularize infinity specifically prove lebesgue constant therein invariant kernel show lebesgue constant commonly satisfying include radial mat ern compactly support kernel shall future numerical experiment sparse learn exponential rkh restrict number loss shall compare model eq numerous solve employ close minimizer equally q compare measure number c sparsity max three type gaussian random noise run mm section thm thm thm thm il usa mail edu part national grant dms banach space property possess norm banach integrable measure continuous linear bilinear example construction reproduce banach pursuit brownian bridge know sparsity extract dimensional datum live apply compressive sensing refer pursuit purpose establish research rkhs many reason account success rkh hilbert evaluation functional usually model therefore space hilbert rise functional rkhs kernel theorem rkhs determine many one evaluate theorem find point desirable force regularization rkhs unknown result banach go case aim technique functional represent kernel possess continuous point semi inner bilinear product natural substitute banach reproduce via semi space fr banach space guarantee inner represent functional shall approach point functional bilinear briefly paper let prescribe start hermitian construct admissible construction set banach respect note every sequence namely constant kx prove section admissible banach evaluation functional bilinear scheme nonnegative continuous q coefficient conversely organization construction banach reproduce kernel describe study brownian kernel final reproduce start banach necessarily definite norm functional explicitly construction uniformly fr differentiable banach fr impose existence reproduce kernel functional accommodate alternative point evaluation functional vanish everywhere banach evaluation plan check abstract completion consist bound functional yield let sequence functional denote cauchy give let pre need invoke space function every cauchy nx x make pre necessity norm validity consistency define sequence f follow f banach banach dense moreover immediately banach limit side kernel end bilinear well define reproduce bilinear whole kernel functional functional begin suppose functional k fx functional take evaluation functional necessity suppose let functional observation satisfy consistency discussion endow extend bilinear tell evaluation functional satisfy property hand side need direction arbitrary g q note sufficient reproduce functional form banach bilinear kernel banach reproduce norm bilinear thus reproduce banach proposition convergent consequence combine dual clear bilinear form necessary property mapping proper nontrivial necessity proper closed subspace kernel proper closed subspace enable find ff f present x property hold complete evaluation functional consistency complete reproduce via bilinear embed subspace nontrivial follow bound imply kx finitely many clear functional check find imply assumption satisfie necessity cauchy sequence hand suppose cauchy sequence nx n finitely definition suppose direct f nx complete satisfying later hold remark banach programming observe say remain check norm consistency shall consistency requirement calculate function supremum hold kx j cx xx xx g x satisfie cauchy nx go infinity eq imply n proposition conclude bound satisfie inner integrable lebesgue sufficient almost everywhere lebesgue point reformulate q lebesgue continuous transform discrete measure imply nontrivial regard wiener therefore fourier transform nonzero everywhere except lebesgue measure argument kernel standard norm kernel almost everywhere page b kernel spline give radial basis wu function compactly support satisfy corollary fourier compactly support indicate compactly lose nontrivial infinitely continuously vanish expand proposition construction dc proposition q yield procedure compactly borel measure satisfy radial function dirac delta assumption regularize construct requirement useful consideration give rise regularize scheme say unique possible acceptable regularize lie dimensional x kx convex fr reproduce kernel semi inner rkhs rkhs purpose interpolation prove follow minimal namely bf obtain
precision error small parameter tend formula consistently whether accurately say numerically absolute scale logarithm scaling stay decrease picture error compute although absolute tend consistently rate logarithmic application good tail tend rapidly uniformly good bound increase look support approximation formula median quantile zero increase varies rewrite fast approximation median introduce relative shape increase keyword median readily formula closed relationship exact median incomplete inverse review ha median value show scale median beta form satisfy symmetry relative median variate ratio unit look median median denote
ns ok issue seed occur submatrix seed seed without represent assume rank complete observation column assume detect candidate secondly still complete successfully draw seed seed redundant subspace completion column subspace candidate lemma determine assumption single subspace span span subspace otherwise neighborhood set column seed near contain certain span union small candidate true contain observation subspace describe sort small large subspace candidate contain sequential strategy far determine span proceed order list identify subspace subspace column span subspace ease span matrix determine subspace determine correct subspace subspace incomplete close bind matrix completion surprising require recover internet connectivity importance load poorly internet recent send passive internet internet resource measurement internet passive internet ip address observe link locate count incomplete fill infer characteristic analyze subspace structure rank internet exist behind probe send particular count address offset relate distance ip rank completion topology measurement count passive locate randomly subspace observe completion procedure experiment improvement miss completion approximately element completion h count result synthetic network cumulative show observe element internet delay ip address imputation significant performance matrix completion imputation ip address university university cs university complete miss entry multiple subspace rank completion union miss version lie union assume mild assumption perfectly incomplete usual incoherence subspace numerical experiment internet topology identification consider assume subspace interested situation total propose novel matrix completion mind arbitrarily focus quantify perfectly complete column complete translate extremely suppose entry mild perfectly probability computationally procedure depend incomplete entry location exceed also rank determine consider matrix subspace yield theory constant completion bind per column significantly conventional matrix rather must subspace subspace challenge provably subspace algorithm column highly dimensional application internet matrix identification device record computer monitor internet determine network computer point traffic portion internet use burden place network disadvantage monitor normal control observed pose completion incomplete distance potentially rank computer cluster access internet limit submatrix computer distance lie distance idea completion one ingredient completion particularly require element reconstruct build assume subspace differ concentration incomplete near paper involve intuitive step outline detail section column call seed neighbor reliably even portion subspace span seed completion agree seed discard seed neighborhood enough accomplish show complete observation find correct use detection complete section neighborhood subspace illustration sample subspace depict small dot seed observe seed neighbor seed near seed overlap incoherence play coherence main make subspace coherence coherence pair intersection distance away belong ball draw column subspace key seed formation algorithm suppose column perfectly methodology apart total sufficiently need fix definition end perfect factor individual low final completion perhaps choose per column detail methodology four outline entry uniformly replacement without replacement assume relation note result unique let size random pe pe pe pm pe pm pe column select random nearby neighborhood design probability assume seed great one seed chernoff select belong contain column yield note seed accomplish seed belong column must column bit challenge determine seed show seed index reliably column common denote probability sum replacement bind assumption replacement note variable change equivalently yield observe let proceed seed column observation seed precise moment uniformly choose ensure seed need uniformly seed return later requirement seed column ball seed pt seeds probability ball expect chernoff bind r seed requirement procedure neighborhood observe entry seed per input discard seed find seed form column seed produce least seed seed lemma column uniformly seed probability seed within seed entry observation seed certainly observation index seed least within show must determine low common theorem assume since seed entry imply seed chernoff bind suffice guarantee enough take index random index subset n following size
fourier sparsity w cp variable selection journal american selection average statistic average manual research american li penalize institute nc mail university nc mail edu department national mail com proposition pc pc pc pt pc double shrinkage minus inference jeffreys prior spike tail behavior straightforward role result prior word phrase dimensional relevance robust prior plus great work selection rich analyze fu fu fan lin yu li article popular coefficient normal ii develop relevance sparsity come drive student fact drive jeffreys hierarchy jeffreys prior prior variance propose maximization maximum laplace jeffreys jeffreys lead substantially improve shrink coefficient shrink improper shrink light tail laplace property yet analytic design zero heavy tail jeffreys carefully mixture spike shrinkage prior analytic alternative shrinkage facilitate straightforward result joint good job quantify uncertainty propose double pareto mention analytic possesse appeal student characterization mixture normal straightforward sampler inference prior penalty regularization rule continuity estimation motivated application genome wide association study lee prior include double pareto limited contribution beyond I formal introduction thresholded ii pareto central work jeffreys limit hyper along incorporation sampler vi detailed analysis double pareto thresholding rule analytic maximum posteriori posteriori case step connection generalize generalize pareto contrast parametrize generalize pareto reflect assume symmetric pareto cauchy tail compare cauchy standard pareto zero double pareto density suggest property posteriori cauchy like tail illustrate different represent lead shorthand proposition reveal analytic form bridge two laplace jeffreys setting imply place jeffreys yield normal jeffreys item dirac delta note tail size understand observed level proposition tail grow variance hence cause though strong increase around increase converge density shrink typical specification like tail desirable robustness motivate default shrinkage factor et shrinkage towards generalize pareto upon proposition double pareto complementary prior behave behavior strength small tends unbounded suggest signal standard prior form prior adjust value prior function jeffreys observe conjunction unbounded effect well clarity increase shrinkage place posteriori hyper parameter trade robustness jeffreys variance normal augmentation sampler gaussian mix absence prior datum pareto hyper prior location scale shape median choice exist transformation suggest pareto posterior embed form interval calculate normalize set obtain fixing treat unknown determined establish tie frequentist approach analytic generalized pareto analysis hierarchical straight posteriori estimation double pareto induce imply follow fan li denote minimization fan li unbiased thresholding coefficient instability penalty first term estimation appeal desirable desirable rapidly imply shrinkage coefficient get fact convergence jeffreys prior control pareto tail bias regardless fan li thresholding minimum imply thresholde p derivative negative two root absolute elaborate root j fan estimator unstable change result signal minor hard thresholding unstable stable fan li minimum ensuring create tail robustness region wide yet nearly hard thresholding double pareto posteriori likelihood formulate expectation maximization expectation respect kk let refer estimator exploit laplace integration prior mix laplace require expectation ease c respect step li via representation mode li resemble algorithm use refer path form bridge reveal use representation I laplace scale representation illustrate iteration mixture representation mix integrate li estimator possess oracle normality selection generally hold n deferred estimator prior average fan li denote pareto prior hyper parameter provide table prior obtain hyper j j calculation fan package default package ridge scale default place parameter shape package default package employ et tuning lin li cross validation report median model calculate median calculate standard model three show great flexibility adapt signal computationally use sampler procedure user let may dense constitute similar add thresholding ability although take value adjust method laplace inference hyper take inference hyper restrictive posterior posterior error higher move particularly prior quite room small accommodate dense sparse I I median retain estimate hence model yield well somewhat worse design obtain improve attribute advance good treat hyper appeal flexibility sufficiently error account uncertainty appeal contain model predictive box acceptable practitioner interpretable mixture normal prior pareto thresholded prior inference feasible provide uncertainty estimation prior allow frequentist posteriori argue model mix hyper
hundred thousand knot automatically canonical knot knot sample another fix knot low adaptation value one fix knot make infeasible run markov sampler loop finite cholesky triangular satisfy I give construct recursion remain time first triangular negative cholesky approximation knot result process ii therefore incomplete factorization available knot burden identify training variance need case stay restriction employ process stop acceptable check tolerance tolerance increment repeat produce update change increment subsequent clearly reduction expect proceed new th row give approximation finding lead package cholesky produce permutation triangular permutation cholesky result cholesky covariance process knot comes replace computing fraction compute need element incomplete cholesky factorization user tolerance absolute tolerance tolerance kk mr lk lk kk nr lm illustrate forest contain axis vertical axis reference surface forest covariance function xt orthogonal relate landscape projection location encode range decay knot knot need meet nine vary hold top leave middle project right column project axis indicate pick knot spatial small give canonical metric surface directional topology knot along horizontal horizontal knot axis project show nine vary take shape rate landscape fix record slope value location top row row large make close narrow south west variation slope feature pick knot figure choice tolerance meet tight coverage entire maintain throughout knot nature try pick bias greedy offer highly attractive knot indicate meet tolerance grow yet available relationship knot extremely bayesian markov computation covariance sparse exploration space fit show efficiently adapt change cause change regard adaptive clear advantage handle knot datum independently eq exposition assign normal independence across knot clearly infeasible place knot axis count place knot learn drastically reduce likely poor ff gibbs knot suppose knot find knot density knot would look figure strongly knot b line interval dash scatter along th gets adapt knot value restrict search knot covariate tolerance likelihood py py pd predictive adapt approximate likelihood compute knot formula posterior py explore sampler figure summary metropolis sampler exploration level united relate education autoregressive relation respectively population high education number log response three relate spatially regression x ii vector formulation zero tx jt ji normal priori fit use count exploration py pp distribution unimodal knot two country unlike substantial meet bind summary fit bottom predict combine figure scatter fitting dot remain one red dot predictor find relate home spatially vary influence positive relate education moderate effect relate weak absolute predictor interpret ordinary include equal methodology substantially conceptual contribution emphasis approximation infinite priori provide approximate posterior distribution theoretical knot fundamentally deterministic knot drive stay limit close process approach model knot stochastic well task discuss process smoothness employ autoregressive lattice approximate case need approximation yield smooth pose additional irrespective ill computation solve algorithm ill norm bound z packing easy knot define calculate integral assumption cb bind side sd st knots predictive cubic complexity process crucially knot retain approximated knot domain present calculation coverage process place knot change change control covariance present algorithm toward cholesky concept already lie implementation predictive result offer substantial fitting keyword gaussian knot cholesky sampling nonparametric gaussian ability incorporate smoothness spatio computer quantile etc thorough overview theoretical bayesian therein view value finitely many characterize involve possibly conceptually straightforward
continuously divergence take quantile simple limit relevance leave open keep good provide leave problem divergence open investigation argument argument remain establish arbitrarily small latter satisfy second show q exist term q hence p tend pt theorem bayesian estimation th universit paris paris com technique estimate establish divergence markov particularly attractive increasingly attractive handle difficult divergence use bayesian concerned measure contaminate outlier difficulty encounter dual posteriori major require estimation smoothing produce estimate contrast empirical purpose estimator reason commonly mcmc outline property divergence discuss estimate section law remark dual divergence general recall respect well divergence divergence chapter divergence problem underlie attention strictly satisfy class define satisfy represent elegant prove divergence connection convex condition represent supremum replace measure q class indexed specify instrumental consider sample ratio robust robust divergence dual censor introduce test copula performance estimator turn function estimator integral type equivalent generate condition access asymptotic use criterion risk incur form density integrate rest bayes expect form loss choose loss square dual estimator integral exist familiar mode divergence define divergence dual divergence divergence modify keep get upon section posterior evaluate use mild satisfied circumstance derivative integrable derivative integrable nonnegative convexity open neighborhood taylor expansion tend usual integration regularity require remainder
surface meta often impossible essence technique aim robust regard though demanding implement reader book develop base hereafter apply involve model world speak oppose physical consideration coherence order retrieve space attempt indicator function problem meta able interest spread lack observation two paper krige extended svm krige interest base essence krige path would model stationary read class autocorrelation unbiased variate stage set observation group solve analytically assume thus solve likelihood solve turn give surrogate reliability consist meta meta second taylor surface surface may accurate quantify substitution failure probabilistic krige instead since express close read follow note quantity deterministic propose probabilistic concept basic limit represent krige meta build represent meta triangle misclassifie fail also red blue fail safe deterministic decision classification smooth subsection propose refine indicator mean confidence surface area blue line margin safe point margin uncertainty due nature prediction read maximize bring statement many author literature decide use global algorithm name equivalent propose add point regard normalize constant either choose pdf difficult usually structural original reader refer mapping pdf indicator chain monte simulation generate sample maxima predict uncertain property center span function propose real indicator failure rewrite definition argue latter equal sum random hereafter g approach probabilistic classification function conjunction technique efficient technique indicator bias failure interest instrumental dominate failure rewrite subscript recall mean read central limit estimation read variance estimator instrumental pdf instrumental pdf many allow reduce variance strategy propose instrumental pdf specific use normal order failure probability may krige build instrumental suit extreme variate indicator optimal instrumental propose quasi pdf read quasi instrumental pdf instrumental follow failure augment ratio real function krige fully correction unity augment propose krige failure account monte first quasi optimal instrumental limit normally might propose monte normalize instrumental interest present use slice simply read calculation coefficient final thick north east cycle thick north cycle reliability inspire concern reliability mm diameter equal mm end stress suppose elastic behavior due boundary poisson though modulus model variation assume square autocorrelation field strategy propose independent group simulate order retrieve von stress meta base importance reliability krige predictor build generate section new refinement procedure stop estimation may provide stop refinement procedure function respect krige estimator failure table compare form estimator implementation matlab
entirely consequence place functional shape highly undesirable limited available crucially select parameterization lead shape underlie uninformative viewpoint approach interpretable distribution context mixture transfer distribution precision assign hierarchical variance necessary space shape however entirely methodology construct metric space regression simulation would compact p ideally define reasonable closeness seem adequate distance parameter euclidean plain impose notion metric construction describe packing application jeffrey prior adapt situation notion need distribution distribution limit discrete spaced notion equally underlie constructive apart lattice calculate unclear whether notion packing packing metric lattice packing pseudo minimum distance aa packing space constructive explicitly practically finite local approximated root imposing hold early distribution obtain consideration riemannian manifold case riemannian consider riemannian manifold parametrization impose parametrization special case taylor appendix calculating transform twice transformation obtain technical metric space limit sequence grow integrable develop general like uniform continuously function back obtain subset transform td apply end desire jeffreys approximate fisher situation residual lead density interpretation jeffreys rule rare among jeffreys rule principle viewpoint jeffreys universal prior weight density statistical model depend covariate undesirable application directly numerically space easily triangular simple yet jeffreys density parametrize hellinger px obtains observe uniform hellinger reference uniform kolmogorov result nonlinear shape reality usually unclear prior suit metric compact vector derivative result functional equal fx improper extend exponential example calculate square normalizing base figure mass desire advantage uniform parameterization particularly choice shape cover small shape extend uniform choice variety choice could fact jeffreys case situation jeffreys surface weight measure obstacle functional fact analytically integration however need observe modelling situation propose nonlinear trial trial variability large often prior test assess formally regression take range compound treatment four adjust score total complete balanced allocation assume response relationship maximum effect give percent clinical exist illustration improper uniform prior necessary improper bound boundary cover account one extend density perform apply introduction induce shape large almost shape shape induce shape shape iterate implement see observe distribution bias linear happen particularly shape fit might expect functional acceptable investigate detail prior study conduct report use hence subject per use scenario power behaviour misspecification scenario compare jeffreys nonlinear uniform prior prior jeffreys proportional analysis tune suit simulation mcmc case mae cp jeffreys power power posterior median mae mae maximum display repetition interval interval simulation length simulation run jeffreys functional improve uniform jeffreys slight jeffreys interval jeffreys keep level linear uniform achieve power interestingly large estimation estimation message parameter jeffreys roughly equally jeffreys simulation conceptual covariate design sequential situation cccc cccc mae mae mae cp model design lead design optimize prior restrict space optimization design functional design whereas essentially look efficiency opt design shape plot shape functional scale observe shape uniform shape efficiency shape motivation practical jeffreys prior want calculate find trial distribution uniform underlie function achieve early simulation reason uninformative metric distribution prior uninformative particular aspect reflect nonlinear argue functional shape depend consider assumption regression situation adequate occur extremely often adequate reasonable rather build uniform potential shape might priori theory
response around neither series causal effect enable direct strength remove resample depend autoregressive resampling effect equation significance g however cutoff remove effect absolute quantile instantaneous acyclic figure graph direct edge orientation dag edge member undirecte direction influence figure price seem connect two price instantaneous difference direction edge go price information flow keep reservoir wind already causal small remove effect lag small cyclic dag equilibrium find contrary generally important market price expect market generation grid price local pay price partly price volatility price naturally peak demand influence affect market partly price play role european price game view advantage previous identify might properly instantaneous forecast decomposition impulse instantaneous dag combine implicit error residual common create see scope deal could include investigate price fine price work statistic innovation innovation particular grateful helpful comment help methodology dynamic major generation series advance answer reservoir price price adjust price vector correction market causal direct acyclic non data market market use advance causal model relationship market govern line different market similar market integrate market dynamic flow major peak price price among european market major full european market market pool dominate pool price market hand dominate production complement assume price causal price dynamic water reservoir level production treat logarithmic direct acyclic dag instantaneous dag joint dag indistinguishable influence edge dag class direction paper acyclic recently allow identify dag able causal relationship dag identify approach direct influence var describe instantaneous causal effect random depend basic previous coefficient integration stationary case var possibility correction derive difference correction long relationship vector relationship series make sense equal inconsistent compare general causal center cause early corresponding estimate dag observational receive infer dag observational greedy implement software fit equivalence reach dag distinguish alone equivalence assume term triangular imply unobserved sense causal let rewrite ica find statistically limit pose use entropy vector density h entropy non efficiently see estimate constant scaling find show estimate available difference var instantaneous effect correspond strict contain autoregressive correspond cyclic var representation coefficient residual find time flow causality causality reduce prediction combine significantly non cause thorough market physical partly explain wind production water level ideally far back wind secondly six time quite rapidly integrate european market market available could include transform common induce exchange price fluctuation influence include exchange series use week try accumulate likewise accumulate consider overview give table price represent united market market expect important formation close market treat wind reservoir influence wind term wind water water ideally price correct log reservoir logit transform price subtract square ht l description pool average price price price exchange price national balancing price uk price exchange average water production wind production htb htb use schwarz lag var even stationarity series perform lag ht rr rr difference also series series reject significance reject level test trace schwarz determining indicate significance schwarz criterion indicate inside schwarz constant within outside lag var repeat lag var outside conclude critical r schwarz combination possibly component combination series stationary since test suggest several reject indicate consist value fit test test normality univariate
markov maker directly access relate state minimize problem optimal find mapping space various controller stochastic quadratic programming complexity know computational target achieve controller controller period class action np guess controller incur linear np even easier observable version extend deterministic allow controller article show hard long controller choose action recent observation np probability action considerably effective deterministic generalization blind article np controller controller unobserve mdp deterministic blind controller action regardless history straightforward evaluate deterministic blind controller action one chain deterministic blind trivially check action good controller action article allow construct np discount factor horizon mdp states cost probability action mdp pair bellman map consider constrain allow stochastic blind distribution simplex mdp blind state relate controller encode cubic graph cubic three cubic column construct mdp matrix action indexing state transition quadratic graph cubic maximum satisfie contain system tight attempt list counting recognize hard np resolve computer argue similar reduce let construct action self start state input state action transition bellman read rewrite jensen inequality note cost reach minimum tight achieve large application jensen establish blind clearly whether reduction case controller trivially correspond transition doubly cost depend proportional scaling arbitrary bilinear real eigenvalue definite see also complement definite simplex corner controller deterministic take operation policy researcher turn tractable show membership np resolve np assumption recently address case publish stochastic relate address grateful point author
aspect present cf algorithm prevent feature perturbation derive form mis rate mixture cluster membership stand variable matrix specifically decision let loss define take measure define distance cluster seek population call strong feature exclude interpretation include cf noise proof calculate expression equivalence recall similarity spectral compute laplacian diagonal degree notion similarity way decomposition cluster form accord eigenvector e eigenvalue partition criterion meet analyze mis cluster rate cluster affinity random cluster generality mis affinity produce block diagonal four mostly perturbation matrix insight cf mis e proportion assign wrong cluster characterize perturbation model main analytic expression perturbation let operator express contour exclude interest mis formula simplify mis derivative hand side unimodal minimized clustering size cluster unbalanced finding play mis analysis cutoff base second shift correspond mis nature perturbation affinity mis cluster two synthetic design several several metric subsection subsection simulation feature capability assume generate observation generate I test include clustering criterion growth exclude clustering rank accord contain five scale eps include plot eps plot first noisy feature identity next generate finally noise useful coordinate occurrence note plot produce competition total accuracy conduct segmentation heart breast cancer robot execution failure lp robot provide feature eight partially satisfactory ccc heart instance cf metric respectively indicator set label idea performance assess cluster different one metric favor observe rp rp calculate cluster bc rp accumulation implementation adopt little try cluster agglomerative rp linkage match throughout run package unless stand base scaling difference feature time grow vector possible empirically particularly rp target conduct proceeding suggest sometimes result replace linkage suggest bc robot table report take produce five illustrate figure competition feature competition chance cluster instance boost table table feature competition h cccc dataset rp angle eps eps histogram strength feature plot eight dataset inspection present choose good thus could feature could measure discuss leave possibility work robot row heart robot row explore vary gain case cf base cluster pre grouping neighbor however statement extensive future empirically compare rp cf base compare angle eps angle heart eps angle eps eps angle eps scale eps explore compare cf cluster eight robust run cluster project automatic search different cluster metric table cf almost outperform eight cccc mean heart robot propose new ensemble include bc accumulation rp cf base cluster boost level provide support formula mis spectral particular spectral devoted deal population cluster e explain competitive certain completeness rule sense figure mixture shift invariance invariance rotation transformation preserve geometry optimal determine eps eps overlap component gaussian mixture star population mean cluster underlie calculate ss ss assume feature exclude two spherical affect addition feature simplify lemma eigenvalue eigenvector eigenvalue curve eigenvalue eigenvector mis dt ii ti ti ii ti dt pn pn let column one verify pn pn conclusion pn part direct rely law omit matrix proof omit forest rf context forest randomly cloud clustering via dataset local clustering cluster cf resemble world cf measure possible desirable close mis aspect cluster dimension include dimension reduction dimensional beyond full subset separate choice attribute projection reveal probe space detect good aggregate separate membership combine many different component direction involve recover projection potentially useful tend huge feasible conduct whole exhaustive randomly probe projection explore compressive however forest rf classification rf improve root start collection variable root become strong strong split cluster perform control achieve eventually produce pursuit therein methodology achieve make recent year treating define notably spectral vector form competition heuristic plot grow cluster corresponding clustering regularize matrix regularization thresholding level less nonlinear grow base vector partition construct
eq follow contradict strong optimal action frequently contiguous construct construction play sequence use identical eq playing subsequent time step let environment arbitrary discount environment r asymptotically later weakly take action function previous maximum sequence algebra contradict exist explore consecutive infinitely often choose step time definition algebra asymptotically infinitely function weak optimal policy exist ucb bandit know explore detect choose environment probability accord explore history consistent policy countable class deterministic environment pz ti kk agent optimal definition weakly deterministic computable environment remark necessity easily generalise arbitrary technical difficulty construct policy l deterministic environment extend introduction see go computable reason rely second operation environment asymptotically computable access number exploration construct environment use require difference environment history different exist satisfy make inconsistent history difference n definition ii pe infinitely ie tp finitely probability triple play policy follow reward recall environment approximation ready environment tt explore first environment inconsistent inconsistent claim bound monotone history optimal asymptotically suppose defined probability start phase explore h I k h lemma history accord inconsistent contradiction indeed policy least recall independently h k therefore close policy environment lemma equation theorem difficulty discount satisfy exploration ensure policy asymptotically discount insight eventually environment change discover weak policy limit turn part asymptotically optimal computable computable policy weakly unlikely even weak asymptotically discount theorem policy discount trivial surprising discount often contiguous dependent discount theorem problematic intelligence construct asymptotically problematic seem reasonable counter able asymptotically optimal assume would counter environment world environment result complexity arm exist weak asymptotically example stochastically environment recursively computable already satisfy markov process mathematically behaviour behaviour usually result formal intelligence satisfying accept even optimality strong something weak order complexity quite similar policy whereas show eventually environment believe asymptotically strong rather actually optimality self possible version n existence self weakly weak self either discount function suggest restrict environment satisfy theorem lose present challenge discount ensure countable section countable reasonable analysis computable environment thesis computable stochastic environment computable discount discount kolmogorov nice question discount prove stochastically discount environment modify policy class stochastically believe possible complex extend part discount admit policy environment weakly augment intelligence valuable feedback early research cm lemma research school national ex artificial intelligence aim agent capable interesting two optimality agent exist asymptotically intelligence reinforcement artificial intelligence knowledge eventually learn mean play car consider existence precise optimality environment unknown necessarily explore two agent play optimally weakly optimally difference asymptotically optimal eventually explore optimal agent decrease asymptotically deterministic computable environment deterministic sake already sufficiently thesis hypothesis compute physics universe stochastically computable large environment universal agent weakly computable environment prove agent weakly class behind proof eventually stop somewhat surprising assume principled way passive extend bayesian computable weak optimal agent class computable environment computable reasonable discount agent exploration component learn ucb bandit explore sufficiently adequate environment exploit balance depend agent discount enough environment surprising exploration understand case decision process various satisfactory ergodicity else satisfactory similar environment return finite alphabet string alphabet set denote string length define condition agent asymptotic interesting discount sequence reward start I limit discount computable computing discount function horizon computable note represent effective horizon agent stand optimal policy tv x guarantee appropriately geometrically strong q weak strong asymptotically obtain environment policy asymptotically bad mistake must fraction optimality imply weak optimality appear somewhat vanish serious infinite believe environment would serious error would notion optimality still discount version
aim agent instead learner observe whether within multiplicative nearly tight gap bound furthermore reveal target number learnable finally tree leave learnable establish bind much construction small class provide approximate class classical class learnable factor start information theoretic showing learn approximation polynomial set pairwise intersection size via construct probability bound union follow large ia ia j nf nf polynomial sized mass great learn claim factor ss know equivalent fs recover optimize direction involves tailor formally therefore quantity lp indicate dual exceed coefficient equal optimal proof fact pf ft correctness structural sketch briefly appendix idea claim linearly try learn technique linear must ensure achieve relate first flip fair coin I coin tail mention find example come correctly function key fs sf non empty chance nan sample see dimension imply reason f value outcome fair nz fs polynomial number complexity easy learnable principle interestingly show within learnable example simplicity structural root leaf ji maximum value power value ls follow tree hand side immediately subset item formally set multinomial expansion tree add parameter prove correctness theorem fed lf least set ms f desire approximate appeal function submodular tree function learn extent interesting namely tree factor location stand small good function tree learnable small non demand hypothesis construct set prove approximate function inequality definition imply desire finish ff demand clearly target result sequence example section start arise item example present plane room requirement leave example company component large million company type company leave tree tree guarantee learnable large take learnable example learnable n proof represent leave leave class learnable use argument unit demand training see unit demand tuple constant pac learnable show representation approximate linear within ground define satisfie demand fix item map ks last reasoning consider fundamental price set item price customer price demand item price price collection prefer gs e gs property prefer old price old price early apply quite economic point query even tool gs characterization marginal fs sf gs maximizer among f f sf sa sf f sa sc sc f previously via concept concavity query relevant set able polynomially target value polynomially output fs fs et submodular restrict lower interesting consider factor gs learnable query learnable query leave tree tree submodular query family uniformly leave value analogous fourth nf concern cost apply submodular low al learnable family simple introduce paradigm many application learner random query framework learner learner price set confidence set value price agent demand price price model variant offer learner respective clearly bound still price reduction learning approximate theorem learnable factor increase convenience family power approximate ss reduction function long set price price always assign price way many induced hypothesis high target specifically l lf l f l notice decision q lx l always q nz lf learn linear hypothesis output size specifically bundle handle separately draw price mistake pn occur thus predict equivalently pn yield fraction pricing algorithm probability recover everywhere bound sequentially price item item depend learnable price tree leave tree leave commonly economic quantitative encoding concern approximate upper factor bind algorithm representation size time highlight importance new submodular distributional leverage structural provide new al query introduce realistic economic decision price observe directly upper continue despite receive less david discussion support part grant grant fa microsoft research fellowship correctness provide allow separately analyze zero subset convenience idea reduce standard binary passive passive unknown u sample flip use linear u moreover think draw belong construct classification linear program find fs correctness feasible early use fact remain show approximate point easy fs fs f finish fact k chernoff give argument inconsistent probability hypothesis approximate prove pac I pac learnable example show solve I reasoning demand sample eq fs training mean generate output one tuple function tree train example uniquely set formally q closely relate item demand item meta pac pac meta guarantee example rank learn everywhere leave warm simple ease fs fs tree leaf every element reader different let completeness tree get reader element leave imply leave fs fs r ts rt fs rt conjecture sketch wang core item rather often complex relate consider hierarchy submodular al realistic setting learn focus submodular approximate level distributional pac style nearly important show complexity representation polynomial time establish hardness class prove distributional setting novel everywhere et finally realistic economic bundle provide bound distributional understand customer assign price bundle product carry company understand preference economic model function know company distribution give customer estimate customer pay package become survey customer price internet particular class bundle henceforth terminology class submodular class hierarchy analyze natural distributional learn polynomially goal q use pac predict high view approximation represent e g g alternative essentially depth max sum leave expressive represent submodular function item include include class present hardness provide finally application receive monotone
non show p f column decomposition use lar software proceed prevent entry becoming stay normalization project update function function well function p project become orthogonal single fact extension admit indeed ii method adapt extend project accelerate work nesterov attract machine processing prove first ability nonsmooth whereas gradient descent solve smooth convex difference maintain iterative past theoretically rate often practice simplicity search scheme indeed fast gradient descent increase exploit nature iterate illustrate good initialization initialization rescale ratio invariance dictionary task structure dictionary come sparsity induce penalization change introduce small overlap easy form rest stay unchanged simplicity single shape dictionary indeed patch define large slightly shift invariant dictionary regime image denoise optimization question already mention flat initialize dictionary overcomplete dct admit choice otherwise initialization collection common initialize pass filter extract use denoise qualitative first move visualize image contain arbitrarily rescale work figure initialization may initialization framework image size seem aspect figure experiment learn patch detail large induce structure show area case regime use white denote pixel arbitrary dictionary patches approximate patch match omp clean patch address clean pseudo norm pixel clean estimate estimate quantitative scale six image five level noise denoise peak pair regularization mean db single scale house I second bold report three elaborate dictionary bm competitive compare seem explain flexibility state art exploit sophisticated new extend shift regime possibly invariance key achieve good invariant mapping partly european grant appendix equivalence define moreover projection vector matrix binary take patch operator create pixel contain correspond indeed entry number overcomplete representation orthogonal paris france call adapt prove task traditionally dictionary flat dictionary choose shift invariance propose learn illustrate image introduce generative call summarize patch image image reconstruction task domain object removal face sparse framework call denoise image combination sparse highly redundant dictionary overlap represented prove useful texture synthesis invariant shift pattern appear time signature idea main contribution framework establish dictionary dictionary shift invariance image signature collection manner q encode overlap patch integer context traditional flat generalize wide structure admit use far relatively family exhaustive naturally new characterized image
speed many order factor implement requirement approach output technique advantage depend solve overhead art per evaluation number zero apparent identity approximate exceed rate additional notable bandwidth radial find convenient feature summarize hyperparameter tune additional hyperparameter let implementation technique k g current consider stop stop operation implement efficient expensive precisely assess identity simulation simulation matlab numerical cpu ghz ram memory window proposition normally consider intractable conclusion application tuning role achieve capability nonconvex demand must usually gaussian hyperparameter dataset identity spectral employ reduce quantity involve computable represent several state art problem solve hyperparameter resort identity exploit optimize verify computational new appendix follow simultaneously mean q hold diagonal additive efficiently derivative consider orthogonal entry q recall square read q element q matrix whose entry derivative n mit em em depth em national scientific setting strongly datum would capabilitie candidate distribution additional need achieve performance operation efficiently presence optima resort optimization respect hyperparameter implementation every novel identity jacobian prove identity computation hyperparameter notably advantage kernel study validate hyperparameter tuning process regression apply wide spectrum notable easily find economic advance science generally intensive application reliable representative probabilistic characterization unobserve former latter valuable qualitative aforementioned probabilistic yield probabilistic unobserved paradigm trade see model bayes rely process critical achieve distribution maximize respect achieve score demand nontrivial maxima employ overcome challenge require apparent combination remarkably decade reduce demand equation develop jacobian hessian exploit notably make propose solution amenable aforementioned follow hyperparameter regression derive art aid conclusion proof result present probability let variable entry suitable definite parametrization rkhs kernel beyond interested observation likelihood derive notably describe well ridge yield combine marginalization respect value constraint transform reason minimization notably derive resort obtain minimizer notable example include particle usually rely minima approximate exploit jacobian score converge evaluation span space iteration characterize q inverse jacobian also bottleneck due store storage local optimization pose severe dataset employ hyperparameter art commonly rely likelihood maximization complexity application represent constraint dataset section identity quantitie specifically perform step describe different first order define let order eigenvalue cost identity summarize main support th eigenvalue proposition exploit pair matrix proposition jacobian hessian valid equation include appendix remarkably employ identity jacobian initial overhead convenient propose whose simplicity
approximation differentiable enable adaptation auto lag paper involve trivial main practitioner bayesian argue optimization freedom strategy design trade exploration exploitation optimize use set adaptation find randomized policy optimization approximate minimize hybrid mix adaptation adapt constrained space sampler state discrete sense sampler hamiltonian hybrid space sampler typically tune even expert require often often want ise boltzmann optimal significantly topology sufficient ensure adaptive vanishing grow large prohibitive reason mechanism adapt run mixture mcmc lower choose building mcmc draw propose move parameterize q influence example discrete connectivity random different setting approach adjust connectivity propose discuss refer certain sake simplicity adapt proposal distribution successful propose restrict adaptation proposal motivated covariance restrictive sampler approximation need later replace mh algorithm ig observe field mcmc base stochastic typically surely adaptation also clear proposal use auto certain lag seem task used assess difficult objective section adaptive adaptation phase accord randomized phase function however value mcmc set approximate entire history noisy gaussian noisy obtain value show review htbp chain noisy evaluation objective I gp sufficient optimize acquisition accord adopt ard intractable invariant mcmc work hypercube hyper however quadrature integrate objective run index indicate implicit q distribution observation restrictive smooth sophisticated reader high candidate acquisition asymptotic alternative thompson upper portfolio strategy combine appear recently several newton method acquisition evaluate location gaussian construct space step several way sampler transition kernel parameterize generate tend ergodic optimizer unnormalize boltzmann metropolis sophisticated version greedy policy account optimizer strategy bandit auto lag respectively characterize value state energy experimental slide minimum l ai move I recently sampler tune boltzmann machines boltzmann boltzmann z e x e e normalizing rest regularize I propose state avoid walk determine control experiment sampling differ pick state flip flip I manually draw naive draw uniformly optimize I parameter sampler parameter baseline ensure bayesian optimize I well naive contiguous come parameter bit three study note ham none bit machine application might percentage regularization selection strategy grid ising arrange rectangular boundary boundary grid exactly interaction weight cube ise arrange two periodic boundary bias visible exactly correspond filter run additional step adaptation consist overhead involve additional far I sampler tune manually ise trial burn correspond burn phase include function begin phase sampler fair mean suffer many long string consecutive evident length parameter choose drawn figure imply landscape cube make manual trivial tune intervention length essentially perform performance confirm ising find rapid inclusion show however
q biased construction overall round conditional index proportional metropolis hasting index equivalently notational pmf distribution eq positive sample conjugacy hasting proportional proportional random may use approximation component index sample variable plus minus plus pt minus bayesian individual combinatorial infinite appropriate posterior beta asymptotic exhibit present domain segmentation traditional goal induce latent class use mixture associate capture mutual observe pattern population variety domain model build model treat exchangeability within domain intra marker marker probability population document individual underlie topic encode image individual characteristic location extract car distinct probabilistic different particular draw draw proportion hierarchical share treat effect focus context literature bayesian model component focus method inferential modeling generally concern although repeat draw situation perspective trait characterize trait possess mutual focus individual exchangeability natural mixture count mixture wish count wish linkage count base survival hazard function latent draw collection coin obtain description conjugacy inference countable coin binary total modeling variable conjugacy conjugacy involve binomial beta conjugacy analogue negative binomial current investigate negative binomial rigorous conjugacy nonparametric beta hdp model allow individual justify modeling choice characterize binomial empirically generate infinite vector count nonparametric gamma connection stochastic process far clear process connection beta process remainder framework measure discuss bernoulli process conjugacy devote study beta binomial summary attack domain automatic segmentation measure tie construction variable present construction use dirichlet present additional independent component atom complete randomness poisson randomness infinite sigma light criterion extension independence describe specify ordinary beta purely atomic purely separate mass notational convenient description deterministic infinite finite parameter atomic poisson sigma number specification allow though ease exposition nonetheless draw process union ordinary atom beta name beta atomic poisson intensity ordinary improper beta infinite parameter countable process interpret though beta appropriate context atom index feature occur form beyond classic law say discount parameter atom infinite finite atomic ordinary intensity focus homogeneous straightforward beta beta process determine part poisson beta weight atom problematic beta associated atom constant restriction position reason prefer component relation dirichlet concern let depend parametric perspective specification classical atom conjugacy parameterization strictly atom purely atom simple specification process introduce exception parameterization atom intensity discuss favor intensity exposition extension depend location beta parameter value quantity rather purely atomic atom atom weight parameterization gamma purely measure ordinary process gamma express l classic dirichlet atom frequency must normalize gamma particular poisson process finitely many atom mass gamma process almost infinitely atom divide mass thus finitely fix choose fact almost atomic gamma far dirichlet provide countable set view countable cluster introduce prior value couple real process weight natural binomial frequency nonnegative rely survival model context construction construction bernoulli analogous conjugacy use couple single process matter bernoulli need even poisson may process bernoulli countable atom differ feature equal weight individual derive ordinary component completely number feature total mass bernoulli parameter atom ordinary finite number important finitely constrain beta conjugate process component conditionally j note posterior beta hold integrated recover condition independent l apparent parameterization atom conjugacy binomial conjugacy preserve conjugacy couple classical parametric conjugacy negative binomial two measure say negative binomial atom construction geometric interpret assigning particular assume binomial process datum word binomial beta process three beta ordinary component infinite number atom atom binomial process fact negative specification conjugate binomial likelihood scalar member scalar measure negative draw atom r j j j post n l piece model recall basic g belong binary one component class zero integer view vector count word draw finite mix view component draw component integer component data parameter individual point draw individual alternative couple poisson convert base follow individual atom location individual might complex perspective focus consider individual becomes achieve remainder bayesian nonparametric individual proportion mix couple achieve via range note global drawing measure therefore weight share atom individual specific proportion specification draw index proportion might th stochastic generate couple count across construct beta base measure independence count multiple individual flexibility mix proportion proportion proportion flexibility hierarchy level atomic coincide atom specific structure decompose decompose draw beta beta explore bernoulli propose prior continuous associate process atom failure beta contrast offer assign individual beta weight leverage hierarchical beta suited come analysis integer factorization motivating lead challenge algorithm develop inexact monte approximation include conjugacy behavior belief interest datum diversity atom process component size cluster asymptotically dirichlet extra parameter size generate grow indeed popularity prior attribute power highlight subtle priori number datum potentially effect dirichlet use process mass determine however treat case number number number case case far establish full statement table cluster find cluster expansion term asymptotic expansion upper table cluster growth basic power growth expand model theoretical simulation law beta simulation perform binomial evenly space generate beta atom binomial sample count number note finally simulation behavior law figure scatter tuple triple tuple leave case agree upper agreement contrast exhibit law cluster negative simulation see right cluster plot asymptotic behavior plot red red law exhibit logarithmic growth cluster deterministic grows generate see yield yield asymptotic growth group claim al al al party experiment model posterior code remain default mcmc confusion obtain model identification notably document group group stem distinguish name c group actual c actual group great difficulty usage frequency across plot density document heat usage frequency pattern nearly group align majority make result similarity ten probable intuitive salient vocabulary organization organization name l organization claim armed claim claim force claim group claim claim group claim party claim front united divide meaningful semantic core content tracking jointly image comprise patch generate image patch label infer object tackle inferential performance patch modality complementary modality texture fourth opponent angle descriptor grant invariance variation geometry dense sift histogram orient single patch cover patch index cover opponent descriptor discrete visual raw descriptor sift opponent descriptor mean four component descriptor modality conditionally object patch define generate microsoft wise image wise ground labeling pixel label truth label lie boundary learn ground task remain nine semantic pixel space pixel image visual vocabulary assign represent patch perform divide label evaluate infer image object hyperparameter dr dr sample patch label high lda standard variational semantic generic figure pixel patch center accuracy report generate training prediction every save object showing provide classical c actual label tree actual car test misspecification hyperparameter hold summarize hyperparameter maintain predictive performance vary order model hyperparameter specification segmentation recognition hyperparameter vary remain hold report test patch average class randomly patch predict posterior sample hyperparameter accuracy latent count characterize relationship asymptotic mcmc document latent vision aim many object genomic seek underlying event responsible repetition segment acknowledgment support national science foundation national science fellowship beta process arise model section deep new stochastic construction lead novel comparison nonparametric clustering conditionally random poisson process intensity negative dd e motivated strong construction cluster random stage consist gamma mostly highlight classic relation study change prior derive process distribution start beta distribute gamma distribution though rest result process gamma process new proof nonnegative define measure specify location atom typically sigma ordinary define l name reflect improper beta beta atom distributional result beta derive transformation atom weight process process suppose ratio construct variable gamma suppose analogue construct beta variable gamma gamma component derive connection stochastic emphasize perform process alternative mix ordinary tell collection tuple come draw intensity variable atom since completely ordinary component measure assume poisson associate tuple mark distribution collection intensity match ordinary intensity component manner inverse change component gamma process tuple process tell marked process uv intensity classic distributional eq start briefly establish cite move concentration satisfy cluster size discount growth range discount beta satisfie beta number beta mass concentration parameter number next asymptotic growth discount parameter growth cluster end establish growth number growth combine asymptotic diversity expect point asymptotic growth growth cluster cluster proportion dirichlet draw measure describe appear asymptotic ratio result statement apply line iff equivalently iff poisson intensity intensity result q follow conclude proceed poisson find atom binomial integer integral desire draw consider let monotonicity atom beta associate binomial count equal previous r process intensity intensity completely atom component atom atom variable binomial input correspond process measure three repeat atom atom old atom normalize q new atom normalize deterministic measure beta bernoulli conjugacy completely measure sigma algebra atom associate sigma induce let marginal counting measure introduce process special finally note ordinary atom total atom atom component atom location together count measure location distribute independent identically case ordinary weight ordinary atom atom restrict location restriction proceed marginal may recall distribute location distribute yield describe next particular atom occur ordinary locate atom locate note atom break eq atom count notable special write single atom likelihood evaluate measure analogously calculate quantity
randomly generate environment generate employ softmax policy addition error bar display b function show underlie mdp percentile bar policy demonstrate mh metropolis alg mh hybrid alg sec first examine derive reward domain mh near mh reward estimation reward inferior nevertheless track suboptimal baseline mdp set attribute poor demonstrate see policy increase plot state space model mh consistently mh significantly rl perhaps attribute mdp optimality policy walk rl suboptimal performance performance theoretic slightly much never possible preference reinforcement procedure estimation flexible policy agent preference although require adjust closely sampler outperform inverse demonstrate simple discount promise hard difficulty maintain see give belief mdps hard preference prior firstly environment different preference tie reinforcement learn easily achievable reinforcement interesting promise already useful modelling agent experimental preference preference addition many automate opinion complex planning experimental useful acknowledgement thank anonymous partially bernstein project I fp project result principled inverse preference relation statistical method learn experimental respect obtain demonstrate preference whether event among event preference determine event prefer utility hypothesis numerical event preferred maker gamble utility relevance cognitive behaviour reach direct user model customer apparent task expert set preference act obtain reward environment functional optimally respect preference framework allow inverse structured reward significantly fairly main contribution formulation reinforcement preference prior determine policy obtain policy inverse reinforcement compare relation preference estimate give behaviour theoretic reinforcement preference setting relate abstract preference discuss preference conclude preference dynamic specifically environment control process environment action convention observe act action wish learn infinite horizon try utility discount choice correspondence reinforcement markov utility decision denote distribution define chain subscript shall notational paper similarly denote mdps mdp abuse different speak reward discount agent belief shall task additional structural policy obtain well assume reward policy easily relax additional define reward policy policy reward reward reward policy datum right reward reward reward reward policy depict basic model introduce model reward obtain leave allow draw policy give arbitrary pure reward agent one valuable quantity disadvantage k ts preference attract lot uncertain preference problem design query generalise additive relation application multiclass generalise multiple user decomposable multi utility discuss introduction act model static agent finally experiment program order good action elegant suffer hard demonstrate visit state frequently mdps gap example could equal every structured implicitly policy exponential correspond entropy approach rearrange computable metropolis employ mh walk exploratory examine determine crucial consistent initial trajectory obtain n guarantee consequently guarantee bounding side high neither suggest require statistic feature entropy policy efficiently particularly notable aware probability demonstrate low bind may link try expert b namely random mdp discrete four action arrival uniformly state arrival pair define
symmetric justify every somewhat complicated good empirical rademacher linear functional vector n sample represent rademacher mutually uniformly distribute uniform error result loss let slightly main finite generalization regularizer mx I nx logarithm course finite conclusion priori pass recover exist considerably rise novel datum retain condition replace corollary eq key result independence dependence turn rest organize present result mention conclusion comment give great simplification range member norm simplify occur simplify remainder generic iid point iid member operator factor dimensional I agree bind dominant whenever corollary need remark lasso replace supremum elastic net penalty priori appropriate operator th coordinate q far almost r eq let j projection span orthogonal regularizer vector whose know desire previous lasso moment moment x previous always mutually give appearance range orthogonal complicated almost linear functional complete lasso satisfying disjoint writing rp quite loose structure nonempty convex show supremum attain extreme closure rademacher class simplification hilbert arise group lasso definite background read finite agreement infinite eq auxiliary hilbert collection recall schmidt letter tuple object ny inequality bound back state write random let iid simple lemma read positive homogeneous homogeneity w finite linear combination subspace therefore also consequence iii infimum together refer bound norm v member orthogonal range member onto take reverse bounded concentration linearly random vector variable iid transformation orthonormal satisfy j q f triangle k difference jensen ii calculus rt r part e integration inequality essential compare hand give insight fine appear infinite variable eq notation part dt inequality variable use dt follow lemma version substitution small since mx calculus imply theorem obvious completeness let norm fact omit version paper analogous recover multiplicative sample somewhat set normal q q jensen give dt
measurable average filter usually would actual actual suitable measure generate weak convergence multiscale signal drift fast sde diffusion slow represent slow change brownian motion suitable multidimensional convergence periodic use probabilistic chapter nonlinear representation determine limit behavior give precise sde theorem multiscale diffusion give marginal unnormalized filter filter sde possess fix filter nonlinear variation component apply variation obtain convergence technique terminal govern dynamic stay time tv nice x behave notation work brownian therefore write instead facilitate read key article expand rigorously terminal converge term formal expansion term omit facilitate read solve terminal exactly uniqueness solution equation apply superposition showing achieve backward doubly differential estimate us component vice existence stationary distribution semi matrix unit coefficient diffusion g finally introduce norm usual supremum metric generate prove section conclude converge backward solve explicit precise allow convergence idea precise probabilistic representation form brownian motion general equation denote equation characteristic advantage permit would characteristics independent solution measurable forward doubly differential equation bx ds dy x ds gs ds cover random side remainder give precise reading able existence result classical theory square smooth degenerate see uniqueness continuity define introduce adapt jointly measurable outside inequality every coefficient time continuously partial derivative continuously partial uniformly time differentiable polynomially classical nan integral indistinguishable derivative bound theorem corollary claim give space deduce thing need verify polynomial growth coefficient pt dx combine analogously neither measurable z ft ty write multidimensional one ultimately show f ty ft ft z associate differential strong sde bx ds associate assumption unique reader suffice q solve gradient x z sx drop sx sd w ds p ij sx ds sx iv z sx p ij sx ds depend p tx x du sx sx sx x run hence pt sx z sx last inequality therefore next sx ds integration sx ds kp x du pt u du kp u b iv sx sx kp ij sx pt du since center ordinary cf sx sx sx ds x sx ds kp du kp u tx du du sx sx follow result high derivative tx existence derivative sx z ds hx x sx hx drop measurable b note independent jensen inequality sx z hx sx sx schwarz combine ds c z ds z ds hold theorem track condition solution state polynomial growth satisfy polynomial growth proposition combine homogeneity proposition obtain eq right hand calculation facilitate read transfer result cauchy schwarz inequality long brownian isometry b moment describe lem x dx dx filter follow exactly chapter sake completeness bound p completely analogue jensen tb bound test eq third pt lemma since replace q countable strongly separate every ix iy sequence topology measure follow metric determine generate topology complete development particle filter state estimation multiscale systems approximation grain coarse grain use particle filtering accounting dynamic incorporation datum coarse grain slow apply multiscale though deal incorporate realistic separation process govern condition vary flow cause fluctuation cause fluctuation whereas drift fluctuation slow limit distribution away solution develop exclude example convergence filter existence solution bound smooth work entirely rely argument might restrictive however explicit terminal avoid building stability filter wish express anonymous reading manuscript comment presentation p n engineering grateful conclusion recommendation express necessarily view support grant ph nsf grant nsf school universit universit e multiscale dimension converge filter equation correction practical weather prediction consist enough class minimize call filter distribution effectively simplify observation filter realistic evolution measure impractical dimensional finite approximation extend kalman filter approximation observation extensively quite strong science provide comprehensive issue see use particle problem dimension completely difficultie particle signal observation filtering capability result order magnitude scale science engineering field example evolution govern slow dynamic address effect multiscale equation canonical way study present filter attractive equation filter useful hence numerical partial rigorous support numerical stochastically qualitatively develop low demonstrate filter kalman differential let
minimizer opposite point regret point act recognize discard recognize great either discard large similarly center since require cut ensure simple center incur start device loss total let l l rl rl lr x rl r proceed feasible region aim portion work suboptimal point round point know stop query suffice ensure regret epoch subset l guarantee epoch point portion work contain point discard suitably exploit query equally separate identify either right contain region discard algorithm continue depict look confidence around identify contains discard possible ci early algorithm sufficiently work feasible region hence epoch query interval three example ci case rely function contain avoid fx remainder condition maintain point query run lipschitz subgaussian least algorithm adaptive noisy apart mid work feasible order section key idea show epoch bound per flat never show point end round epoch terminate l x former latter imply convexity q mean analogous replace fact lemma incur single epoch first incur epoch trick epoch incur epoch continue incur round detailed incur fact incur round epoch suffice continue round round iff contain analogous epoch incur entire regret epoch lemma incur round know query recall query incur round final epoch incur q feasible epoch epoch perform proof epoch would confidence lipschitz epoch end definition give claim rearrange follow combine epoch hold incur epoch end round eq overall epoch recall thus condition event construct round make hoeffding epoch uniformly query upper algorithm dimension would construct cover unit along cover know along scale exponential encounter optimization direction polynomially query define pyramid point form pyramid base see graphical build capture direction early approach fail idea case regret combine center feasible noisy box allow r construct simplex vertex let continue k pyramid epoch angle define cone ellipsoid contain pyramid region optimization domain begin begin epoch epoch end discard set way retain optimum epoch apply affine volume ellipsoid define constant within epoch round successively let round center query ci pick depict diagram pyramid epoch angle diagram successively successively construct identify region discard book angle pyramid orthogonal pyramid always sphere diameter sphere angle pyramid depict pyramid center pyramid ci pyramid include large denote ci illustrate know next pyramid continue pyramid cut event depict figure center pyramid sample current pyramid terminate step pyramid illustration pyramid ci angle pyramid let pyramid show pyramid step correctness perturbation conclude epoch b pyramid base angle define center pyramid cone thing isotropic position ellipsoid ellipsoid brief discussion regard clear step cone cut isotropic analogous ellipsoid know furthermore update ellipsoid correctness epoch fx sample remainder condition correctness proceed cut indeed couple construct center least cone discard epoch round ci pyramid epoch assume construction pyramid graphical see pyramid pyramid ci pyramid convexity simplifying know dr therefore complete guarantee discard mistake discard final check correctness lemma new pyramid form round epoch cone center angle vertex center dx ff since enter know fx line use suppose incurred address early theorem scale noiseless random walk idea question future incur together round epoch play pyramid encounter pyramid operation base angle construct center reach pyramid ci net incur evaluate consequence inside pyramid pyramid vertex upper reach value brevity shorthand center convexity rearrange bootstrap vertex center center incur sampling center pyramid ci complete total face b dr ball center net incur round substitute value complete critical us pyramid pyramid exception lipschitz visit round case bound lemma modification fact evaluation little round pyramid construct simple geometric exploit pyramid equal pyramid case enter regular simplex radius contain enter simplex round simplex guarantee unless simplex pyramid construction enter one b terminate unless ci insufficient resolve thing construct sufficiently lemma focus control construct convenient incur round every net round round ci terminate net incur round construct instantaneous pyramid note pyramid cause know pyramid construct round function bound instantaneous incur sample pyramid simplex sample vertex point query point ci geometrically total put together proof state incur regret round terminate ci net control round description terminate exception round ci proof simplex pyramid round instantaneous regret incur pyramid end case bound lemma ci general ci pyramid query net pyramid construct complete put piece show incur ci come far geometrically regret incur ci get regret epoch epoch end ci bind show cone cutting understand volume discard reduction prove method suffice intersection ellipsoid discard cone exactly origin cone height inequality suffice distance origin ellipsoid intersection sphere hyperplane volume cut ellipsoid statement epoch show end contain regret least next ci least terminate condition case epoch proceed cone cutting show equation terminate pyramid dx lemma point complete lemma number epoch play total ball radius around instantaneous volume epoch instantaneous regret epoch maintain radius algorithm algorithm claim together observe guarantee analysis far condition round design complete proof theorem present possible build low point algorithm crucially demonstrate optimal rather dimension dependence dimension noiseless walk successful improve noiseless investigate scenario acknowledgment work aa support google support grant nsf nsf grant dms pyramid dd pyramid center base r center vertex base figure inductive note vertex distance claim node node black north black node south circle east east pyramid mass prove claim q lemma ccccc ccc microsoft california berkeley university new berkeley pa cm address lipschitz bandit function interest value query demonstrate generalization incur classical armed formulate arguably sequential decision arm maker observe accord associate performance algorithm sequentially cost expect much bandit action rely cost reward arm make tractable paper space
frobenius translate tail proof lemma function dependence use loose upper weak explicit directional deviation additive hide kx g v u observe minimum size inner mm black hide minimum size sep mm draw name v uv inner draw circle minimum sep name w condition eq matrix singular large value case relative observed style fill circle inner sep name name h orthonormal km cm circle minimum size sep mm z name h minimum cm fill style z name name name hide r name z uv schwarz also strict pair return must hold I z z imply ji combine return prove useful allow suppose induce iv return allow imply return next characterize root algorithm subroutine return main maintain loop terminate achieve show loop claim properly parent subroutine whether child claim definition sample condition threshold test return event need hold union henceforth next ignore direction subtree ignore direction root subtree imply child relationship child etc exception root rise start subtree subtree say subtree fact super tree maintain super collection disjoint root leaf super next relate opposite bottleneck correlation bottleneck induce z scale observe style circle cm style observe name observe name z notion accord edge direction effectively depend relative exploit z cover size cm sep black style circle inner z name z hide name name z orthonormal inner fill sep name z name name h g choose column u sep mm fill minimum sep black name z name name style circle minimum draw black mm black name h h choose orthonormal u two lemma case subroutine collection disjoint root leaf far relative leaf pair u u x return note neighbor relative path undirected node u u follow observe style sep style minimum name observe name hide name g style circle sep black fill circle draw name name without undirected node node pass root undirected leaf x v topology observe cm inner sep mm fill size name circle minimum draw black style size sep draw black name observe name x v x upon return disjoint rooted root leaf exist share neighbor least neither v u u u hold neither common leaf neither uv uv inner hide cm mm black u dash least moreover xx xu yu induce mm draw draw name name hide observed observe name u give follow style black style circle sep observe name hold inequality analogous x undirected circle inner fill minimum cm sep mm observe name name name hide u x claim second undirected observe sep draw black fill circle sep draw name name since sep black fill hidden style mm name name name x u prove hold respectively root must v u uv observed style inner draw style mm hide name u dash leaf u yu u xu u yu topology sep fill black hide black name name name observe name observe observe u p u observe style size cm sep hide inner name v observe name name observe observe p x suppose iii generality leaf root uv inner sep fill hidden mm name u name hide name name dash argument prove finally loop consequently object loop u appear subtree moreover loop prove initially cardinality lemma final subtree root loop initial inductive start terminate failure loop begin iteration u pair neighbor leaf adjacent component neighbor existence two neighbor component leaf child exist u pair v claim terminate subroutine satisfie v return claim relationship common u iv share subroutine first leaf claim child leaf relative u u u q adjacent leave leaf loss generality argue subroutine symmetry subroutine call construction one leaf component leaf collection pick leaf child loop subtree imply u except change add degree one leaf consider continuous markov markov evolutionary tree certain goal tree recover sample recovery reveal furthermore sample explicit algorithm applicable dimensional heart determine relative topology graphical central tool learn methodology success ai language processing vision bioinformatics statistical challenge graphical model rich parameter understand involve certain expectation em recently understand learn np greedy focus model tree hide binary evolutionary tree set multivariate observe graphical relevant vision scene co aid structural reveal tree configuration four additive metric induce unfortunately ultimately robustness estimation quantify various neighbor serve basis method work area mathematical focus evolutionary basic effort deal allow oppose polynomial recent learn extend evolutionary domain observe style inner hide inner name name z name hide z scale circle inner style sep mm draw observed name hide circle sep black black circle sep mm observe name observe style mm fill mm draw name z observe z extend model scalar address multivariate may scalar handle wide need characteristic discrete multivariate latent core multivariate classical test spectral canonical spectral success useful test recursive tree confirm hypothesis relative topology properly probability efficient manner also considerably restrictive address provable spectral direct every term internal moment either though configuration possibility return degenerate fail motivated singular km test true tree condition strict deduce correct topology q confidence sense reliability z z z j j z spectral singular lie length event hold remark valid variable vector iid copy return induce topology important redundancy correct among gap z z return quantify observe tree ii condition iii return use structure iid globally grouping procedure build fashion root discover
immediately carry notation proof find proof markov substitute geometrically mix show essential mixing basically identical corollary come form boost make clear nontrivial space strongly individual function scalar old inequality example regression latter strongly logarithmic loss guarantee per discussion logarithmic regret requirement modification et combine forecaster sample prediction specifically iteration receive suffer al logarithmic appear drastically encounter update stochastic recall fact q difference stable require outer matrix mahalanobis stability linear generate derive follow assumption online apply linear prediction specifically regret remain q assumption argument minor martingale previous measurable difference adapt modification involve guarantee noting complete proof conclusion ignore size space probability geometrically follow geometrically analogously build largely identical bound excess extend martingale hope proof without require expect believe open question work raise guarantee underlie necessary couple strong regret bound acknowledgment thank careful greatly aa fellowship google national engineering fellowship bit simple form intuitive turn bind nesterov begin take supremum continuous mahalanobis g z z definition update plug bind complete norm note result give begin thus convexity apply h old organization accord recall outer construction proof q update inequality consist achieve begin eq inequality divide proof ex em berkeley edu sample dependent online computable probability error assume loss sharp bound problem logistic least svm application martingale convergence need rely attractive point fix predictor hold assume statistical regularity sequence ask something probabilistic function example algorithm good distribute et al output predictor loss play probability ask ergodic online regret fix predictor natural lead researcher online change distribution computing fix process fix meaningful reasonable practically include problem learn difficulty encounter researcher setting scenario generally mix imply yu law direct approach convergence stationary natural localization self bound exploit machine statistic sequence extend bernstein inequality due generalization dependent particular predictor favorable regime geometric loss lipschitz loss hypothesis prediction fast regression boost expect predictor zero show demonstrate generalization guarantee online answer give unseen say question regard dual averaging broadly establish suitably sequence history area book dependent probability guarantee markov bound mirror optimization non build martingale random bernstein geometrically mix process version prove far relatively convergence argument datum stationary suitably convergent receive sample play algorithm suffer generalization goal low algorithm respect hypothesis require variation distribution density respect suitably converge grow absolutely weak mixing respectively assume mix entire slice result either mix strong mix regime mix mix markov chain recurrent example process example arise metropolis hasting sampler lower state instantaneous boundedness common literature boundedness first final center sequel strong presence bound strong strongly lastly generalization iterate assumption stability least analysis quantify algorithm possibly sequence produce sequence dependent measure sample conditional loss sample I excess risk course set slightly definition assumption place suitably substitute completes remain consist probability throughout algorithm result satisfie depend add sum give divide f jensen recover theorem generalization leave development reader stability make indeed ask analysis insufficient guarantee p rule trivial consequently expectation cumulative result expect guarantee guarantee online expectation would like strong setting martingale argument use concentration remain predictor guarantee bind arise sequence around mirror algorithm idea martingale ergodic block random variable dependent previous directly moment generate sum different proof index random associated define martingale adapt subsampling define first boundedness difference hoeffding term representation well concrete corollary begin corollary mix assumption assume constant mixing process generalization set somewhat slow define et al concrete error mirror extend convex function average satisfie stability immediately universal least obtain corollary assumption weak nonetheless quickly predictor apply proof treatment piece require care observe via see though thing seem make stochastic geometrically corollary geometric mixing corollary unless desire argument polynomially mix somewhat well regret learn due reader recall gradient mirror average stability f term fluctuation martingale scale sample property construct martingale self bound martingale
even marginalization search correctly even propagation could energy mc propagation physics long rapidly topological distance bp pair conditional cavity instance probability remove assume neighbor message fix normalize iterate reach point take time free bp z ij c aa stationary likely fix maximization group several free decrease phase diagram phase transition non arise benchmark agree except effect plant actual assignment configuration energy module soon close message fix typical first phase fact locally stable location easily small random perturbation point dynamically attractive hard ground marginal exactly hand impossible hard know retrieve hard unlikely realistic world test common benchmark fix correspond energy large node initial good splitting actual identical marginal term finite tree principle asymptotically strict phase free energy landscape infer community problem number exact size generative performance variety acknowledgment grateful mark foundation universit paris paris france computer science department asymptotically detect community module noisy plant cavity transition assignment split easy translate practical module underlie hc network genetic community module social connection community propose connect letter generative modular provide popular module cavity develop module sparse network assignment nod intuitive topology retain membership algorithm unable previously approximately size addition unlike minimize entire boltzmann distribution believe exist propagation size bp however ability module marginal marginal aspect approach applicable discuss briefly restrict elaborate block consider node specify member label group choose model leave rescaled affinity block assignment literature plant fundamental cavity block however monte carlo crucial contribution compute cavity bp roughly network assignment prior available maximize assignment network generative
send receiver reverse symbol probability receive ease inferential active symbol symbol digital frequency variance vb conjugate remain unit circle serve conditional location directional signal processing literature phase loop von explicitly von hence various normal symbol exact basis symbol wherein apply htbp symbol marginal symbol give characterize incoming get simple phase vanish symbol ambiguity symbol map decode distribution multiple period sequentially decode output model reasonable inter consequence period employ kullback intuition functional posterior force independence learn artificial intelligence recently result posterior latter partition node kullback vb consider situation transfer period infer symbol phase symbol expand unknown phase posterior integrate summation resort discuss include author assume make sequentially infer marginal symbol symbol independence symbol assume combinatorial step von symbol expand unknown eq section iii apply transmission characterize vb figure describe proportion various develop batch symbol regime db db algorithms well decode use exact ignore fast offline though general ambiguity pilot get main paper primarily able synchronization generalization resolve ambiguity show independent phase thereby apparent concentrated presence unimodal phase invariant symbol relatively long overhead angle scheme pilot symbol invariance simplicity observation deal essentially delta significantly low snr snr vb wherein virtue modelling problem ambiguity lose multi would grant minus ex ex ex receiver noisy message receiver receiver rotation receive plane impossible process synchronization important aspect art phase synchronization vice decision soft decision consider aim fully flexible varied uncertainty procedure variational
decomposition ccc linearly sparsity example keep compare toolbox point interior point method centre tree indicate tolerance converge obtain tolerance grid bring remarkably increase overhead pass constraint grid reach well vary iteration contiguous value contiguous whose noise tree use recover coefficient reconstruct image method h background right new algorithm else sparsity form contiguous proximal penalty derive computing result highlight advantage greedy indicate study convergence whether sparse institute college college university college structure extend group incorporate straightforward prescribe contiguous overlap optimization limited build point algorithm class norm rely proximity operator subproblem regression optimization statistical question definition regression parameter result interested word structure sparsity certain configuration prefer several denoise discussion structure recently nonsmooth vector auxiliary constructive sparsity specifically formulation involve variational incorporate magnitude sparsity contain vector optimum extend formulation sparsity terminology sparsity contiguous embed scene learn limitation technique describe easily associate regularization proposes coordinate feasible limited contribution general combine inspire fista recently significant research subject grow reference therein mainly penalty former focused extend possibility hierarchical structure penalty upon improved form whose computational per iteration comparable descent simple combine acceleration gain certain importance admits form proximal appeal proximity operator computable structure sparsity organize section structured numerical method induce otherwise structure vector prescribe positive attain infimum attain auxiliary vector term sign insight upper namely tight fact arithmetic particular encourage function norm encourage desired well understand contain sparsity pattern indeed since pattern reflect moreover lead end case convex whereas possible convex cone shall constraint specific understand pattern choice encourage hierarchical regularizer explicitly overcome difficulty may unit ball encourage subgraph correspond penalty recall conjugate function subdifferential subdifferential nonempty compact set differentiable subdifferential generic computable deal end recall proximity value proximity operator r proximity operator define minimum various way regularization reproduce hilbert regularizer write reproduce indeed eq view function belong prescribe encourage convex interested cone norm sparsity early ff cb solution hold duality condition supremum attain solution f combine norm duality yield interest quadratic norm yet equation solve denote constraint p problem variable norm impose cone mention penalty dual automatically employ end see example solution q condition self dual solution p x accelerate first scale respect computation proximity special computing operator apply wide variety constraint proximal r solution attain equation simplify define ts solution proximity composite solve seem difficult proximity map task notation eq iterate despite fix modification point fix point iterate proximity give require vector state domain attain set make root motivate numerical problem nonsmooth nonsmooth respectively replace linear specific lead computation case w simple
et underlie simplify merely furthermore kalman complexity grow time behind improve start whose linearization non wise analytic path adequate analytic grow multiplication output substantial multiplication dimension well dimension dimension whose denote deterministic ordinary differential omit behind firstly wiener approximation differential secondly replica couple henceforth index space replica input input calculate conditioning solution replica assume couple denote strong replica couple stochastic differential path read determine green chapter path order exponential function translate adjoint green read function two replica two find symmetry consider dimensional subspace component replica replica shall matrix suppose integrate degree eq square matrix whose block equation determine replica simulation therefore split replica online note replace remainder chapter recursive ordering eqs imply variance former relation latter recursion relation calculate pre h relation order use recursion multiplication dimension interested recursion order analytically case piece analytically q derive eq last section call vector water store water fig equation bf dp fraction model equation read intersect respectively eq input dependent euclidean metric eq reasonable noise diagonal entry obviously calculate conditioning statistical solution series dynamic path analytically resort wise present integrate step piece additional quality exact present eqs numerical multiplication smoothing need multiplication disadvantage one huge I many r code presentation apply sciences simulation intensive calibration ordinary develop
shape part extra part assign shape define value gibbs follow sum omit unary potential unique additive transformation consist potential normalise important notice call homogeneous marginal coincide present reveal verify converse chain homogeneous admit homogeneous appearance assume colour space shape popularity ask identical respect occur imagine complex structured latent show variant expressive want able model unsupervise informally task necessary segmentation task task assign penalty number misclassifie pixel lead max calculate probability currently infeasible belief propagation overview guarantee exact sampling potential task comprise unknown latter applicable situation distinguished format element format supervise consist learn cope well e event potential train logarithm substituting give derivative potential co occurrence random exact calculation expectation ascent posteriori replace expectation calculate potential completeness mention simple view posteriori perform interaction far neighbourhood unfortunately know option interaction despite complexity lead discrimination neighbourhood give possible variant likelihood exhaustive search would prohibitive successively include structure successively remove start variant greedy texture start unary potential edge include previous potential assume gibbs neighbourhood likelihood orthogonal proposal vector kullback divergence euclidean estimation proceed neighbourhood successively remove impact impossible gradient variant situation supervise potential gibbs potential denote nothing family expectation among remove neighbourhood see write arbitrary latter nothing potential sub subgradient uniqueness potential potential unique estimate removal neighbourhood neighbourhood gibbs potential small relation segment relation segment input segmentation middle bottom priori code colour image full neighbourhood whereas ability capture neighbourhood structure show training scene appearance segment assume multivariate gaussians neighbourhood neighbourhood baseline semi fix rectangular priori potential appearance model learn observe show wrong neither simple simple investigate shape generate neighbourhood standard neighbourhood vector gibbs short potential long edge modular potential short anti modular heuristic structure discuss generate approach iterative run show grey code histogram may essentially correct shrink neighbourhood estimate conclude neighbourhood least model composite shape row shape capability capture simultaneously shape spatial manually corrupt accordingly model appearance model gaussians experiment growth variant estimation describe fig upper row object part part shape capture time segment correctly number edge relation grow bottom necessary relation neighbourhood neighbourhood reflect adjacency learn potential occurrence vector responsible potential red express characteristic potential potential mainly anti correlation position part composite shape encode experiment demonstrate possibility segmentation conclude appearance learn fully unsupervised shape combine model statistic eq bit detail shape fig obviously potential generate easy shape detail us shape background label joint set shape vector fig informally correspond joint shape statistic extend component latter gibbs statistic joint correspond shape shape type appearance part final object composite share shape often understand object property nature follow modelling shape expressive segment shape aspect simultaneously shape understand shape recognize whether simple part desire spatial part explicitly important issue useful particularly valuable regard structure education project support grant author education new distribution two corresponding difference potential q function fairly general modular unary function second step claim non edge respective coincide vertex equation eq hold tuple sum unary consequently omit type value vertex coincide eq vertex eq subset span whole space analyse shape modelling expressive already shape simple part motivation goal characteristic major visual colour pathway cognitive way human simpler coherent spatial part principle formulation assumption concept convexity stage visual feedback layer lead whether composite shape learn aim question mathematically principle vision divide two shape
violate close approximation insight simulation plot dot smooth introduce concavity covariance implement concave classification estimation correct incorrect breast cancer aid diagnosis future potential breast instance defer theoretical concave study concave overview enforce study hyperplane assume eq piecewise affine tuple distinct think plane attractive derive cite introduction sample base moderate appear datum typically good idea log come corollary concave likelihood estimator moment however modify variate smoothing automatically concave log concave analytic agree nx dx nx dx aim smoothed concave preliminary compute convolution transformation region integration set furth u distinct use argument small similar onto unit simplex integrate simplex integral propose apply note one variable see consider integrate simplex concave estimator straightforwardly quadrature variation package method close zero way slow case inversion complex fine grid transform convolution multivariate draw observation draw return probability sufficient existence unique semi continuous density fact certainly kullback concave density play behaviour smoothed suppose dp x nx impose theorem ensure slightly version close concave concave consistent exponentially variation sublinear despite turn quantify bind moreover new insight enhance understand behaviour estimator result show concave preserve component concave writing smooth concavity definite statement concavity develop density large upper log instance concave log projection density mixture integrate smoothed concave likelihood replication log kernel concave offer substantial analogue particularly outperform considerable estimator violate smoothed concave however fact smoothed log concave remain situation concavity violate cause misspecification variability concavity another simulate none multivariate motivated procedure pt density distribution follow compute test justify concave draw instead run small first simulated fx setup proportion reject report critical omit critical trace critical replicate setting bivariate unimodal log correspond present table ht c confirm trace error appear concavity compare bandwidth note critical bandwidth due bias estimator quite outlier also region somewhat issue well density support change slightly previous section independent identically pair conditional random measurable assign q classifier risk define small analogue coincide classifier concave density classifier give class concave smoothed analogue classifier reveal bayes encounter parametric classifier quadratic discriminant parametric modelling concavity x log somewhat practical analogue outside hull datum positive avoid consider study recommend curse dimensionality circumstance dimension remain applicable part breast cancer fine breast aim aid capture variability figure curse dimensionality facilitate plot ht density class smoothed treat serious may seek incorporate class notion generalise incur assigning continue observe modification require smoothed loss package demonstrate boundary vary purpose illustration plot course considerably set interested simplicity exposition return situation accord log concave usually apply sampling outline describe denote norm q probability whose approximation strong continuity functional measurable associate two anonymous grateful support fellowship early theorem empty follow notation dominate independent let total probability let convolution surely yield log concave generality semi generality seek semi concave hand uniquely semi concave quantity x uniquely semi continuous density side write independence structure give dx ty follow denote generality may suffice f px xu moment density must integrate part calculus continuous upper continuous log concave concave semi density metric
occur sentence title etc structural svms predict structure structured input map submodular scoring structural example input document summary typically document annotate multiple manual summary denote summary optimize eq call determine I structural formulate ensure score objective learn trading qp sentence polynomial via cut plane step document worst solve quadratic step require care appropriately loss intermediate slightly target normalize train structural maximize manual summary easily different loss tie structural prediction test empirically conduct contain document four manual set article use determine good report use single recommend performance organize group construct stop sentence contain word far refine threshold weight form variant pairwise cosine similarity word word create coverage coverage consider sentence e instance instance look cover word sentence combine feature word tune baseline use weight cosine follow result evaluate score work apply function evaluation predict summary supervise approach hand summarize pairwise result increase manually pairwise result performance test conjecture dataset note coverage hand report model pairwise feature strength greedy largely reflect model pairwise model perform coverage coverage limited flexibility move document adaptation manual evaluate effectively make give amount already example increase improve rough estimate scenario manual summary inter subject disagreement hold summary reasonable restrict summary use select drop drop may summary look report drop finally come drop overfitte fidelity promise l human affect strength final score remove basic remove result role I basic one improve rich essential feature sentence make intuitive usually group none location send stop feature actually score later reliable clear whether use manual manual arbitrarily four summary label otherwise experimental subsection document slightly show figure similarly drop conjecture manual effort summary expensive component help multiple benefit svms scoring pairwise coverage base solve plane empirical evaluation svm conventional ability provide building fidelity member nlp feedback nsf computer usa ny edu score structured optimize measure learn apply demonstrate effectiveness coverage art enable fidelity beyond automatic short text describe summary news article paper summary hard sentence art lin inter sentence hand score reward summary coverage hand summary sentence optimize however address select inter sentence similarity leave select trade redundancy manual overcome problem propose similarity lin apply model due function designing develop learn parameterized scoring document summary maximum unlike approach learn optimize method dependency learn submodular tune particular tune hundred fidelity tune counterpart show investigation source identify approach start known approach selection consider redundancy extend support incorporate pack centrality graph system identification use sentence sentence extraction scoring pair graph include inter document sentence diversity diversity summarize set document integer encodes compression removed allow part preserve structure explore general model document document sentence sentence maximize give score subject character restrict submodular iff say submodular section return reflect redundancy submodular q similarity include summary sentence sentence scalar amount summary exclude repetition simple cosine similarity show document retrieval apply document notion coverage
optimisation decompose separate mostly provide collection put restriction follow lead consider optimal distribution mh mh mm mh q kl variational optimisation need directly family conjugate em restriction optimal achieve optimisation weight proportional result approximation posterior conditional estimate sampling equal posterior weight vb although come classification hide chain emission normal emission whereas collection mixture datum see al semi density approximate hmm latent variable hmm hide binary aim estimator wang mixture model wang algorithm hmm emission article author variational rate obtain likelihood belong family state markov involve despite modify framework decompose stationary parameter belong update hyper parameter collection distribution vb pe average mostly estimation eq variational estimator focus average classical hmm throughout paper describe probit within configuration homogeneous simulation mean alternative consistent exponential conjugate proportion performance consider reference provide vb pe variation quantify pe population well separate simulation pe estimation approach mix pe among tend average similarity vb method clearly appear vb estimation pe become vb simulation estimation get close estimator become easier low hmm vb include misclassification misclassification calculate bold correspond smallest pe similar rate rate averaging average seem table show vb result equal base optimal low misclassifie rate vb provide good hmm averaging bring estimation weight entropy vb pe seem select take account oracle term hmm highlight similar vb provide mse pe misclassification closeness misclassification condition furthermore average vb compare dramatically focus real dataset collect health surveillance al fdr false compose show group group moreover describe hmm aim population want interest influence within varie c cc ccc ccc cccc ccccc ccccc every transition article due keep four model low weight influence classification component propose al differ correspond approach vb display estimation comment approach estimation great comment averaging refine probability area consider difficult estimating probability mainly binary within variational approach selection theoretically prove average mse modification log require datum show estimator mse highlight approach classical moreover weight close problem high significant carry refine epidemic remark corollary proof sketch paris france france cr cr cp cr cp france want retrieve precisely know distribution develop fit sense estimate less relevant information estimation name aggregated collection provide approximation dependent study public health surveillance system average bayes unsupervise classification want retrieve wide want equivalently pure noise situation observation precisely retrieve unknown label associate label label far know perspective model consider finite likely information bayesian averaging develop provide improve et demonstrate provide gain term fit determination weight ingredient reasoning stand calculation schwarz laplace integral size monte provide joint
jump reconstruct slight fig cut short mask spike mask cutting estimate interact side cutting trajectory inverse branch two branch associate respective distribution interact variance large pointwise global jumps value reconstruction reconstruction reconstruct add jump therefore short even centroid mode mode strictly case range traditional method map close miss successful missing possible miss application density conditional present rise dynamic search exceed miss finally seem nonsmooth mapping correct mapping nonsmooth define g forward end robot region end constant length transformation joint angle analytically general trajectory formation sophisticated derivative forward map continuity configuration robot angle position angle toy uniformly mapping normal spherical see train mlp unit space result isotropic mlp toy mean regression continuous sample point obtain trajectory apply obtain sequence toy perform often close bind high occur hard pt pt trajectory trajectory dot space end choose branch trajectory jump report method alone allow incorporate length path planning operate stage candidate whole p sec p continuity constraint normalise random term np continuity period attack directly reconstruction logarithm variable n n standard fitness weight elastic sophisticated prior miss take mode restrict operate right reconstruction section think unimodal ambiguity reconstruction maxima many maxima mode n certainly difficult perhaps optimisation anneal helpful another candidate reconstruction mode weight mode towards pointwise reconstruction expense tradeoff reconstruct datum separate construct pointwise reconstruction find short joint offline reconstruct many programming complexity always mode finding confirm crucial amount miss affect mode accelerate mode discard speedup increase mixture potential mode variation shape large mode spurious smoothly spurious happen rejection spurious search wrong mapping mapping spurious mode spurious give prevent spurious mode locally component width globally smoothing cost mixture centroid loss density log low mode remove relate gaussian represent method likely effort reduce nonsmooth note mixture consider avoid mode mode make gaussian mixture framework long outlier attempt density aspect posteriori application trajectory respect approach differ markovian kalman filter speed trajectory play make experimental trajectory reconstruction depend trajectory sensitive slow speech space time series n useful constraint would lead besides reconstruct appropriate sense several branch around branch branch multimodal value take reconstruct concentrate possible select would decompose mixture kp model gaussians attain mode probably average missing operate suboptimal miss want compute acoustic viterbi feature miss example thing pc miss variable unknown give reconstruct contain delta superiority context recognition reconstruct classify speech frame benefit wide reconstruct whole segment classify reconstruct n continuous variable noise latent latent expect correspond sequence first reduce pick representative map onto multimodal besides dimensionality observe equality conditionally convenient continuity observe countable happen variable geometrically joint value span approach nonlinear system variability inside manifold embed observe take care need conditioning mixture provide working mode thus act manifold appear exponentially mode variable miss save possibility computational centroid principle lie area space fine centroid problem reconstruct long receive reconstruct speech reconstruct pass operation unbounded horizon greedy unbounded problem recommend programming datum stream request reconstructed risk get stream though long reconstruction effectively subgraph whenever dynamic computer point detect experimental recovery field generalise multidimensional experimental elastic net high curse multidimensional algorithm exponentially feasible dimension efficient multidimensional use continuity sequence experimental intrinsic pose challenge modelling difficulty dynamic search cause reconstruction robust local past aspect approach learn use mode reconstruction definition geometric aware attempt reconstruction full perhaps set error test model inference population come inference small interested inference e variable miss imputation imputation multiple instead single repeat imputation carlo require ignore dependence take constrain result candidate reconstruction pattern multiple imputation method reconstruction section difference imputation whole imply avoid exponential dynamic multiple imputation average branch mapping method branch identification mean conditional joint use random sample centroid high well mode miss predictor mapping use invert train net implement must descent provide minima gets find return mapping square reconstruction drawback need mapping combination separate mapping grow result branch review extension approximate forward mapping describe miss solve approximation mapping member attain inversion task partition subset mapping try set restricted branch separate branch restrict global costly run inversion mapping disadvantage getting cluster difficult difficult neighbourhood branch without priori branch detect geometric manifold curvature sensitive wrong region computationally costly density represent branch implicitly determine topology branch mode mixture combine net multinomial logit model expert process expert expert network expert large mapping forward inverse robot result map conditional solution nonconvex convert inverse satisfie first forward result unchanged result portion network description whose train branch multivariate method map kind constraint particular never trajectory contain branch incorrectly reconstruct inverse mapping contain jump branch similar feedforward net mlp depend recurrent net feedback loop unit unit delay n attractive modelling net high feedforward net learn point reconstruction drawback feedforward net reliably unit apply autoregressive datum greedy expect suboptimal universal mixture spherical variance etc e centroid high mode rest conditional ignore value pointwise propose method joint curse dimensionality function treat asymmetric miss reconstruct mapping construct codebook codebook candidate onto return codebook vector codebook reconstruction codebook programming continuity forward programming option configuration produce acoustic many huge high codebook time construct codebook difficult among reason codebook manifold space codebook search really several reconstruct finite though codebook limit require neighbourhood crucially preserve reconstruct missing exploit temporal smoothly propose miss secondly reconstruct obtain candidate give plausible reconstruction actually plausible one constraint analogy miss pattern treat unlike treat predictive distribution deal mapping branch always branch average branch consider inversion high simply input general extremely constant miss reasonably contain spurious suboptimal reconstruct inversion mode forward distribution definition insensitive arc length geometric invariant also reconstruct mode offline pointwise distribution pointwise reconstruction ex c greedy net apply unknown situation relationship mapping inversion problem arm inverse estimation pose movement video inversion decode activity miss reconstruct speech acoustic perhaps grouping principle scene multimodal speech want read vice versa type feature pointwise mapping variation datum applicable case construct make mode every independent exist I original fine joint though use replacement mapping efficient grateful helpful discussion cm college road email reconstruct multidimensional contain independent involve take give vary redundancy redundancy live one continuity constrain reconstruction mode present obtain programming toy reconstruction mapping inversion programming reconstruct component speech corrupt sound instant band corrupt consider problem band whole give net usually mapping mathematical determine give one invert forward map robot arm uniquely determine angle vice versa mapping occur traditional miss compatible flexible generating mapping subset via however impossible break ambiguity prior choose result kind redundancy component pointwise redundancy vector rest paper explain explain mapping reconstruction section experimental version appear conference publication part reconstruct examine definition give rise consecutive point sample point observe necessarily collection redundancy mobile wind field wind colour several problem vector live want reconstruct verify walk trajectory region allow take trajectory live dimensionality look relationship since use sequential unless note generalise write sequential case mean variable scalar measure otherwise say whose reason multiple measure lose depend rather uncertainty recognition speech stick miss classified though associate set act mask complete problem approximation mask varie miss reconstruction mask missing describe pattern missing datum completely pattern miss mechanism take account missing may miss reconstruction provide reconstruction information terminology seek reconstruction set single pointwise reconstruction measure reconstruction vector component index pt convenient missing miss set reconstruction base functional relationship idea define functional relationship conditional pick representative multimodal mode reconstruction pdf contain fill perhaps mode several peak functional relationship mass concentrated construct mapping particularly general probability low spread say principle bl joint area dot density manifold mapping multimodal obtain define probabilistic extension manifold dimensional curve surface concentrate near purpose distribution discuss issue measure conditional informative uninformative require unimodal single depend domain differ median mapping usually via see section mean multimodal lie area bad may support since even unimodal pointed mean common representative respect point ease variable locally probability general calculate know computing analytically find bar locally though absence information keep mode likely representative pick sample manifold effectively attractive mode much sense unless conditional return corrupted may fall area mass body reconstruction serious set certainly miss correct wrong affect global via constraint consistently bad generic missing model observe relation variable appropriate p estimate long estimator density offline training even pointwise criterion greedy tendency point jump reconstruction offline gaussian mixture density time sequence n miss gaussian reconstruction candidate reconstruction easy traditional method map one gx xt gx see forward mapping sometimes reconstruct square reconstruct five mask missing miss miss regression apply mask complete miss sample curve additive train b perceptron mlp layer square reconstruction descent mapping baseline basis separation function interval see implement reconstruction pointwise mode mode likely pointwise euclidean course lower achievable tell reconstruction conditional pointwise reconstruction pointwise mode continuity base unweighted conditional unimodal intend skewed unimodal pointwise reconstruction follow slightly branch branch chance contribute without appearance compare coincide unimodal density training trajectory density manifold lb lb lb mask mask mode bc pt bc pt bc bc pt bc cm cm r view colour panel panel follow text detail dot manifold dash contour joint fa mode circle fa fall reconstruction noiseless original several mask trajectory trajectory mask e reconstruction fail centre corner occur point trajectory gaussian nonsmooth box panel several red line circle widely reconstruction noiseless nonsmooth panel mask confirm reconstruct far full gaussian mixture mlp report aspect draw conclusion bad mask forward practically true mask mask mapping inverse branch branch symmetry branch happen branch symmetry inverse mask predict theory
curve simulation outperform pair positive definite entry quantile aa mn equality remainder sample implement assume comment section optimality choice valid dimension case provide intuition competition separate large difficult discriminate desirable reducing tend bring close hard fact measure term kullback divergence consequently seek dimension preserve pass transform distribution dimension project classical project statistic k draw haar manifold k invertible probability consideration property projection k eliminate variability projection refine average quantity remainder consider p p k diagonal indeed confirm theoretical quantify relationship integer p data random nominal asymptotically order p kp possess nevertheless concern k g qr distribute algebraic section generate number depend well precision eliminate average fluctuation illustrate simulation case roc statistic average characterize statistic theorem comparison relative past work section asymptotic sequence index size mean implicitly vary although derivation restrict condition hypothesis assume ratio tend divergence alternative shift satisfy asymptotic power result let denote order pair define exact twice kullback project determine satisfie correspond blind advantage blind kl divergence discrepancy du rp function form entry denote alternative similar tend arbitrary compare function van term inside define ratio term explicitly asymptotic compete test advantage theorem covariance formula interpret gain insight rp natural interpretable shift power likewise natural extreme adversarial orientation direction haar sphere du equal compare encode idea shift direction direction easily follow factor emphasize power relative usually context asymptotic lead clear proposition theorem comparison state scale power appendix maximum minimum quickly eigenvalue shift spherical limit hold kp ok include variety sparsity entry support pattern capture shift shift scale pt demonstrate dimension proposition indicate scale fluctuation formally lead maximized provide condition optimal class along limit optimality limit straightforward numerical roc several refer case statistic compute projection construct slowly decay spectrum eigenvector haar induce shift vector accordance proposition curve fluctuation sphere similar proposition five setting correspond matrix draw uniform sphere choice equip chen theorem end fix inequality p serve reference asymptotic asymptotically interpret pt structure long grow orient test pt measuring indicate rapid decay project map low subspace chance variation great instance contain mass interpretation eigenvalue contain total mass calculation satisfied grow effective another decay occur connection fourier p dx pt computation integral easily hold decay rate associate lead competition hold must theorem direct proposition recall k k k event tend event tend replace turn sd recall shift pt limit k pt pt condition unlike play role sd test frobenius large datum large small uncorrelated let prevent small effect think statistic take correlated datum enhance relative suppose block structure block diagonal entry dr imply q consequently conclude copy dominant assumption theorem example illustrate involve powerful test bs correlation generate eigenvector haar orthogonal impose sufficient condition non dimensional e pt low grow fast confirm gaussian matrix eq involve norm oppose frobenius norm condition interest pt hold write r z consequently combine conclude inequality replacing compare broad compete simulate theorem roc normal roc curve datum dimension roc curve reflect choice choice c decay block shift shift theorem draw draw describe haar orthogonal group see whereas select spaced slow control rescale fixing amount variance plot spectra figure last use block along diagonal interpret negligible greater slow rp implement bs sake completeness procedure mmd kernel label statistic curve rp test procedure rp independent power gain slow diagonal covariance fast qualitative compete compare panel panel roc unchanged correlate similarly advantage diagonal panel agreement remark slow spectral versus panel compete insensitive spectrum take remark theorem quantitative assessment curve determine c c decay decay pt pt pt r involve obtain average theorem sd pt average figure set average increase remark second n projection novel implement addition derive interpretable correlation compete specifically well interesting regime furthermore realistic case condition comparison work dimensional discussion suggest case fellowship de er cancer partially support grant dms proposition proceed state main lemma concern p kp lipschitz gaussian pt eq substitute pt obtain second use tail namely eq eigenvalue k pt p variational cyclic letting eigenvalue jensen spectral follow ki diagonal column square equal kp pt guarantee ensure bp pt pt pt gives let precede equal almost eigenvalue zero semi matrix p ok imply sufficient write lemma limit prove limit converge expand notice realization kp kp kp kp nan gaussian strategy conditionally pt pt hold turn later next pt precisely eq follow theorem deal pt need kp k simple definition orthonormal invertible upper positive thin qr factorization white wishart claim substitute verify limit cyclic trace jensen inequality extract exchange since may note paper check uniformly variance mean bound integrable formula jensen twice yield norm reasoning work irrelevant piece tend uniformly ensure k uniformly integrable pt imply eq q adjusting event k c k ce piece tend upper piece replace frobenius schwarz inequality k pt
impose analogous computer image discretize image call picture mean array number observe alone detection object colour colour colour pixel within value assume object image pixel call pixel always noisy transmission medical example image scan always body observe noisy image optical begin pixel noise special noise normally normal mean know noise symmetric image rectangular behind present various analogous possible related role mostly analysis need plane black pixel interior point like interior pixel colour boundary type noise explicit method complicated situation pixel colour different formulate formally denote therefore accordance stress dependence assumption even general variance adjustment quantity proceed quantitative estimate black pixel correspond omit notation identically colour original standardized black colour ready describe main thresholding colour colour pixel grey grey procedure colour observe grey add reasonable pixel colour white pixel black end call analyse real follow small hold obvious principle formally first transform picture pixel call colour white vertex edge vertex side connect connect point black picture collection lattice call graph efficiently picture graph lose consider presence get gray make sense overcome definition goal probability site get picture black formation cluster estimate size shape split vertex correspond pixel original background subgraph denote observe black detail form cluster little black difference method derive efficient randomized happen relation issue observe black white cluster outside moderate value make look make colour several way suitable maximizer use white picture decision possibility test h nh object maximal probability picture fact terminology working algorithm picture image object detection problem example even make completely side pixel eq obvious relax replace shaped figure two asymptotic character valid nevertheless remark asymptotic consequence assume resolution infinity mm pixel therefore could detect fine formulate false run thresholded find report detect step find pixel satisfying connect rigorous us work assumption linear exceed nc implicitly mean comment think work place detection devoted crucial convenience predefine operation see analysis chapter operation save pixel cluster position complete analysis show false q white pixel theorem constant event black lie mark black site theorem book explanation terminology measurable large obviously subset lattice side increase measurable decrease ci ci p denote coefficient nn ci k assume part true site lattice rectangle vertex leave side number right rectangle slight lattice pp picture satisfie thresholde picture observe site less take certain happen
customer make history customer response write customer restaurant need collect prior pr l belief ni beside customer decision effect customer customer use l recursively ir n customer response customer induction two make decision restaurant part paper restaurant game research area four response strategy utility dynamic spectrum cloud deal application formulate chinese restaurant response system simulation well random customer table probability signal customer purely signal regardless customer reveal table give strategy extension learn system reveal customer strategy belief customer maximize customer decision signal reveal previous grouping except bayesian subsequent customer evaluated treat rational behavior traditional dynamic access identify available spectrum sense potential enhance efficiency available sense share member within collect access individually spectrum increase detect user primary user access secondary user access network decision secondary estimate primary secondary cognitive secondary user access access primary active user slot primary transmission secondary activity slot secondary individually sequentially channel going slot loss make decision secondary access user user policy slot transmission interference transmission cognitive model chinese restaurant hypothesis primary user detect activity activity detect addition sense probability updating belief sense point access channel slot slot primary secondary choice choose secondary user channel secondary choose high follow response recursive simulate cognitive channel primary access user slot secondary primary three beginning slot primary channel false alarm channel channel channel within fig response secondary factor user secondary channel loading channel equilibrium loading secondary make secondary secondary secondary secondary difference channel successfully identify channel offer utility secondary expect number secondary secondary six make decision utility secondary early secondary secondary user decision subsequent secondary eventually benefit make early user case mistake make average utility secondary secondary scheme average secondary agent action user large scheme take decision secondary make decision later likely primary likely choose channel look secondary hand good response high secondary effect secondary access channel channel hand scheme since tend available channel spectrum phenomenon consensus primary make finally interference fig involve learn scheme primary signal efficiently channel primary application service cloud reliability availability major concern cloud storage two enhanced platform software hardware reliability availability affect transmission capacity reliability platform may decrease software hardware storage platform service potential platform growth platform let cloud storage say maximum pricing long term platform since due service availability binary respectively platform platform reliable probability party platform service however prior platform platform form platform receive public discussion know assume decision short compare service ignore availability decision service end platform receive platform otherwise platform platform respectively platform vice belief rule function platform probability service platform service reliability let choose platform response recursive simulate storage type reliable low reliable offer failure per one first decision simulation show utility decrease decision take negative effect utility need predict decision platform reach number reliability low platform effect dominate reliability platform advantage choose dominate platform platform platform case advantage platform reliable collect utility decision however difference scheme load balance platform reliable provide high serve reliable platform exactly different scheme fig see high balancing reliability platform platform reliability platform platform high signal scheme reliability reliability well load balance reliability platform without loss platform network social show business offer especially platform significant discount deal deal limited purpose deal network facebook twitter observe successfully likely receive response service product restaurant restaurant serve huge customer restaurant exist customer possibility service quality deal mean customer deal customer contrary restaurant customer visit restaurant review internet binary model high uncorrelated customer deal customer along public customer customer accord review customer positive restaurant represent quality customer collect belief restaurant review customer restaurant customer customer discount degradation customer function significantly simplify due linearity number customer derive belief customer customer customer customer response customer recursively customer response customer compute j deal customer deal offer restaurant restaurant crowd factor restaurant equal quality restaurant conditioning restaurant customer receive positive restaurant restaurant restaurant receive restaurant customer receive choose deal reveal customer sequentially response four result customer random scheme customer customer quality high major determine advantage collect review deal utility contrary restaurant determine utility case early customer choose restaurant customer customer expect subsequent customer customer customer become indistinguishable scheme customer also reflect phenomenon social deal eventually reduce decision consider network high learn customer make different since separate crowd prevent severe quality degradation however well network social finally study restaurant price deal restaurant game new restaurant try enter put deal discount customer give review restaurant become customer control signal control restaurant depend signal restaurant game restaurant deal website deal restaurant restaurant restaurant already customer restaurant unknown probability restaurant conditioning restaurant customer receive restaurant restaurant customer receive review restaurant new restaurant deal restaurant simulation restaurant simulation expect see customer always deal price regardless signal restaurant increase customer restaurant high customer decrease
significantly train row flat represent nn train flat flat node perhaps important running analysis processor complete time require nn nn call give result average time appear one train likelihood likelihood evaluate equivalent decrease ccccc ccccc equivalent nn speed prior fast analysis know knowledge nn prediction value variance hessian weight prediction calculate validation initial predict uncertainty calculate error great nn continues nn nn confident make confident enough set justified way bar train use old quickly multiple average bar nn product future demonstrate rapid blind multimodal inference combine artificial network reduce significantly run computationally toy nn surface produce accurately log non flat model case nn sufficient nested predict able speed comparison mcmc f ph ph constrain hamiltonian carlo gr ph w ph ph r p j ph ph r p gr I quantify user manual entropy white h neural ph ph ph theory machine von nest york laboratory jj cb uk institute road cb algorithm combine nest artificial network blind multimodal bayesian nest learn function expensive rapid approximation order begin ability complicated surface ability probe observation valuable increase addition obtain train function combination problem expensive particle task traditionally exploration tune accurate additionally efficiency affect multimodal selection need accurately chain multiply expense evidence peak fail multimodal degenerate situation nest method design provide allow implementation nest multimodal particle physics evaluation million able call ideally task approximation precisely nn give widely technique function variant conjugate gradient optimum hessian towards blind accelerate multimodal inference combine use train function predict likelihood original make accurate place original future sample second user obtain network train evaluation near nested find good weight show statistical estimating hypothesis write posterior dimensionality ignore estimation inference un normalize achieve monte hasting hamiltonian publish operation begin live likelihood live remove replace new high removal replacement live continue difficult find discard point algorithm go volume small inefficient problem new decrease live surface contour contour constrain distribution able multimodal separation calculation evidence substantial physics equation run output node activation smooth purpose linearity training mapping original additionally percentile training calculate mostly original require network accurate multimodal degenerate surface toy peak must able likelihood problem narrow region lastly long degeneracy require reproduce nn problem require simple analytic expense dimension must able region peak structure run live recover method value analytically agree analytically compare return two lowest remove identical reduce log nn gain would thin circular section magnitude prior dimension live obtain manner return nearly identical nn end explore use degeneracy dimension present function log dimension figure prior perform sample live node live hide return analytically value analytical compare posterior dimensional distribution identical posterior nn decrease computationally expensive cc solid outer contour toy example usefulness surface critical parameter standard code paper performance require evaluation temperature value code second compare computationally limit factor likelihood function full benefit particularly physic peak around location set parameter range table parameter range flat analysis alone galaxy df survey ia datum resolution website live training likelihood
kt signal case preserve process sensor exploit exist straightforward refer result svd svd operate block finally svd cast source power spectrum source power modification estimate expectation modification base method location index peak k detail inversion update h nonzero correspond location source entry correspond location set zero truncate hereafter kn zhang direction arrival development reconstruction advantage conventional one practical situation estimation study modeling develop base joint exploit assume prior snapshot snapshot simulation maintain high grid sensor array research decade focus far wave front assume angle music conventional estimation prove realization large uncorrelated source advanced year development reconstruction compress cs candidate assume unknown grid formulate constitute recover vector snapshot guarantee recovery prove recover isometry property rip column highly incoherent measurement sparse share support exploit average incoherent learning method cs formulate perspective exploit laplace posteriori signal possible recover g inference heuristic extent offer accuracy compare array include compressive combination outperform music cs though exist show still practical grid necessary gap near point estimate constrain dense coherent signal standard coarse notation conjugate transpose vector trace entry denote element take real part estimate organize grid introduce conclude far delay represent shift observation q tt ty sensor reader refer discussion unknown mapping relationship kn approximation measurement write validate approximation error noise measurement gaussian closely observation approximation small modeling error adopt grid grid higher dominant grid adopt considerably comparable find formulate bayesian perspective develop complex value simply jointly white denote precision pdf circular symmetric assume conjugate prior broad among share two row
expansion subgraph first fuse fuse eigenvector graph eigenvector subgraph indicate value eigenvector amplitude slow subgraph half show plot fuse become eigenvector graph impact spectral time vertex slow belong subgraph relatively fast term subgraph vertex subgraph value expansion decay slowly large belong subgraph slow come heart section close slow fast realization fast mean vertex fast valid nevertheless ratio similar within fused see fast graph separately slow fast subgraph fuse subgraph axis subgraph fuse result confirm embed concentrate vertex much experimentally average patch graph study graph clearly transition fuse slow slow slow slow subgraph slow graph fast order magnitude transition display slow separately component fuse b scale lastly transition slow dynamic slow confirm embed fused vertex subgraph effectively divide subgraph away subgraph preserve demonstrate identify subgraph fuse implication anomalous rapid away baseline concentration anomalous patch happen patch fast embed segmentation become patch local extract requirement exhibit geometry fuse section synthetic prescribe autocorrelation low frequency yield signal smoothness argue type change class quantify time fast portion show four four compare numerical regularity autocorrelation autocorrelation control local partition autocorrelation keep create regularity homogeneous poisson intensity adjust autocorrelation frequency increase figure signal appendix give decrease regularity figure covariance parameter time describe brownian motion specific maximally patch time regularity compute embed eigenvector frequency principal show signal display extract high segment slow blue extract section regularity signal patch smooth visual confirm color frequency embed code match plot signal patch patch patch leave quantify noticed patch ratio end therefore study lipschitz patch patch slow mutual short eigenvector design gradient measure indeed eigenvector therefore minimize quantifie restrict see whether fast however belong fast patch compute square either slow study ten realization slow regularity shape patch remain compute choose show smoothness exhibit rapid local associate fast concentrate parametrization analysis true patch realistic signal realistic probabilistic argument walks provide explanation analyze patch base reveal presence contain rapid change another leave slowly change along low dimensional interested local scale become live curse informative large patch continuous patch nontrivial graph patch weight distance requirement intuitively reasonable patch represent discretization nonlinear manifold close another distance conversely poor approximation geodesic information available us patch trust observe contain walk fast patch jump avoid walk distance large choice random large avoid connection choose mutual project sphere allow local force irrespective distance consider vary adaptively patch self weight matrix consist patch patch local behavior involve indicate optical also exploit construct dictionary representation g embed slow fuse dataset exhibit concept clique correspond subgraphs texture synthesis super reference patch patch near analyze patch use provide evidence provide justification experimental embed organization natural patch extract patch contain intensity work provide explanation success patch denoise author filter interpret evolution diffusion patch duration rely properly toward backward eigenfunction one perturb patch assume image piece experiment anomaly frequency content change furthermore et derive time use energy existence high patch consequently low finally patch argument add slow interpreted function support manifold potential narrow agree reach present investigation present work use reference therein area physical neuron fire instead process slow fused rely aware decrease reason rely loose upper provide geodesic tight could number path length able spectral choose ultimately vertex q last inequality approach bind slow compute pair vertex bind standard bound nash inequality nash electrical adapt introduce concept disjoint separate path connect edge weight loop walk two disjoint slow attention edge show leave removal edge prevent connect diagonal green entry walk move connect connected edge generic edge entry figure go height therefore l sum sum put everything nash summarize slow observe graph divide side simplify green self show rely relationship electrical random walk graph begin assign connection consider potential difference result electrical flow equivalent effective connect necessary maintain unit current express side choose obtain weight express eq connect distinct rewrite binomial effective geodesic distance scale expression os graph utilize geodesic distance yield euler simplification give electrical network node parallel add edge decrease autocorrelation nonnegative fourier equality apply binomial frequency fast frequency identically unit check autocorrelation linearity mean together problem understand success base analyze art technique denoise work explanation metric model geometry patch graph parametrization graph time mutual patch correspond patch correspond expand concentrate rapid local change would otherwise patch numerical diffusion patch address success patch metric patch signal organize connect patch similar reasonably patch measure distance patch edge short geodesic associate walk analyze art denoise consequently geometry geodesic two vertex geometry analyze study walk metric derive efficiently patch manner reveal notice base concentrate rapid rapid contain furthermore contain spread metric main contribution explanation model geometry patch patch smoothly patch exhibit anomaly rapid change parametrization graph vertex relative change effect result explain parametrization patch eigenfunction concentrate patch would patch phenomenon exploit classification large analysis indicate develop intuition patch study several example patch allow parametrization embed experiment finish discussion without sequence patch need extra notion patch contiguous extract collect patch patch patch organization organization presence patch delay specifically theorem allow replace dynamical system equivalent form observable organization throughout think several originally series think also keep mind understand patch set connect patch patch patch graph vertex connect along vertex sensitive change measure detect change smoothness scaling define edge weight rapidly topology appropriate two location explain analyze patch h h become patch region patch edge patch along weighted weight diagonal reader intuition geometry associate help motivate provide sketch plan signal image visualize patch patch onto large variance patch analysis display size around image quantify local patch color magnitude local red blue encodes temporal proximity arrival wave proximity baseline arrival illustrate detect patch maximally third horizontal collect patch image image local variance patch patch distance patch correspond intensity vary varie little concentrated surface visual mutual patch explain remain overlap close coordinate vary slowly curve argument rapid change neighboring argument allow understand characterize accord patch appropriate normalize transform rotation patch region blue contrary content change middle try organization patch represent activity expect rich content slowly baseline mutual distance compose part expect sufficiently long generally patch extract distance another associate display indicate close gain organization patch patch note diagonal close dark near left slowly vary end diagonal variation column exhibit rapid local center tend lack prominent diagonal entry concentrate column correspond center intensity smaller extract apart indicate relate preserve indicate patch concentrate slow patch blue green align along surface display embed patch map patch low part concentrate region figure define patch time measure goal explain patch embed figure characteristic observe compose patch subgraph fast patch extract confirm analysis applicable model graph without generality patch compose patch affect conclusion interpret proximity proximity patch corner weight edge patch diagonal slow fully connect self connection require distinct regular last regular since signal periodic comprise patch demonstrate size throughout away weighted graph model os self connection weight possibly less one edge green w n slow appear low right fast right patch exhibit region fuse size subgraph probability weight ensure connect validity parametrization edge allow patch combine create subgraphs subgraph connect patch construct adjust edge connection subgraph vertex fuse connect distinct binomial vertex pn n vertex equal vertex realization graph vertex vertex understand fuse subgraph slow tractable fused complement fast slow provide subgraph fuse confirm study see understand within subgraph study subgraph would appear straightforward compute reason low time sufficient rapidly even rely connection electrical electrical circuit vertex connection circuit circuit key state moment rough slow fast quick version slow self self time compute path add decrease nevertheless walk distance lead conjecture average average regard graph analyze clique vertex
prove differentially query release packing extend wide mechanism date query error machine perform database draw particular result view draw predicate sufficient proportion half main order preserve construct approximate respect yet distributional interactive handle theorem computational release domain vc dimension algorithm query usefulness release privacy create answer alternative definition relationship database tuple database string length hypercube tuple endow copy mechanism arbitrary synthetic concept notion definition differ symmetric access differentially neighboring database section alternate strong privacy simplicity privacy proof notion difference neighboring database release evaluate fraction counting query neighbor analogously predicate restrict count paper linear query vc predicate vc predicates query predicate collection one counting predicate vc exist cardinality abuse write collection predicate et al give mechanism sensitivity differential privacy laplace query return variable draw mechanism mechanism answer query level query answer composition useful tell differentially private et differentially mechanism string mechanism access shoot interactive public arbitrarily seek preserve privacy imply query seek release database usefulness output satisfie derive net class net cardinality net general release mechanism map quality score state fix user prefer mechanism select proportional mechanism privacy large implement super preserve exponential net mechanism privacy differentially private net usefulness necessarily net mechanism useful quality exist definition union output n gs c prove condition therefore differentially query net query question mechanism mechanism count upper net count prove crucially finite query eq q sample database mx standard chernoff database satisfy complete exist prove class count straightforwardly analogue counting strictly strong utility mechanism useful sufficient set concept preserve privacy view draw extra usefulness guarantee result discretize efficient remain section give mechanism respect count tight namely count differentially private fix counting query predicate denote universe vc database answer simple class line many follow preserve database contain value would able percentile point usefulness answer impossible privacy answering class generalization answer percentile fall percentile real privacy database contain element contain value answer must point must mechanism query class generalize high axis align differential privacy database preserve differential privacy construct answer impossible differential binary since around definition domain section another relax definition interactive mechanism query mechanism mechanism non mechanism second margin private database note without utility query denote margin introduce fact projection norm dimensional purpose preserve projection pairwise integer matrix entry apply theorem ax ax ax ax ax ax ax ax x identical ax complete proof synthetic collection structure random mapping query write evaluation u h ji project projection collection net choose approximate answer guarantee shift answer projection bind need discretize uniformly vector j privacy composition mechanism differentially call database except dd polynomial let margin quantity structure corollary iy iy u apply take plug choose recall distribution assigning draw except therefore condition union plug conditioning privacy universe say draw database example treat patient disease guarantee privacy informative actually motivation definition particularly perspective sample draw reveal inherent whereas whether strong differential provide single guarantee two database hand distributional privacy make element exponentially probability probability database draw nevertheless possess distributional privacy strong database neighboring satisfie privacy mechanism satisfy distributional meaningful database satisfy differential privacy draw mechanism preserve distributional privacy preserve privacy differential impossible preserve privacy note imply differential distributional draw distributional query behave draw preserve distributional preserve differential privacy neighboring database theoretic existence accurate differentially private mechanism query query database linear dimension class discrete interval main left algorithm utility comparable net database question query thank david theorem definition database query size net represent particular counting query guarantee grow query query simple generalization preserve time slight relaxation guarantee release capable represent answer query dimension notion collection privacy increasingly might useful sensitive correlation cancer collection medical financial reason sensitive release dataset quantify privacy mechanism interactive interactive differential privacy removal element amount intend information private reveal cancer almost nothing information motivated preserve mechanism conjunction count say class answer change building et show discretized admit net release release count computationally efficiently interval axis dimension guarantee range discretized definition impossible differentially private even quite simple natural usefulness release information answer approximately correctly approximately correct relaxation notion margin theory hyperplane fraction concept mechanism reveal database database change probability distributional privacy satisfie accurately satisfy database element distributional privacy formalize separate privacy issue outside database learn database notion privacy database connection compatibility formalize substantially linear call adversary original private mechanism fraction remain privacy counting answer query preserve differential scale linearly mechanism et consider learnable model learnable polynomially sized interactive differential privacy query perturb laplace sublinear number query answer counting grow class allow analyst query linearly learn private database ignore pac learnable pac learnable build
easy solve difficult derivative follow partial derivative eq get equation solve arise yes element zero matrix definite definite force product univariate normal possible center corresponding since model simply acceptance jump tune address obtain model effort require mixing figure show jump frequently compare implementation within almost identical minor attribute jump posterior model probability jump sound mapping reversible apply selection problem particularly plot ratio year year exposure ratio number loss unit exposure collective collective exposure value normal denote time adopt process precision inverse evidence introduction loss confirm behaviour sort ratio negative could theory way restrict ratio within adopt serious failure ability detect provide useful check equation use gibbs posterior parameter distribution implement full distribution gamma ne q shape scale variance q full use dependent turn conditional difficulty general metropolis rejection stationarity flat interval trace small mass marginal reveal union disjoint contain interval identically conditional density sample whether reduce effective integrate nuisance difference implementation first largely integrate parameter result reduce posterior make posterior show conditional density likewise depend conditional form integrate density conditional gamma q conditional readily scheme variate current implement updating proposal interval fine proposal difference improvement table mean interval variance important diagnostic tool mcmc autocorrelation figure autocorrelation autocorrelation great except lag big lag significant autocorrelation significant lag plot trace implementation integrate could indicate indicate proposal small current chance b integrate b plausible alternative paper examine detail discriminate adequate algorithm notably simulate iteration compete vector jointly simulation construct include reversible jump diffusion birth remainder jump specify reversible countable index great probability estimate modelling include reversible reversible state space problem eq move move achieve deterministic matching ensure match discussion function acceptance probability q respective jacobian acceptance chain reversible balance sufficient existence distribution reverse move transformation use uv acceptance probability move regard run reach move augment model mixing improve increase address reject attempt proposal generate new previously proposal mix reversible propose extend extend give motivating reason term reverse move govern result easy density unknown use uv centering center expand uv proposal new identity acceptance probability become convergence assessment chain parallel diagnostic include chi square kolmogorov hasting model within context particular reversible jump algorithm simulation order propose square kolmogorov kolmogorov compute critical value kolmogorov significance reversible jump selection additional distribution exactly simple eq distribution remain posterior conditional necessary describe correspond simplify respectively simplify posterior distribution equal probability propose ease addition reversible parameter keep remove density ensure dimension matching density simulate reverse move move jacobian u density parameter theoretically choose scale poor acceptance move try result posterior marginal mean posterior jump posterior estimate jump candidate reverse simulate q change move description similar introduce density marginal reverse acceptance move w acceptance move transformation notice probability well discrimination posterior like place posterior instance place increase algorithm probability since offer table great
computer propose decision take account learn duration decrease fusion scheme compare propose decision fusion suitable problem drift track change non describe decision scientific technical grant european grant fp network project multi network risk extreme weather department university email university education city image interpretation foreground background segregation functional adaptive framework vision compound sub yield decision real around zero decision combine fusion projection set describe sub algorithm human feedback fusion video detection oracle security present test active call propose compound yield final decision take real number level sub perform projection projection onto describe sub projection computer decision classifier combine useful difficult especially procedure projection convex collective recognition middle last decade lead large classifier achieve improve employ individually classifier recognition characterize simultaneous individually effective base face pixel texture significantly integrate texture window technique evaluate give texture window article automatic video five video object colored wavelet iv range camera separately together adaptive fusion forest video projection hyperplane monitor case security whenever false whenever occur security decision incorrect update decision security security support tool help attention typical security minute monitor feedback interval run without feedback organize entropy decision section previous part propose entropy update sub section security five detection experimental fusion universal weighted algorithms method uci repository nn training decision compound compose sub algorithm input decision center detect otherwise event type sample region vision pixel incoming vector value algorithm step simplicity rest define input step advantage propose feedback produce incorrect correct iteratively specific subsection scheme ideally decision sub eq q dimensional hyperplane convex next determined project weight hyperplane orthogonal hyperplane close hyperplane formulate minimization solution multiplier next weight hyperplane define plug obtain hyperplane new decision project hyperplane iterate intersection hyperplane rate introduce guarantee set hyperplane track oracle support establish point successful sequentially h yx iw ix x reconstruction compressive sensing everywhere approximate minimization bregman algorithm widely bregman provide globally problem framework cost represent convex bregman start successive projection perform hyperplane step iterative generalize projection onto convex cost problem hyperplane tx yx projection equivalent point e orthogonal norm projection hyperplane functional hyperplane lagrange hyperplane equation hyperplane globally convergent update hyperplane notice orthogonal equation reconstruction positive code fusion error yx ix yx snapshot typical capture camera km fig like segment gray pixel indexing curve instantaneous histogram decision deviation energy currently available decision propose camera different spatio nearby far nearby describe accordance weak intelligence ai ai ai address engineering area move ii colored wavelet region smoothness pixel incoming select combine determined produce false number around camera output sub confident first four describe add review sub deal color discriminative calculate pixel location pixel horizontal use filter horizontal vertical region vector covariance region formula total component covariance half first region predict actual label train actual confusion matrix image software library use posterior library sigmoid large descriptor contain compare result point final pixel threshold issue alarm alarm oracle give decision alarm variation area temporal region weight evolve manner operate mode scan constant adjust classification reduce scheme intel cpu ghz processor test surveillance capture forest weather stable throughout entire happen successfully detect forest also independently california false snapshot present forest ht projection linear decision linearly update assume decision govern minimize cumulative loss normalization indicate n scheme approach compare table hour surveillance video video range km capture good knowledge stage european project mention method forest detection comparable hand fusion reduce alarm feedback describe fig alarm move cause background appear algorithm ht depict bound false alarm video frame km frame rate universal base weight v c video error duration universal weights v v data video contain cloud cause especially frame except low pixel classification scheme compare video weight stage gradually frame number software forest propose method
satisfied design finding solution system compress lasso various monotonicity position denote use equivalent stand notational simplicity review sufficient x differential ty optimality preliminary remark simultaneously c conversely deduce belong p hand index argument imply index sign complete monotonicity important function use condition begin lasso introduce vector definition x subtract together desire submatrix ty establish always uniqueness general position resp resp equivalently remainder proof rely linear programming ensure extreme solution completely extreme basis unique way couple determine x immediate deduce therefore assume uniqueness part give equation uniqueness either assumption equivalent exists decrease since infimum concave singleton thus moreover last assumption surely non divide increase boundedness notice ty norm continuity boundedness two sequence converge problem b uniqueness prove goal fidelity compare penalty vice intuitive tend tend decrease tx immediate consequence interval lemma divide small let sequence converge recall l write union subspace dimension deduce small thus x imply tx moreover hence tx tx interval continuity result partition sign pattern nonempty interval multiply ty tx calculus step increase nothing prove result continuity deduce tx tx continuous moreover tx tx non decrease france penalization crucially fidelity provide general proof well know basic generic denote assume briefly scalar denote notation submatrix whose large discover sufficiently estimator due absolute stem least estimator remain nonzero predictor interested overview relationship concern extension strategy instance uncorrelated several oracle may oracle often emphasize support ahead
cm cm home page http ar intelligence information national increasingly important network widely formalism handle technology sound whenever representative work survey state discuss limitation produce quality efficiency realistic domain store world support refer literature popular reasoning uncertainty statistical calculate model represent assign add abstract domain tuple limitation requirement variable domain mathematical nonetheless restrict variable second usually conditional variable representation semantic pattern proposition represent distribution people influence may figure table row assume mutually n mutually table parameter address exponential storage requirement several simply formalism compactly joint compose numerical structure encode present numerical defines quantify detail graphical encoding distribution network distribution dependency influential last decade graphical wide field present image give address complete exposition field domain area datum random field present spatial random wide biology economic diagnosis heart disease propose discover gene application graphical describe individual fitness recently propose retrieval dependency markov contextual dependency propagation list summarize example reader solution certain cm computer vision example cm object cm cm method biology spatial economic environmental disease biology discover cm search network term dependency modeling dependency support capability highlight graph provide conditional efficient computationally tractable compact graphical exploiting understand semantic give graphical highly task marginal usually marginalization compute conditional posterior prediction sub routine learning task elementary sub exact working structure magnitude fast joint extensive topic popular work open approximate recently return graphical model really knowledge author tool use drive provide automatically large claim approach purely review many core challenge technology independence infer independence representative target art discuss current limitation potential current document discuss relative advantage conclude open domain overview network consist qualitative quantitative denote qualitative know domain distribution quantitative quantify relationship distribution two knowing tell I nothing already know independence dependence undirected independence distinct node random domain encode represent first regular spatial typically spin model mathematical mechanic read conditionally variable conditionally conditionally give correctly represent graph short disjoint conversely map connect map empty map characterization relation separation graph concept basically graph graph distribution say graph necessary condition axiom axiom network omit l p eq disjoint stand axiom valid strictly list axiom represent relationship hold summary represent encode well inference graph undirecte represent acyclic direct case encode introduce predict application concept denote domain influence identical graph strictly markov variable state every axiom strictly axiom section quantify relationship encode address learn quantitative network distribution subgraph whose call maximal clique among edge clique clique clique clique graph assign negative potential measure compatibility configuration assign clique show include clique gibbs normalize product combination system eq compute use general second calculating discuss difficulty historical condition analyze format machine file per assignment solve know computationally challenging technique markov parameter perfect encoding every learn represent desirable densely exact inference intractable method limitation reason less main goal learn underlie evaluate distribution use perform never improve answer fraction segmentation image feature discovery structure motivation correlation discovery assess prediction extent identify medical diagnosis discover certain disease markov usually tune propose require estimation computation combination although possible guarantee optimum result estimation simple ascent sophisticated unfortunately partition across network propose alternative variant another robustness belief propagation avoid scoring need good running stage validation broad learning network approach generally suit inference complete structure mind independence suited goal discovery independence feature since algorithms domain purely tool section structure work complete one base perform accurately high space markov learn search start atomic potential create candidate currently model potential model second potential model compose candidate assume remain parameter candidate evaluate much log end perform search candidate structure automatically induce log however report variation highly optima couple induction force select potential reason involve clique long potential evaluate potential approach markov match nearest drop generalized improve score incorporate loop end slow space intractable evaluate necessary require numerical explain constraint discover semantic constraint input underlie goal encode query variable assertion dependence assertion example independence practice information pearson bayesian test test compute statistical triplet decide instance actually assume true rejected reject statistically significant elegant several independence call strategy generalization previous outline theoretically sound step ix piece together construct learn al bayesian local learn global variable member exist publish algorithm pc independence appear work network series appear ks incremental association parent mb mb parent algorithm child mb important conclusion review cm sound specify mb advance gs sound first phase grow variant cm sound variant gs implement poor variant well mb cm sound trial enhance efficiency sound add candidate speed poor mb sound make topology much compare slow cm sound theory efficient topology information distinguish distinguish mb algorithm distinguish distinguish parent child algorithms learn markov algorithm publish markov grow inference independence appear network particle algorithm framework improve independence algorithm detail independence contrary sometimes complexity sound statistical correct correctly represent correct outcome reliable third algorithm learn sufficiently sampling underlie incorrect good contingency table example must cell count less another disadvantage learned approximation publish literature independence review independence appear review grow independence structure network algorithm adaptation learn show algorithm gs outline h gs maintain initialize empty line contain markov variable line gs association unconditional proceed two dependent conditioning contain false positive positive conditioning prove theoretically correct independence statistical reliable efficient structure disadvantage use cascade incorrect outcome generate incorrect test markov bayesian incremental association prove gs modification sort positive grow reduce phase condition conditioning exploit union axiom disadvantage prove quality gs literature propose network evaluate grow work gs inference theorem reduce learn structure expensive introduce triangle axiom sound infer far eqs every triangle rule triangle rule test independence triangle check independence assertion infer infer store determine visit inference run obtain comparable particle filter structure independence review improve work iteratively select major bayesian select test correctness test distribution converge domain independence case domain respect comparable algorithm triangle unnecessary test outline show reduce dynamically select state knowledge select inference help test require evaluate markov subroutine comparable section independence focus structure statistical reliable deal reliability independence error incorrect test axiom direct axiom depend target power axiom knowledge incorrect propose resolve reason robust call markov network evaluation simple statistical accuracy discovery gs disadvantage formalism first proposition
store cycle constitute active allow another cycle weight update begin iii discuss sampling depend pathway direction region extend sampling initialize phase less flow indicate obtain unfold reconstruct unfold pathway interestingly reaction pathway pathway occur break around occur around pathway pathway pressure path trial path suggest probable close two pathway locate place first encounter pathway even unfold pathway region unfold pathway equal pathway total unfold ensemble histogram unfold ensemble peak peak state histogram ensemble peak histogram pathway unfold ensemble path respectively unfold ensemble e flow clear pathway contact entry unfold pathway construct contact difference subtract contact entry list ensemble contact entry unfold contact map figure characteristic structure figure divide group b along associated agree order unfold rate counter path increase unfold flow flow gradient surface loop apart unfold decrease flow trend comparable average time go amount time also make trajectory list compare flow ns relatively flow increase rate field illustrate enhanced unfold trajectory start unfolding reach usual increase probability observe straightforward trajectory ghz intel processor simulation processor parallelization processor employ trajectory unfold unfold event grain far unfold rate mechanism significant flow effect flow suggest contribute moderately state lack rna prevent unfold transition extremely occur slowly dynamic unfold pathway one secondary break break p loop one parameter sampling pathway precise competition pathway pathway flow currently parallelization increasingly space grow acknowledgment thank award sciences engineering acknowledge resource fusion compute laboratory center laboratory support office contract ac unfold reveal freedom system molecular phase separate solve enhance divide phase integrate region method facilitate parallelization trajectory aside initialization termination enhance suitable drive coarse grain nucleotide rotation dynamic unfold long dynamic unfold range rate pathway biological force induce unfold complement follow evolution distance force optical experimental probe degree molecular dynamic simulation position assumption prove tool elementary condition representative computationally costly employ extreme alternatively computational effort priori prevent enhance sampling without rely relatively uniform different region space degree freedom acceleration convergence trajectory portion space relatively develop paper version improve significant association weight copy motivated segment integrate independently make strategy processor simulate unfold grain nucleotide rna flow single study loop rna enable rapid lead question dynamic compete pathway depend flow compare reversible unfold simulation straightforward coarse grain enhanced need study end overall describe phase simulation parallelization ideally parameter degree relax relatively quantify unfold backward degree freedom separate unfold pathway enable simulation order parameter region uniform region evolve natural average copy region configuration list neighbor choose choose list partition active copy active copy continuous incorporation motivated partition branch ensure set lattice situation study know configuration apply flow configuration principle simulation configuration unnecessary correct sampling start run record configuration serve entry visit configuration unconstraine region employ copy describe visit activate region activate neighbor trajectory activate result directly study account differ require thus region parameter token boundary forward distinction importance dynamic lead rapidly principle similar permit control include detail branch prune weight slow weight large region convergence issue fact space number region procedure accelerate initialization phase interface interface scale weight scheme region total element value nontrivial solution simulation copy require limited communication parallelization implementation high computer parallel array implement global address access side sided communication address storage set entry node region store enable atomic update modification instance prevent global array communication initialize collective rate break cycle end cycle region segment choose trajectory step region per trajectory region list mean computational end rna simulation tractable coarse grain atomic interaction nucleotide rna treat potential potential potential connect pairwise potential namely nm ensure prevent integrate velocity fs construct full coordinate isolate full carry coordinate mass structure coordinate add relax without force add form loop contact model rotation dynamic method particle group cubic cell comprise stream particle update velocity particle random likelihood combination water calculate equation move flow show allow include particle make ml rna particle use periodic boundary cell cubic lattice employ generalized cell extra locate prevent move along cell boundary condition periodic sphere flow direction acceleration cause extension rna direction flow particle box rotation acceleration parameter range fs figure flow ratio
information mathematically validate model insufficient general conclusion stage opt handle exploratory bayes evaluate carlo evaluation selection measure comparison stress repeatedly validity testing setting operate follows produce distribution abc call summary distance tolerance indicator justification tolerance necessarily insufficient sufficient dimension loss information pay computable quantity provide identifiable handle abc arbitrary specific posterior probability tool marginal eq valid naturally choice proceed create parameter model denote straightforward challenge available abc represent choice mutation justification behind abc acceptance proportional probability practice posterior straightforward derive widely e illustration model analyse north initially subsequent area abc genetic belief decade process categorical comparing involve logistic approach widely another illustration popularity availability abc abc smc abc biology include propose smc implementation mc population molecular mc g incorporate principle correspond summary statistic concatenation elimination pseudo model follow motivate base software fundamental posterior say representation q simulated simplification without generality uniform f converge namely versus current abc contain use abc odds bayes method convergent regularity condition bayesian inference insufficient impact indeed contain sufficient namely statistic gibbs field detail sufficient immediate mc approximation discrepancy limit inference therefore abc wrong express answer general arbitrarily grow base example si abc may differ derive bayes possibility theoretical model currently costly must standard numerical gibbs detailed case differ inference bayes derive approximation ratio setting necessary bayes factor choice absence stress validity model choice abc instead abc frequencie inference select suitable complexity likelihood abc mc comment apply software whole validation markov extend rely observation contradiction computation gibbs competition abc converge field allow vector exact field statistic property exponential set sufficient statistic concatenation statistic sufficient compare optimistic complex beyond family happen realistic normal si insufficient loss happen wise bring light huge abc factor formal understanding discrepancy discrepancy impossible expect illustration wise statistic come geometric counter gibbs across give poisson geometric negative binomial geometric bayes show produce nothing produce factor log versus binomial negative binomial replication discrepancy fact increase selection iid sufficient create sufficient mathematically provide choice outside formal structured devise mechanism across detect rather specific abc introduce apart since increase biological study include choice insight study specie g module population importance sampling order discrepancy abc base scenario two content problematic enough order evaluate divergence abc experiment population third scenario first second pseudo individual assume evolve mutation occur increase mutation effective population assume choose particle scenario provide posterior computation reference million simulate provide simulated close allow summary subset reduce collection yes average population allele allele var yes allele population population average sample sample population allele population allele population yes allele dm yes distance dm yes population two two third population simply pseudo experiment individual mutation contrast analysis include high scenario abc analysis experiment evaluate abc decision boundary discrepancy demonstrate opposite conclusion experiment fig first genetic experiment summary experiment validity sampling obviously display across different proper per see si distinction abc provide model conclude evaluate carlo genetic experiment evaluate independent carlo evaluation simulated dataset population two scenario sampling abc extensively involve realistic exception respect sufficient conclusion analyse bayes statistic sufficient situation insufficient rapidly increase worth estimation need produce approximation stage
fill outcome six six player must manner cause player real world performance advantage implement transition ease twice home run basis entry probability home run block every player possible base event represent c represent event play represent th th transition zero calculate calculate game current single calculation represent already therefore whose run column maintain cause run distribution calculate eq subscript nine nine row run end occur apparent heavily datum outcome stop player goal compute team establish need player instead use scoring run use nine production nine index similar ensure distinguish scoring markov becoming score triple scoring somewhat offset matrix getting walk double triple home run markov refer score score way ability optimal write difference omit computer permutation produce team near criterion team high high scoring fourth scoring position second position position low scoring position score scoring four low scoring four seven ten possible show comparison team whether yield run consequently method require near regular determine yield test datum index nine team team article affect significance abundance promise evaluate difficult output consider different team want maximize affect situation play surface play away home eight common situation player average opposite ground ball ball home versus play versus count count ball refer count score first half star player whether play home away home performance home th contingency denote home player home bernoulli success accord binomial respectively transform q contingency express player representation logit represent attempt player situation flat nuisance nuisance effect priori use distribution mean degree reflect lack effect assign construct home variable representative upon home away simulate primarily datum spread give situation shift ten day reveal behind count ahead high opposite point eight home result pattern player individual nine extreme year play helpful exist mention availability visit measure evaluate depend question take attempt exploit whereas bayesian evaluation phenomenon heavily sample statistical limit easy spatial movement decision ball fix reach base shot infer player shoot complete six game shoot chart shoot center shot shot player shot purpose covariate certain apparent reader article lastly success shot location convert angle shoot al shot attempt game explore affect could shot attempt shot attempt column effect select variance gamma proportional ig analyze confidence median time early develop relationship angle shot location shot location divide shot shot often drive net directly shoot location take four shot cell share effect shoot multinomial shot assume multinomial ap jj regression logit coefficient car normally neighbor towards neighbor share realistic establish cell adjacent cell angle smoothing quantifie type normally car mean angle respectively angle distance angle computation stable lie distinguished distance angle sparsity beneficial issue amount location successful shot attempt cell represent behavior covariate cell change team strategy team point production due player player covariate percentage regression smoothed shot draw distribution shot article methodology spatial player shoot et effect shoot selection drawback entirely thereby method affect smoothing grid insufficient cell angle smooth analyze accurately aspect separate poor jensen player ability current metric rating evaluate play player state difficult accurately player ability area insight illustrate roughly player front behind evaluate metric jensen et success fit idea et al playing relationship distance angle home name spatial fit two model ball model direction velocity incorporate plane depend velocity direction measure angle dimensional arc position reason formulation intuitive position almost base base model fit respective remain position rf lf aforementioned fit author fit year separately result total article previous instead location shoot ball responsible fails similarly shoot previous mathematical formulation model play play success outcome success model model distance indicator direction move forward move cumulative equation control directly backward recognize probit regression permit direction angle bernoulli change location velocity move leave right still quantify previously share player position issue draw specific share player position contain player component posterior lastly player wish find contain player position separately unknown sampling sharing across cell share focus shot shot sharing cell et representative rely share extensive safe safe safe seek evaluate number play observation player respective yield surprising standardized constant add player maintain consistency noise safe position second base safe average position safe correlation surprising safe elegant intuitive another pose safe model adjust safe lf ss lie ss positions b adjustment account ss eliminate ability case article statistical equip article cover perform paragraph safe year computationally infeasible another beneficial artificial intelligence artificial intelligence numerous algorithm analysis review explore study intelligence see modelling use programming medium infeasible reason computationally due discretization article seek represent also sampler single infinitely team maximize team team goal number reach either team terminate new fail play team begin achieve receive team achieve number achieve team random depend diverse play maker option attempt field attempt reward minimize net mention state mind goal abuse triple vice lose team state reward chain minus score team start position yielded ball arbitrarily obtain varied attempt make wide recommend pass attempt large enough fourth diverse team goal play goal goal attempt recommend produce agreement employ programming show optimal plan play opponent advantageous policy policy heuristic good reward point even policy result simplify version combine technique intelligence hold policy realistic play different goal player situation model promise representative require validity aside previous article computer possess ability greatly statistical possess elegant elegant property amount computation analysis situation effect base situation precise quantify chain th transition p correspond upon mathematically k x dimension refer period state give chain important paper review difficult sampler generate act generate sample enough say converge random enough say deal general reward maximize state immediate state start long discount state looking often refer bellman achieve bellman determine system value start guess define successive ki intuitively speak function approach way start evaluate policy converge long evaluation article discount provide quick frequently count four base appearance end team score team team consist least appearance pt algorithm people come fan serve fan detail eliminate various idea chain player environmental could dynamic programming control paradigm theory idea theory winner team opponent opponent contribution non scoring team team category player refined reflect retrieval player ball team shoot team retrieve ball player playing seek player make heavily team assess result difficult assess team situation investigate impact team outcome loss team could
gap vanish obeys generated obeys imply entry r ii also condition make give quantity w ready thank geometric hold result bound explore performance match disagreement independently observe run optimization successfully equal underlie cluster vary repeat experiment time trial rescale curve align show indicate one predict control curve fashion run follow figure note parameter choose scale align figure quadratic tradeoff I p compare popular adjacency zero first adjacency generate fashion probability imputation scheme particular scheme consider zero b symmetric reduction graph show guarantee find optimal disagreement clustering succeed miss exist effectiveness method validate application expensive consider deal connection lemma non inequality give auxiliary operator entry operator p tc high indicator ij bernstein provide symmetric appear define symmetric n large constant random symmetric relate entry provide fix random satisfie obey bernstein obtain yield norm bound equal otherwise row u u sx inequality assumption right hand u j complete argument size equal size size approximation hold partial unobserved denote p piece chen xu chen xu observe know edge remain edge want node relatively dense observed yet focus cluster edge cluster use optimization disagreement minimization recover sparse partially performance plant block cluster characterize tradeoff density equal observation undirecte unweighted disjoint connection mean pair know field across engineering search query phrase yahoo represent similar background find well document database focus application facebook accept user pair thus obtain e regime interested mathematical potentially performance relatively provable clustering observe graph additional input require section different pair total type note specify formulation problem bad compare section result aim combinatorial disagreement splitting section represent matrix corresponding cluster sparse correspond return cluster cluster natural strong matrix presence field course vast survey therein scope correlation cluster plant stochastic block partial observation mathematically graph lack define correlation np work relaxation relaxation rounding emphasize focus yield without round enforce triangle inequality relaxation tight constraint program however argue tighter practically since deal graph practically small high complexity cluster plant block assume inter fully recover often directly comparable area provide table cluster density cluster difference needs exact plant soft notation recover except handle partial directly miss probability apply side require condition wise thus restrictive observation show recover nice consider adjacency entry similarity show similarity hierarchical seem structure split cluster tree active control observation know convention pair find define analyze observation study derive algorithm either b never suboptimal present describe already ideally cluster edge clique second augment sdp graph interested produce correspond ideal cluster validity output rank optimal clique column order validity check elementary identical block insight yield disagreement solution valid cluster adjacency graph initial empirical fraction pair develop tailor splitting take observation solution contribution provide find minimize among entry characterize plant describe plant partition observation partition let rank adjacency otherwise fraction across k cluster support recover show condition universal cluster q disagreement sufficient gap observation parameter side condition impose decrease require mention cf demand weak vanishing tradeoff density side mean four time gap consequently treat weak agree handle miss entry easier whose unknown would point capability handle isolated node connect classify identity low row correspond appear theorem consider establish fundamental density gap observation correctly cluster plant partial clustering correctly complexity characterize logarithmic significantly improve use complicated picture know rigorous recovery impossible requirement gap probably rigorous tradeoff unclear regime theorem appendix produce e node minimize feasible valid semidefinite obey disagreement zero hence subject edge relax feasible minimization clustering
induce usually apply co use available proper subset way initially largely machine literature split flexible lead co size hence divide sized also call slice unless state refer random split special construct self classifier co randomly pick partition alternative split co training meanwhile classification section optimal partition heuristic test future accord mit essential ingredient classifier label closely strength coupling classifier determine preserve connection list supplement gaussian mixture indicate ratio separation original quantity preserve q carry substantial fraction contain u indicate quantity feature separation closely relate rate know result loss column write subset see suppose wish affected general classification power vanish I assumption I technical later proof let cholesky triangular matrix possess locally dependent may avoid assume assume eigenvalue permutation assumption induce indicate supplement mainly work closely state gaussian separation mixture center original difference carry nontrivial transformation separation euclidean original separation mahalanobis comparable empirical tree randomly half construct formal justification half provide support aspect comparable classifier meanwhile set tree good result co training conventional reduction redundancy feature become nuisance get clearly redundancy redundancy coupling co carry splitting feature type redundancy assess stanford microarray see http stanford edu er protein cancer er marker express assign scoring represent definite dark positive cancer cell unbalanced assess proportion split first various option image set section relationship train co experiment additional experiment default reflect image er marker occur accord moreover feature derive co derive experiment range far expect substantial distinguish subtle gray ensemble node value suggest package fed rf set report rf find apply score blind set image two evaluate self necessarily grind truth serve test set argue example represent neighbor rate refer theoretically good training rf around twice bayes around sample variation bayes original close pt image patch image conduct patch achieve indicate report score salient detect salient highlight highlight verify highlight pixel oppose identifying highlight score two copy along score reference accuracy evaluate proportion score see assess proportion repeat score consensus issue obtain image avoid range course one desirable automate rf svm rf alternative network set boost boost svm naive adopt idea maximize posterior ix seek detail rf make conduct co correspond combine split combined number example fix design make class carry reduce without interesting training rf consistent sample partition lc rf marker algorithm marker surface additional stanford image marker cd large miss exclude experiment automate refine recent er cell present manner include count statistic spatial regularity heterogeneous image salient score incorporation patch cell type achievable set pixel scoring patch insight co slice whole classification power coupling co train natural lie population seek evaluate potential marker may score poorly manual score hundred hour score marker manual provide type appear subsequent statistical determine clinical potential marker regard score two pilot reveal observer accuracy define reference image inter observer attribute include lack lack surely upon highlight inherent provide clarity confidence marker localization er marker characterize cell ease marker software request associate make project acknowledgment thank anonymous constructive suggestion proof theorem separation skip skip grant gm microarray technology medium throughput diagnostic study grow manual limitation throughput greatly variability expense co occurrence quantifying phenotype base regularity summarize inter pixel relationship train tune salient pixel contribute score via training marker training train high redundancy flexible scoring evaluate clarity study er marker performance describe throughput technology assessment group image display indicate score begin distance rectangular core contain core block micro array capture display grid fashion separate probe sample rapid dna rna protein expression large number clinical remain common method localization expression quantification scoring validation assessment target clinical panel image construction automate large limit throughput intensive intensity density provide quantification problem inconsistent highlight thus grow rigorous actually without consistent automate scoring sophisticated focus commonly system rapid device newly scoring rely various several reaction detect typically marker nuclear dedicated person propose co pattern texture pattern rely color filter segmentation base expert patch categorization interpretability note design diagnosis tool clinical thousand require score prohibitive automate essential purpose concern necessarily cost time adopt substantially explore natural split readily available fairly support theory slice carry condition organization remainder section follow insight address lack easily quantify cf template retain idea image require common scoring selection expert biological feature involve substantial gain interpretability machine knowledge select benefit manual difficulty setting beyond b svm sigmoid report lc rf svm boost use underlie classifier classifier dimensional rf performance argue rf tree instability create bag node randomly feature index measure bag proportion class rf pruning example fall tree vote class vote rank index alternative permutation base valuable property salient see pixel rf variable since entry count associate pixel pixel associate entry position pixel treat pixel salient important g rf pixel rf salient quick work manner salient pixel
approximate family switch fractional help level sensible evaluation likelihood integration require gauss quadrature computational moment function evaluation mode cavity modal case mode integration limit around hyperparameter monitoring require marginal numerically fraction first cavity remain update care entry become due outlier present suitable option cholesky update laplace parallel site decomposition definite site negative cholesky ep step evaluate uncertainty outlier increase heavily hyperparameter handle regular map initialize hyperparameter small result outlier unable explain induce plausible outlier mode case flexible enough divergence fractional update discuss property increase posterior uncertainty input region priori estimate insight negative laplace I I I regular observation increase far clearly increase uncertainty situation arise hyperparameter many covariance matrix positive posterior ep behave disagreement cavity observation modal moment posterior decrease sequential run smoothly remain site multimodal standard outlier provide narrow small apart significant posterior uncertainty input neighborhood smoothness govern equally much neither label contrary site correspond posterior despite outlier update become problem sufficiently example force unimodal converge case hyperparameter panel increase panel hyperparameter separately calculate analytically draw gaussian unimodal ep estimate cover posterior laplace one nonlinearity observation close strong nonlinearity much strong near approximation localize standard ep mode example modal short py py panel correspond fractional update site discuss illustrate dimensional site update ep start dimensional marginal contour fit around near site get negative expand toward outli site long update cavity precision happen support current posterior reduce ep cavity site first posterior center couple loop negative cavity variance problematic site get small need expand cavity variance keep large uncertainty small site hand induce may algorithm update decrease require sequential convergence fraction include correspond fractional site initialization still modal less distribution consecutive fractional marginal variance site large regular ep small keep divergence measure allow localize fractional ep hyperparameter unimodal panel small uncertainty put objective illustrate hyperparameter marginal ep mark panel site keep parameter value iteration negative cavity become also decrease nearby fluctuation prior amount gradually still compare cavity double loop loop proceed loop ep slow iteration get parameter ep attain much switch drift visible site estimate parameter marginal may slightly inaccurate implicit gradient evaluation ep without apply localize see positive update show approximation ep fractional ep approximate fractional mae region degree contour log contour double loop converge hyperparameter sequential converge marked dot hyperparameter mark ep hyperparameter mark example problematic ep previously ambiguity area problematic large unable strongly region unimodal clear artificial hyperparameter sequential ep fail initialize evaluate lie area visual accurate contour standard uncertainty see fractional property match hyperparameter fractional except fractional ep run dot coordinate la vb four use involve create five irrelevant variable generate house price prediction concrete predict quality distribution fx method fold approximation mcmc predictive deviation laplace ep plotted numerically integrate statistic clear see indicate derive somewhat approximate density la deviation slightly together method hyperparameter depend optimize section mcmc measure mean predictive test variable datum fold validation compressive large bootstrap described ga ga ep vb optimize density em variational augment challenging issue la ep vb vb robust alternative outlier tail ep ep robustness estimation linearly log run optimization hyperparameter approximate integration require speed target variable scale zero mean freedom initialize magnitude optimization initialization direct hmc mixture sample sm combine slice result well give visual calculate autocorrelation burn period exclude begin remain form mcmc central credible observation ga reference student ep scale sm number method stand integration comparison sm credible show sm figure four illustrate figure densitie sm difference student sm sm compressive significantly difference concrete ep actually sm explanation wrong possibility proportion observation pairwise comparison reveal ep ep vb except strength compare perform la vb compressive vb prove challenge sensitive initialization hyperparameter often likely style la vb integration objective whereas la bad la compressive vb significantly well decrease vb compressive pairwise comparison integration significantly compressive strength never significantly give ep marginal line mae student ga concrete give perform hyperparameter optimization ep others compressive sm ep concrete otherwise difference la vb ga ep sm max c c require posterior hyperparameter prediction cpu give time fast compare baseline ga ep slow double iteration achieve difficult clearly ep repeat value slow compare adaptation la vb optimize guess hyperparameter average set exclude mcmc mcmc clearly mcmc randomly show much research ep accurate application problematic utilize difficult mixture modification fractional well loop good ep interesting give interpretation negative site increase posterior site turn problematic update even hyperparameter optimize example ep unless converge discard hyperparameter situation relate cause away unimodal note moderately ep work fine multimodal globally unimodal true underlie think approximation false uncertainty prefer also design approximation ep also cavity loo latent site loo site approximation cavity reason fractional student omit robust mixture gaussians moment analytically adjust contain gaussian tail adjust increase infer estimate turn consistent base modification improve robustness ep concave likelihood ep likelihood package consider regression observation student several ep find problematic contain site student illustrate standard fail algorithm occur type posteriori ep may robust implementation primarily ep update utilize matching loop compare laplace variational bayes chain propagation observation include outlier strongly failure absence robust require study extensively describe observation lead observation however rejection outlier posterior rest outlier becomes give tend state combine normal prior robustness ny negligible student heavily commonly influence matter far outli rejection reject locally nearby already multimodal illustrate adopt student gp heavy tailed distribution degree freedom robust model laplace student model analytically use see show method burden uncertainty assume function variational vb un low independently detail comparison relate describe student freedom divergence kl describe vb special kl extensive comparison performance laplace kl ep choice since kl ep establish student one discuss ep update ep loop primarily update utilize moment loop adaptively size stationary solution implementation implementation laplace approximation carlo three world zero q control notational simplicity variable function produce semi covariance magnitude correlation increase analytical approximate limitation robustness marginal p analytically method approximate review give short description well vb way problem draw represent posterior numerically implement markov student gibbs representation variance slice sampling hybrid sampler may reduce implicit model latent give negative diagonal py I inference hyperparameter laplace marginal hyperparameter gradient log scheme laplace essentially name gaussian integrated field thus later implementation posterior approximation ensure moment convergence share moment bound consequently stationary small free choose first concave maximization concave equivalently substitution constraint respect proper fourth student site moment match cavity keep positive cavity distribution regard loo ep simplify moment evaluation robustness family flexible due cavity f site moment standard recover ep cavity another divergence compare force mass consequence fractional ep tend minimize represent overall reverse kl divergence fractional fractional benefit problem approximate measure fractional update help cavity cavity decrease cavity make numerically cavity become combine
near sequence value part sense interest similar observe order start segment historical price asset pattern scale asset window pattern profile see look euclidean small increase kt construct appropriately ahead asset neighborhood pattern order q adjust represent term value inverse asset process expect worth character sense hyperparameter fit information indicate change provide non maximize log indicate event occur definition asset return occur way horizon contain scenario return price probability price satisfy q therefore evaluate estimate conditional respective value eq expect worse design informative take asset portfolio thus change return base price arithmetic give divide integral price asset quantile normalization dx evaluate get evaluate calculate index recommendation establish currently quantification condition examine daily year value index calculate window e year consider acceptable accord compare financial market difficult agent permit correctly fluctuation thus exchange increase worth would apart difficulty combine access volume computer software advantageous simplification view assume asset predictive center asset without necessity assumption evolution return beneficial ease include benefit evolve similar initial specify evaluate correspond though present correspond whose fall evaluation useful design aspect wish sim de measure financial two asset evolve piecewise process supervision result indicate satisfactory tendency assume series asset density point practice really
meta passive learning cm passive simplifying arrive extra constant vc learnable rate learnable directly claim aside disagreement dependent directly prove kind meta theorem simple possibility bound request meta unlabele disagreement budget formalize lemma note unbounded bf relevance insight arrive follow show generally constant without label appendix exist achieve cm interval improvement disagreement coefficient like improvement passive disagreement subsection least little strictly general subsection active meta analogous disagreement meta dramatically meta meta ms mx mt x k mx v ms mx mt l l variety way long converge fast true result definition meta however disagreement focus conditionally I feed break stage rather share stage chernoff depend complexity wish architecture sometimes complexity corollary replace return termination within value analogous label infer update unlike mechanism motivated contain arbitrarily classifier discussion label step potentially informative sense meta key obtaining improvement label guarantee overall build intuition behavior meta toy uniform ignore fact effectiveness threshold nonzero interval round meta essentially identical meta grow threshold width period I find meta must look round improvement sufficiently process label correctly request reach furthermore circumstance appendix l label interval single nonzero disagreement improve passive round meta provide improvement round reasoning sufficiently label separate consistent argument pf relevant go scenario study analyze number target difference subsection quantity disagreement improvement achievable meta analogous disagreement characterize disagreement core integer define h c far key role concept classifier q disagreement represent generalization disagreement coefficient measure measure f f k additionally target union implication label corollary asymptotic always might define restrict never discuss extension vc mention restriction expense complicated theorem disagreement coefficient describe variety behavior toy example interval see x bf bf consider see f union interval width j purpose h h interval label interval nothing interval nothing interval label element minimal contain bf z z ks j bf bf rr latter point union jk z z ji inspection reveal exact see coefficient particular course quantity able describe family beyond toy fortunately disagreement much euclidean linear c b learnable exponential establish similar argument px px c hx hx bf rx bf pc pz z surface li dt regularize satisfie indicate mass least mass xx bf rx bf r hx yx aforementioned spherical reveal height surface spherical every bf hx little reveal height mass r r bf bf p rr albeit follow reasoning use opposite example advantage indeed disagreement subsection characterize magnitude passive specifically immediate corollary proof vc achieve complexity satisfie passive distribution achieve complexity passive plugging simplify arrive vc c learnable learnable first follow meta actually closure obtain label direct cut return original algorithm obtains bind passive see possible separate classifier return meta let corollary significant improvement represent good gain achievable answer meta corollary strength modification might potentially improve possibility theorem careful analysis instance meta f tight see refined complexity disagreement additionally significant constant quite careful meta possibly exponential learnable countable vc element except instead infinite path one depth path countable slightly replace eq quantity term sometimes possibility improvement ms likewise replace v state meta remain require modification proof gain theoretical arbitrarily room arbitrary summing constant possibly sum furthermore prevent set function large improvement active capable passive access unlabele exception major issue inherent achieve unlabele practical concern unlabeled data efficiency disagreement base method meta introduce trade though trade factor definition indeed desire actual theorem appendix unlabele follow set might ps implicitly ps ms x appropriately modification dramatically example ps costly unlike serious meta achieve corollary consistent classifier returns classifier replace return become run invert clear guarantee case absence meta large obtain evaluation remain corollary previous learn move make interesting perspective significantly version passive complexity nontrivial conjecture regard need focus develop technique noise disagreement active generalize meta meta sake disagreement active subsection develop agnostic relate disagreement relate meta represent disagreement agnostic several result sense less elegant potentially promising direction subsection interesting direction subsection joint hx xy xy xy py xx xy xy context denote xy h xy h ph ph simply c g hx equivalence compact recall totally closed monotone close nonempty learn ix ix xy define request time sequentially request joint specifically definition label em may classifier rate particularly interested passive generalize analogously definition learning label em agnostic set passive meta agnostic active xy cc agnostic vc class agnostic even exist passive classifier nh n xy behave expect result include agnostic threshold expect case leave open family passive agnostic interesting agnostic reasonable excess family passive help value ignore redundant request discussion passive algorithm fact vc active meta conjecture vast grow literature complexity risk minimization empirical effective conjecture remain remainder view label passive subsection interested state active learn explicit initially purely nontrivial depend passive passive subsection commonly use property margin specifically formal p depth passive literature passive imply variety case satisfy p study depth wherein xy xy xy way space realize surface depend surface surface assume xy xy density surface quickly work interpretation study passive risk minimization return classifier mistake label satisfy achieve work nontrivial xy xy xy give infimum complexity learn passive refined active improvement satisfy condition disagreement active analyze generalize specific refined generalize label since entail hope purely nontrivial xy xy xy satisfy value range complexity achievable follow subsection establish disagreement agnostic present originally analyze label agnostic beyond disagreement disagreement analogous meta replace passive also vc namely know find offer general complexity achievable agnostic exhibit dependence coefficient role discussion vc unlike eliminate classifier merely mistake good take mistake underlie discuss present empirical excess learn literature disagreement slightly consider representative disagreement algorithm concern disagreement method label budget mm request generalization passive inspire application learn dependent follow rademacher independent operate label disagreement space mistake since infer establish implicitly round loop update step include somewhat behavior vc suppose condition parameter output least essentially simplify idea also immediately implication concern vc class dependent em result algebra achieve require procedure focus move beyond disagreement active theorem result art label complexity advance understanding capability active next improvement natural meta benefit label improvement passive whether meta agnostic theorem whether f modification analogous see often compare represent active toward algorithm input mx l v I k estimator usual rademacher principle meta proceed repeatedly label size significantly agree infer rate remain equal difference empirical passive literature bound excess eliminate situation subtle principle case space explicitly step eliminate neither classifier meta meta exist version space specifically label active exist active f extension perhaps generalize possibly multiclass structured label particularly problem scenario substantial improvement structure claim passive preprocessing clear instance specific bias semi splitting index interesting characterize number request unlabele active algorithm really adjust example aside many would try unlabeled example reflect label example behavior increase label interesting agnostic exhibit trade concern agnostic vc empirical risk algorithm agnostic positively capability active passive well factor classifier h hx bf repeatedly define l passive achieve meta achieve satisfy p bf n argument end bf chernoff n ng nf imply label choose term identically distribute algorithm pl probability vc p pf nontrivial break essential event f claim note return necessarily decrease define nonempty return minimal completeness ix fx meta know nontrivial since happen small theorem claim definition lemma proof relate meta slightly parameterize state vc behavior meta suffice passive theorem importantly relaxation subsequent namely lemma substantially general version eq convenience convention classifiers iff iff x p ks ks ks ks rate hx ss adopt short wherein everywhere almost variety way definition possibility though access unlabeled partition preprocesse ix statement hold state serve partition three eq indicator every indicator whether define definition eq remain q certainly definition interesting property general trade present instance could probability would possibly drawback alternative priori many expect k h bound might sequence address modification elegant definition follow primary proof set aside request label return end support regard serve core probability zero difference lemma primarily extend basic idea surrogate sub label request lemma subroutine request output least n em completeness total request sum size batch label px large jx x pm en union must h px jx one hoeffding en describe limit see probability establish decrease except zero lemma large except zero p fm hold fix infinite h ix bf rs rs follow ps mf mf rr dominate first event probability note establishe claim event take claim monotonicity law imply event monotonicity extend event give implie note specific implication take bf latter bf eq complete final bf k f bf h h right left equal examine denominator numerator fact side let let random implie equal fx particular result imply monotonicity bf bf h v otherwise inequality claim claim continuity measure generally dependent imply fs fs line bounding chernoff imply since cm em integer define bf k bf bf bf bf bf eq extend constant event q clearly hx bf ks bf hold take h imply lemma imply satisfy eq second kx claim lemma bf bf x event convention proceed chernoff bounding technique markov independent bind total h claim define q bind imply remain first term k convention eq eq monotonicity q establish iv iv iii iv iv g q imply inequality namely inequality combine yield represents set meta specify integer q suppose r begin u w independent chernoff law total union event establish union bind establish run lemma imply n imply f term note l I furthermore always meta infer label eq q f pl know p p establish arbitrary meta immediately lemma arbitrary passive certainly every implication modification replace step kk imply appendix lemma expect request meta unlabeled request mass disagreement although lemma insight disagreement coefficient include throughout fix notational proof hx I ct eq represent index unlabeled request aforementione em prove lemma disagreement unbounded bf bf request make assume likewise implication region bf bf bf bf bf xx mx bf rx bf rx cx equality independence x inequality indicate bf bf inductive unlike lemma mn cx mn last mx mn mn mn x mn inductive note lf lm f infinite ir bf r run upon reach continuity bf bf monotonicity union imply lemma bf bf bf ia bf bf bf bf bf achieve vc particular x h f result replace meta specific take definition appendix lemma previous throughout reference final implicitly continue denote fx repeatedly must consistent meta quantity quantity represent upon reach round end correspond event bf hold lemma monotonicity imply claim also principle hold constraint redundant decide request dependent f f iii f kk kn k last simplify iv k kx km total chernoff imply inequality thus nk k nk probability sum redundancy establish label obtain meta budget data ii em n n ii nk serve base consider value meta induction imply ii choose sufficiently budget exist dependent iii iii f iii e iii event iii truncation f f iii iii eq plug iii thus f ni f fx w n chernoff total ii n fm n union imply sum trivially hold ii ed run passive iii v recall l ff classifier return meta f ii iii ed ed p f l unconditional meta take agnostic show achieve algorithm xy nontrivial xy specifically xy pa characterize xy classifier furthermore p xy exist z q z px h z z z h px hx fact label xy xy xy xy xy pn n p xy achieve label complexity xy establish correspond bind active prove proceed estimate toward useful nonempty sequence first let independent internal proceed bar p combine construct hypothesis test define ib ib p b p ib ip mass imply eq establish nonempty range random independent define ip b side ready result set achieve active learning xy reduce task estimate iid take request already repeat request label give independent return run return context denote z z sequence value k xy xy essentially adapt section fix joint marginal complexity purpose suppose satisfie furthermore continue appendix though proof general threshold replace sequence point sequence run internal denote power additionally establish concern definition define purpose e rademacher excess lemma derivation minor follow borel purpose budget apply independent rademacher I imply claim union existence probability n confidence event satisfy f true serve suppose condition let integer note bf ii induction upon reach claim lemma inductive since reach every ii last combine inductive toward end recall upon h principle budget I fm recall fx fx bf total request constraint event inequality invariant event n fx bf iw q toward end h imply expression chernoff law union exist j ii nj satisfied combine c finally ready break tie remain thus update remove minimizer exist budget confidence fact statement consider passive method union xy xy xy xy xy preprocesse un un shift reference label un u let examine hoeffding least xy satisfie noise xy p xy n xy p xy acknowledgment grateful discussion study classifier vc class learn asymptotically strictly nontrivial characterization magnitude generalization also presence guarantee strict improvement selective sequential design rapid growth datum source rapid extract bottleneck annotation accord instance straightforward collect training significant effort way example refer label expert emphasis design however go beyond allow example label select interactive designing effort toward informative redundancy establish provide significant practical term accuracy understand capability yet sure meet characterize active behave yet identify principle active design design linear decision assumption design passive learning improvement well magnitude improvement toward capability additionally motivate practical concern date decade behave understand hope equivalent amount discover algorithm particular leverage vast passive extent design desirable active algorithm prove variety common even namely improve guarantee passive obvious dominate exist test heuristic active active algorithm passive subroutine passive execution active rigorous look develop reduction study passive general transform significantly fewer result active compare label passive reduction noisy find problem capable explore generality entirely design active complexity quantity algorithm supervise protocol characterize teacher access learn query human expert question nature query simulator protocol type sequential common model set allowed pool request observe label algorithm another unlabeled pool request continue sequentially intend generalize behavior collection label objective return approximate true future previously hope example achieve accuracy label passive many modern learning unlabele abundance annotation pool set labeling examine website survey application discussion multiclass indicate advantage active primarily interested request understand topic largely year progress advance group category know linear decision tree well know pac passive instance strategy request label label example determine phenomenon think line exhibit general strategy elegant refer meta behind unlabeled consistent request inspire disagreement sometimes active never example label attempt seek particularly informative example disagreement far ranking disagreement quite obviously strategy setting know random unlabeled analyze achieve information gain gain exponentially small threshold certain far interesting implication bayesian characterize complexity achievable active learn tight concept analysis match generalization higher homogeneous near distribution notion beyond simple employ disagreement describe pair regardless provide elegant request understand improvement active general particular clearly illustrate quantity vary target issue example come slightly perspective active sufficient natural generalization learn set quantity relate easy calculate though opposite seem next progress toward label disagreement algorithm characterization complexity original strategy state disagreement direct discuss disagreement substantial detail base active sometimes suboptimal disagreement large analysis disagreement coefficient surprisingly disagreement coefficient practical fairly calculate geometric relatively smooth discuss use label learn noisy reason ease work label favor disagreement coefficient label wang significant paper focus extend generalize maintain make disagreement useful particular passive type namely adaptively label request need achieve outperform learn index disagreement certain respective reflect label analysis note algorithm simply label request find good classifier result vanish concept vc passive learn superior target certainly advance active simplify elaborate statement active rather strong remove replace calculation dependent whether dependence claim passive concept theoretically active practically direct access available complexity particularly well passive threshold many leave open characterize improvement achievable even achievable type leave far involve label pick progress question addition advance method misspecification topic root agnostic know perfect set linear objective whose bad well strictly agnostic hope achieve passive agnostic progress publication disagreement base essentially effectiveness show uniform improve complexity learn strategy find extended bound world general achieve express disagreement result hold arbitrary vc dimension case passive disagreement problem soon strategy agnostic set new disagreement base reason establish disagreement coefficient b improve dependence coefficient set computational algorithm develop capable essentially loss large class loss term loss encouraging reflect well constant passive fact label bind depend improvement detailed improvement passive study refined noise well passive special restriction improve passive type threshold class result show agnostic also improvement complexity far bound disagreement apply arbitrary type classifier reflect improvement disagreement coefficient wang later noise identify weak noise improvement exponential certain simplify principle class reference publish learn typically complexitie verification might whether might significantly well complexity label build extend contribution meta active algorithm label nontrivial target exist characterize lead new type algorithm set issue mention disagreement coefficient achievable active learning case small result new improvement passive achievable case include publish literature purpose active base aforementioned complexity achievable agnostic often previously formulate type throughout protocol procedure active superior review precisely characterize scenario disagreement learn desire improvement beyond disagreement set procedure section begin bound disagreement disagreement coefficient somewhat set passive define disagreement procedure us improvement extend allow noise term proceed learn term generalization disagreement coefficient present general concern algorithm conclude investigation suppose borel space usual borel algebra though generalize measurable classifier characterized refer simplify focus counting sequence special denote px informally access access unlabeled example request select request observe integer return algorithm attempt give study sufficient active label distribution originally dependence complexity achievable illustrate achieve accuracy note sufficiently label active come problem need small label label rather number label algorithm request application active label extreme label target equal true mean notion play concept generally design intend relate notion complexity learning include verification interested sequence label passive learning internal passive passive unless state complexity take strictly complexity passive active passive complexity guarantee error also discuss guarantee formulate asymptotic single probability guarantee extract much explicitly employ notation include asymptotic always consider form mean notation follow passive meta passive active call meta passive establish universal exist vc bit explanation interpret strongly little seek learning function toward target trivial passive wish implication strictly nontrivial bind label complexity passive algorithm f serve purpose framework require trivial scenario focus toward nontrivial scenario scenario truly proof give slightly broad nontrivial discusse regard scenario passive algorithm finite intend framework trick aggregate always allow analyst reasonable passive expect c function interpret particular measurable hold make mention notation throughout convenient discussion hx label sequence ph p ph b ph x x h h hx equivalently h problem space space throughout repeatedly canonical although toy important type problem fundamental type example drive understand complexity mind provide passive universal passive sort increase request median repeat update x request least integer request procedure maintain invariant likewise equal active algorithm label example request achieve label simple vc subtle problem b universal request example sequence immediately return constant point nx mi jx u repeat update accordingly request accordingly let label request phase fx otherwise maintain every sequence mean pn n ph bring phenomena analysis learning much strong target fundamental avoid analysis active learn highlight explore verify observe rate base observable essentially issue example handle initial value subtle consider x em sometimes issue effective search perhaps identify point identify perhaps choose request disagreement candidate canonical discuss disagreement subset candidate request request would information substantial attention natural way noise misspecification set disagreement frequently share discuss complexity achievable active let type representative disagreement base achieve passive disagreement property sophisticated complexity passive guarantee vice versa use request label isolate aspect active involve improvement region disagreement formally specify meta passive budget mm request ny mx meta estimator divide stage reduce sample might observed indeed albeit constant property passive conditional choice example second stage trick employ explain implie behave classifier simplify ignore address second phase algorithm request label mass l unconditional complexity return meta f label universal space threshold contrast example one h x h x l label simple passive expect meta passive meta scenario subsection prove meta algorithm tie particular refer reason certain em clear disagreement request disagreement core therefore build perhaps bf h bf z z decision slightly r b disagreement illustrate boundary geometric disagreement core label region derive sufficient condition behavior intuitive request refine interesting reason core correspond boundary intuitively happen surface could probability nearby rest turn case important nearby tie disagreement scenario disagreement insufficient sophisticated overcome along corresponding refinement definition disagreement case meta large expect request around core thus expect request passive request become focused fraction passive factor intuition formalize class meta universal f idea correct also grow classic shrink converge strict improvement passive fed hand unlikely label example represent sketch prove meta achieve strict improvement passive nontrivial f disagreement difficult always hold countable effort class dimension see probability disagreement core particular studied aforementione aside nontrivial disagreement discussion scenario pf thesis wang disagreement core always whether simple could passive vc unlike exist vc handle disagreement request meta become therefore grow speak meta algorithm scenario request specifically meta request prevent improve passive restrict surely large split disagreement fx accurately x perhaps subset long need request agreement label request request shrink asymptotically improvement passive meta describe already improvement address nonzero target address x nearly f repeat argument time infimum agreement surely thus label disagreement repeat many need shrink disagreement obtain improvement passive simply x continue clear shrink determine iteration sufficient shrink probability maintain possibility result kind comparison multiple majority infer generally estimate certain probability detail implementation later algorithm reasoning refer meta state passive budget classifier mm request nh l mt l n jk en jx kx request set meta estimate variety way universal seem appropriate respective state take request batch one third subroutine unlabele chernoff redundant mechanism motivated reasoning check whether new disagreement increase vote request mention property since intersect vote correctly v reasoning dependent
coefficient see conjecture open influence generalization boolean function eq prove conjecture conjecture tight answer consider weak state almost weight concentrated entropy extreme fourier concentrate level fourier fourier entropy bound suggest could decay proof entropy use level discuss section upper bind conjecture influence formulate conjecture conjecture show contain copy complete induce assume conjecture base let concatenation translate q first relate influence second relate fouri coefficient third middle norm describe relation proposition depend since ready equation apply conjecture q random subgraph threshold choose value range graph property show order simplify computation copy nice hold range choose otherwise constant upper necessary union portion fourier concentrate show correspond follow equation consider copy fourier induced measure contribution include contribution equal negligible combining get proof note inverse polynomially conjecture measure boolean generalize universal measure show cube bound implie since preserve equation polynomially statement claim differ conjecture constant factor section discrete study boolean fourier simplify replace character thus simply fouri weight concentrate particular induction assume assertion let fourier degree concentrate low level let discrete th fourier expansion definition thus fourier possibly work strong eq f beyond fouri weight concentrate appropriate use discrete inequality decay assertion important allow observed conjecture q f f prove conjecture power proving deduce sufficient weak enhance decay concentrate around expectation become almost inverse slow majority weak enhance reach conclude mention easily prove deal cube accord first n conclude strong note boolean indeed conjecture g show fourier concentrate one hope fourier resemble theorem conjecture strong conjecture boolean influence fourier contribute grateful recent theorem notation conjecture conjecture seek relate measure fourier boolean claim generalize biased product
first random let suppose follow see pay know situation situation price tp pp past lag additionally dependence gram depend thought complement measure increase minimize reach offset correspondingly increase expect minor interpretation property p zero lag convenience p lag lag significant lag analogously c c c facilitate introduce notation theoretical normality square multiply without confusion lag correspond coefficient lag coefficient consistency appendix regressor consistent associate regressor tend consistency selection assume diagonal assume consistent another challenging cholesky triangular upper entry transform select show affect entry entry set show entry much small correlation weak grateful participant comment would stay california berkeley thank ba proof tp intermediate result show tp random p upon martingale limit p dominate probability local minimizer complete proof minimizer kkt th employ central term equation analogously proof lemma consistent taylor sufficiently normality indicate stand suffice show complete section remark nn economic financial forecasting economic financial lead improvement forecast dynamic lag mixture serial dynamic spatial dependence lag appeal study auto regression estimate treat variable lag different lag lag select lag consequence series consider propose tuning scheme illustrate superiority regularization c e economic broadly speak forecast structural economic theory hence fall year forecast forecasting exploit time series little theory work continue improve univariate series analyze ar moving move autoregressive var many challenging lag omit mistake create omit variable consequence structural forecasting point reaction price forward shrinkage auto improve add additional illustrate example forecast primarily multivariate model e jump switch mostly series bank fed basis adjustment policy heavily variable forecast interaction finance come information variation aggregate default reflect general economic find event affect affect finance finance also example analysis finance grow trend financial series forecast impulse response structural see economic finance economic behavioral finance one imaging fmri brain identify finance dynamic volatility risk pricing field large analysis portfolio management price international finance among low multivariate method might relevant temporal suboptimal forecast rate forecast dynamic deal macro cover production order exchange span summary come correlation temporal moderate lag address simultaneously appeal problem factor datum assumption dynamic represent factor dimensional series well principal auto natural factor procedure dimension high dimensional series variable impulse correspond feasible since var involve lag primarily augment var recent theory regression analysis large dynamic therefore proceed bayesian view grouping study subsection coefficient autoregressive shrinkage relative article regression large regression selection theoretical exist meet practice contradict regularization later propose serial correlation dependence together dimensionality moderate enable consistency scenario reveal selection carry lag lag lag additionally lag repeat procedure lag varied stay drive include vast financial weight seem restrictive base package angle regression package work prior organize next var estimation method motivating outperform contain conclude proof appendix discuss optimize provide lag b p ty tu tu ti j justify variable standardize also carry diagonal discuss see large moderate macro effective much original attain representation parameter avoid fitting follow mild try balance ccc cc loss generality matrix say dynamic lag diagonal term different pattern w associate variable primarily practical choose unit preferable specifically group type penalty impose lag lag respectively generic control lag less lag large assign lag lag diagonal one dynamic drive vice versa reflect different lag small get belief amount distant lag amount shrinkage towards one importance distant lag one decrease however datum drive correspondingly especially multiply side p b fit already lasso type pose realistic economic one hyperparameter control relative lag lag lag lag meet practice correspondingly forecasting average instead suboptimal forecast lag always lag general row implicitly variable group estimate column consider th column use lag lag idea subsection emphasize call estimate b lag lag tune forecasting get disadvantage term drop common panel financial rate time r price index segment segment generality index segment also denote situation lag lag lag segment correspond ideas ii segment b I neighboring view grouping penalization selection hyper grouping scheme real forecast denote tt ahead equation ahead forecast compute spirit forecast forecast relative benchmark walk parameter observation word desire magnitude fit perform grid minimize j group computational first loose grid perform grid use angle regression package provide www stanford edu experience motivate jj w bx b omit generate corresponding solve grouping b motivated one penalty mix iterate penalty standard generate w u j output j minimize step iterate multiple package www stanford edu matlab parsimonious exactly time retain use thus implement ordinary
speak show occur positive suggest generic establish stability markovian expense result expand projection stay surely inter dependency various reader fundamental theorem establish abstract lyapunov satisfying set assumption speak instability infinity point focus noise condition trade solution ergodicity proposition prop final prop smoothly random step comment stability one independent metropolis hasting ce existence subsequent adequate define inspire due page many see setting boundary unbounded contain line twice continuously differentiable subset constant family random whenever w I constant satisfy I introduce lyapunov modifying derivative certain extreme present induce spurious boundary notice field lyapunov projection assume satisfy continuity involve role control ergodicity hereafter relate growth ergodicity establish topic h eq trivially eq introduce infinity enough whenever respectively remains prove eq order achieve deduce x since condition imply w imply conclude taylor pt c since note lyapunov condition quantify drift lyapunov assume hold hold q tail trajectory eventually contain visit infinitely often thank hold shall suppose contrary condition expectation leave hand side necessarily almost surely contradiction unless go na imply claim condition time w I imply conclude provide stability ensure abstract condition expectation poisson require shall verify geometrically case old allow consider old continuity continuity recover c vx call comment relevance expand projection choice must condition efficiency grow slow establish convergence choice constant property easy optimality state simplify decide growth quantity power whereas small weak condition inter proposition admit old continuity appendix constant projection denote different value sum x p last summing term g w turn sufficient notice cv x condition jensen imply yield term hessian condition w consequently x condition c last independence I I condition respectively variable independent term observe term right side assumption side index scenario geometrically ergodic satisfied numerous chain practical regularity see norm kernel norm reader exist evident poisson condition bind clearly establish lemma vx vx vx p vx iteratively ergodicity rate hold finally state condition simultaneous drift condition exponential decay contour irreducible invariant borel set vx x depending follow r proof practically interesting possibly old natural quantitative manner applicable old continuity exist commonly encounter slightly handle start p f f r write p f hold r kf k p provide verify constant hold I p h independence jensen sequence establish proposition imply observe h condition old continuity establish condition involve enforce independent clear exposition power practice period induce manner condition hold condition hold poisson equation I v condition uniformly satisfied I imply imply decay otherwise converge context monte stability stochastic process expand finitely often typically stability convergence strict neighbourhood applicability often lyapunov stability practical lyapunov function yield close possible lyapunov establish formulate convenience set field exist differentiable exist inner h di without sequence unchanged theorem contain finitely projection sake detailed establish noise size sum poisson practice either size must ensure z maximum likelihood hasting order intensity employ measurable give taking assume markov value latent likelihood iteratively open application require one log precisely focus use mcmc particle filter particularly suit state space denote output value output consist variable see hereafter recover function n n k introduce function trajectory latent return complete parameter statistic choose stand computed particle rewrite f whenever integral define note possible within stochastic static diffusion theoretical essentially thing stability apart expand projection random sequence allow consider intensity determine autoregressive determine follow law brevity keep unknown datum lx cc proposal eq convenience artificial initial associated perfectly quantify ergodicity drift shall bind proposal weight determine likelihood jensen recall particle hasting overall generate proposal distribution give ordinary hasting proposal particle eq q number give stand particle bind directly bound establish ratio proposal density ergodicity analyse behaviour unbounded average overall filter one proposal overall density particle finite path jk jk v establish expand projection intensity identify lyapunov statistic purpose constant p x symmetric except enough numerator n convention rearrange write independent u eq overall deduce constant exist leave bound variable calculus q deduce choose sufficiently stability distribute variable independent constant proposition convolution c I hold establish straightforward check generality geometrically ergodic imply drift assume imply soon sufficiently proposition condition establish proposition relax certain fix em dash correspond induced logarithmic control sufficient start final rate run unstable behaviour rely vx consider claim q let vx rx rx c rx vx rt proof sufficient show claim p r p martingale inequality imply employ
commonly find similar control develop systematically vary experimental conclusion term amount relevance difficulty closeness solution assess confident sample synthetic set irrelevant redundant hypothesis scoring account amount relevance suboptimal yield artificial opinion mention properly let cardinality feature call feature refer feature relevance feature jointly relevance carry although way latter keep differentially contrary interested subset likely equally instance optimize account inspire need technical etc call whole set assume imply scenario fix minimum cost meaningful amount among find among restriction subset exist feature policy may solve resp scenario addition shall shall described feature inductive implicit induce logical example place useful particular embed pre mode evaluate subroutine favor mode goodness disadvantage burden briefly genetic exclude review none allow work include comparison equal consider feature count instance equal equal q monotonic evaluate hash particularly set arguably advantage feature filter frame equally set well allow repeat end incremental algorithm describe portion instance inconsistent portion iterate intuitively big add current portion hence similar author suggest proportional experimentally choose may fail noisy datum consider irrelevant htb output portion portion repeat stop else make inconsistent find instance latter instance among separate near separate version original feature htb frame array initialize zero random instance near near end iteratively check nested subset measure simulate type look assessment irrelevant principled discrimination correlate account several similar type sequential generation iteratively try backward counterpart describe evaluate preferable relevant otherwise report contrary practice frame line b output find xx backward generation operate sequential forward possibly essence remove among chart characteristic size backward counterpart backward follow effective popular drawback desire subset generation current sample quick branch hybrid compose branch stop branch branch variant full basic idea point remain search efficiently automatic branch find var list state end begin queue list allow element htb quick branch label monotonic evaluation small element arise design aspect certainly maintain well trade sized solution assess time practice task greatly advance ability respect relevance redundancy family whose influence relevant set provide subproblem redundancy front identify redundancy situation something find report something aware set problem redundant add instance multiply mean depend measure capture obtain match criterion sense similar denote partition relevant irrelevant redundancy let general r x ax flexible divergence redundancy end collect feature enough feature weight redundancy redundant valid feature break correct equivalence define solution every split equivalence form feature redundant define equivalence original redundant give relevant denominator let three redundancy x severe relevant feature redundancy reflect irrelevant miss relevant redundant well one fact choose could precise equation suitable reasonable though depending need twice count difference practically overall section parameter know relevant size function illustrate fig accuracy respectively automatic criterion cut sort great weight discard idea low cut bt total three odd nn odd classic set group follow p boolean condition check divide group explore second explore group different instance relevant indicate twice specifically involve analogously reason representative graphical fig random fig example relevance redundancy vs relevance represent ratio explain vertical represent htb vs relevance vs relevance f c example redundancy vs sample fall dramatically perfect top almost number present good tends poor total feature fig interesting provide increase example study fig score fig b average difficult high algorithm top bottom also algorithm end also useful reading graphic show end fact show surprisingly quite general poor compute independently f picture close view way execution order respective detect redundancy presence size light conjecture fashion well fit option like bring different performance way scale number feature reliably tell also like outcome entirely precise another good deal limited sample dependence feature regard outcome yield outcome resample recommend final specific evaluation know solution expect score performance experiment different use na describe solution well evaluation
focus contain favorable relatively association observation health conditionally limitation independent large health path environmental support etc need health remain essential health predictor provide person limitation support many people survey percent severe rate health hypothesis confirm individual component support health general health variety environmental environmental factor upper favor false negative issue choice team particular individual already appear background restriction macro unfortunately exchangeability violate disadvantage implementation issue subsample choose categorical validity finding mixed seem runtime depend estimation execute core roughly hour child complex heart cognitive behavioral impact daily routine severe heart due heart disease upper expect cognitive conditional validity child common likewise minute latter connect score minute identify insight open genetic factor mostly work child already upper bind plausibility potential imputation utilize forest exclude identify value original exchangeability memory minute random forest satisfactory mixed type exchangeability responsible conservative dag true small poor mixed case feasible modification multinomial response penalization continuous task scale health survey consist cluster macro cluster affect bind connect confirm tendency toward health evident individual many fail identify factor connect risk genetic xx j equivalent thank anonymous valuable proposition among graphical novel estimate conditional tuning graphical obvious constitute help positive health order reproduce health organization international health risk suffer heart perform stability control positive graphical mixed forest selection response predefine predictor association set remain pairwise possibly I appear absence conditional application include largely focus graphical base see binary involve method lost deal mixed ensemble notably performance importance forest allow rank conditional suggest overcome definition response rank derive dependency framework allow specify false random appropriate forest compare performance comprise possibly health health environmental organization international health identify independence conditional realization cumulative dimensional assume proof trivially analogously level express bernoulli moreover conditional let motivate nonlinear regression whether include regression indicate variable rank edge rank mixed type ranking criterion find comparable instead ranking perform local analogous among forest regression decide e ranking select regression tie rank th edge tie remainder tuning outline guide choice allow positive subsampling n thresholded construct impose subset theorem select subset false positive depend precisely give edge vice versa actual minor different fix throughout choose formula specify desired accept false exchangeable note interpret threshold solely assess stability undirecte whose forest date convenient type incorporate observation predictor predictor assess regular forest random fit within relationship response choose package goodness continuous majority fit response local rank response conservative assign bad finally aggregate stability subsample stability graphical henceforth variable many forest favor influential inference unbiased tree overcome implementation forest conditional party package lot drastically reduce become feasible produce positive instability forest ensemble allow fair random forest linear response predictor predictor coefficient also logistic regression median aggregate category discrete consequently suggest estimate lasso remain penalty sequence zero select conservative penalty j estimate regression correspond subsample selection denote stability acyclic statement node represent connect parent entry percent zero encode dag simulate ji present variable least multinomial predictor opposite total effect category multinomial predictor restrict definition link relate previously sample purely bernoulli purely multinomial alternate sequence multinomial ij b ix j ij sa sa scalar choose multinomial association half pairwise dependency ise realization parameter conditional absence see triangular uniformly triangular upper size interaction without averaged repetition give observe false repetition average small achieved bernoulli ising seem figure third many positive rate cover rate return satisfactory poorly cause perform gaussian multinomial set especially procedure nevertheless one indicate potential recover counterpart forest estimation ranking consequently lack various provide selection forest besides enable deduce across hence stability condition mixed setting explain error however return indicate problematic behavior unlikely exchangeability argue appear grow minute hour run core comprise core ghz gb ram package false positive average simulation report average raw counterpart false positive average third report average true third report false rate raw counterpart stability relative stability false average third raw counterpart selection false positive true raw counterpart average repetition similar finding positive arguably small drop surprising ability incorporate whereas explicitly false positive average column rate n average repetition outperform positive positive control especially apparent raw estimate counterpart signal factor positive average third average false raw counterpart remain clear forest par selection apply ml estimation burden positive bound well similar perform somewhat raw stable average column report false positive organization international body structure influence factor gender age environmental include relation support property macro recommend world descriptor conjunction health conduct interaction among environmental health variable conditionally know observational health survey office base private select survey complete percent participant mostly collect far available elsewhere include various limitation respective sometimes ordinal mass health outline consideration plausibility index check module index
fix denote eigenvector furth eigenvector leaf include obtain eigenvector remain two establish taylor minimizes mean error mse conditional distribution q estimation note achieve identically unbiased measurable h x h minimize minimum mse unbiased root finally prove broad establish let vertex vertex root sample correspond top markov topology contain information state high probability distance vector use resp close identity matrix indeed identity variation distance vector total gm couple since statistic structure describe state leave I v vi kt estimate datum eigenvector stationary distribution first eigenvector orthonormal respect eigenvalue description difficulty however note deviation concentration enough ie small use relate eigenvector eigenvector define include eq use proposition length since recursive setup weight child constraint bias condition setup paragraph satisfy far concentrate internal z z establish u equation moment argument apply factor cauchy schwarz e c proposition expand v hold lx x bias accurately estimate take enough bound equation probability calculate x lx observe e lx exponential suitably recursion let eigenvalue e lemma j show ks question conjecture may coefficient technique extend non homogeneous tree weight discretize tree result would control state combinatorial rgb rgb rgb corollary definition corollary proposition theorem conjecture conjecture hide inexact latent tree widely biology efficient procedure latent improve previous requirement regime tree tree transition mathematical processing e reference therein seek evolutionary datum specie assumption molecular sequence sequence evolve independently tree key parameter tree evolutionary branching past matrix mutation alphabet arise biology generality entry encode state branch edge branch roughly mutation think evolutionary involve section naturally context estimation tree topology give fully unobserved statistical theoretical computer science provide insight parameter weight symmetric ise critical parameter section detail contrast tree depth regime know ks connection specifically regime label optimal factor conjecture discrete furthermore polynomial requirement need several reversible threshold may little rigorous dedicated regime prior rate edge discretize require sensitivity root inexact design new ks provably robust discretization precisely ks regime discretization reversible know evolutionary far know previously threshold statement work sample tree neighbor metric leaf follow work dyadic technique high degree conclude root label dyadic integer set balanced leave markov tree leaf variable discrete evolutionary rate reversible balanced rate respect move away root parent special symmetric ed biology generalization include channel asymmetric let eigenvector correspond markov field gaussian signal balanced covariance leave ensure identifiability nonnegative affect convenience leaf edge choice denote extend leave leaf covariance detail close think pick distribution independent globally markov disjoint subset separate independent failure leave go infinity tolerance model model satisfy moreover tolerance establish equivalence threshold tree without sample tree satisfy weight tolerance general critical leave give sketch discuss let balanced generic use framework basic initial sample loop build infer root reconstruct previous step heart step assume correctly correct need issue address topology call ks effect notable ise discretized estimation structure distance estimator eigenvector define define note leave quickly accuracy failure constant replace approximation infer natural mutation estimator flow variance follow strong moment level obtain estimate accuracy lead bias hidden term close exponential moment provide ks bias distance bias adaptively recover estimate eigenvector build eventually eigenvector identically subtract careful procedure prove balanced assume explain building state sample everything else leave build correctly construct convention form satisfy amongst procedure minimize level adjust deviation accumulate branch child big constant estimate seek construct treat weight coefficient recurrence particular weight estimate compute similarly let constraint bias prove satisfied concentration uv markov third moreover gaussian estimate type explicitly note suffice independently assume hold z z z similarly direction hold bias x x x x x line inequality e e e enough recursive estimator q x take enough proposition rely know topology know estimator leave let topology split
connection indicate display sometimes mathematical description take include exploratory algorithm label box connect side point motivate long characteristic procedure within theoretical world point prior treat inference human making never satisfy logical framework purpose go make important look back really high ground inference logical cost thing begin introduction neither avoid connection inference hold inferential frequentist act normal involve conjunction equation speak accord equation bridge build theoretical statistical partly see pure argue succeed data problem complicate aggregate uncertainty scientific investigation context distinction practitioner understand analogous believe figure population concept drop understand option large development terminology avoid confusion offer recognize pearson make contribution introduction behavioral seem think far particular hypothesis upon valuable evidence nonetheless behind lead inference valid mechanism chance small situation quantum physics chance variable carefully head david imagine absence explicit chance strict I however precisely reason vast mechanism fisher jeffreys many cox extent like stanford fundamental theoretical align hand gap seem big offer scientific goodness heart acknowledgment support grant grateful comment frequentist assumption interpretation confidence assumption bayesian frequentist interpret suggest serve foundation statistical argue jeffreys take goal exclusive failure nearly old statistical go way significance aside concept meanwhile part ignore possibility inference event despite occurrence interpretation posterior belief confidence frequency limit use introduce practitioner open view mode aware suggest place logical world frequentist fundamental paradigm take separately understand important student right imbalance describe dominant modern think valuable inferential chance count fair basic immediately mathematic come calibrate fair another say odd mathematically assign fair cover consider run may regard consequence interpretation important use reporting interval statistical inference model illustrate figure scientific distinguish world call world random confidence interval probability live world implicitly datum capture reasonably world real apply statistical construct conclusion theoretical inference implication phenomenon scientific concern scientific implication involve new somewhat model large weak careful statistical note random live thing normally proceed kind shorthand assertion purpose variability variability occur sample linguistic framework distinction aside treat apart inference good way reason inference development elsewhere quantity counterpart mathematical important viewpoint bring center purpose provide interpretation believe practitioner stimulus histogram fire neuron two ease familiar situation issue arise conduct human construct light intensity light vary intensity subject determine intensity subject light observation yes intensity day tool analyze datum maximum light report report al involve fairly large answer bayesian probability interval come spline fire rate study ultimately differential display fit fire rate maximal firing bar follow frequentist posterior base bar similar bar either inferential interpret frequentist paradigm sample mean interval confidence frequentist assumption infinitely random interval cover confidence world nonetheless substitute useful conclusion align world produce remain statement apply variable random prior become eq inferential interval statement remain nonetheless draw conclusion world real produce although world relate object live inference distinction random inference statement introduce happen distribute reasonable describe accurately important student principle notion picture illustration figure extremely population work subsequent drop analogous concept try claim population random analogy independence response suppose student whether reasonably reason
tag tag document share tag proportion tag tag document topic proportion tag tag word replace latent topic tag tag multinomial tag order document largely specific vocabulary tag bag exclude label exclude subsection modeling mrf assign semantic label explain bag vocabulary corpus label topic index word group model often theory mrf assign label posteriori estimation mrf map framework vision specifically mrf attempt topic label configuration word maximize problem latent mrf use labeling reduce labeling configuration encourage smoothness neighboring topic neighboring system exclude denote label associate index exclude lda factor design potential local specifically encourage labeling type hyperparameter complex inference mrf factor connect connect neighboring labeling direct undirected fig parameterized function multinomial smoothed hyperparameter pseudo count multinomial resemble collapse gs treat perform collapse recently reformulate constrain mrf causal indeed reflect lda generative latter mrf neighboring mrf extend visual topic model labeling mrf expressive directly emphasize generative loss factor representation direct world slightly graphical neighborhood function fig neighborhood system connection bp efficient fig calculate turn calculate marginal message normalize efficiently computation message proceed convergence iteration adopt sum infer message pass scheme em infer hidden graphical mixture backward hide probabilistic labeling message base infer almost gmm detail book fig variable message q normalize message turn pass factor message neighbor labeling evaluate dependency topic imply configuration q often cause close avoid arithmetic product approximation mrf acceptable convenience shorthand notation subsequent mrf clique prior knowledge encourage high pass neighboring message document message message word index vocabulary make message comparable across different word word except sum possible message normalize locally message converge multiply count topic message perform figs message derive equation employ multinomial dirichlet conjugacy hierarchical maximize mrf assign topic accord rule p collapse gs variational bayes topic alternative topic model bp gs resemble bp sample token update base currently sample topic label gs word token bp message vocabulary document word bp gs addition bp gs keep complete information vb use objective function maximize maximize variational resemble minimize leibler kl vb differ involve function learn average token document word indice small token vb corpus hypergraph fig bipartite undirected hypergraph equivalent bipartite c denote connect tag factor fig c hypergraph tag connect neighbor solid black tag share tag three relation among document label configuration influence separately relation meanwhile influence neighbor high result tag fig encode order among semantic tag usually account part document image associate document appropriate tag probabilistic likelihood form label base message message factor subsection tag content topic initialize tag per iteratively normalize topic tag notation hadamard wise product notation indicate document tag hadamard similarity tag pair tag encode message tag message pass word tag depict high dependency tag order relation tag similarly base denote total tag obviously triple capture representative order structural dependency contain loop develop bp estimation subsection calculate message subsection involve pairwise high fig constraint factor denote tag current document document except document tag pass message document joint message tag influence pairwise across tag product operation neighbor arithmetic product calculate input flexible message influence play high rewrite balance factor sum term tag message pairwise high relation automatic require work manually tune message fig summarize use relation image engine area paper engine name tag fall broadly category collection contain kind range manually label appear picture colored pattern appearance visual slide window visual word vocabulary index quantization summarize total number vocabulary vocabulary number tag per document constitute constitute c manually relation comparative study model order topic among link experimental benchmark model handle relation document contrast additionally relation induce connect tag lda topic lda tag link tag document test word topic unseen test low correspond topic consistently low ability unseen test benefit rich relational information document pair tag specific pairwise potential capture subtle dependency document specific show compare order role worth prediction performance predictive topic latent interpretability involve basic idea identify topic document define inconsistent word knowledge ten qualitative topic share university third show interpretability top ten moreover link predict share tag link document author name link retrieve document tag proportion link decide link link compare encode document contrast link outperform link reason rich proportion tag differentiable without share unseen word enhance content similarity document alone account additional may well document classification mutually may document proportion feature discriminative ability document end proportion seven randomly select choose water tree people use associate tag image proportion inconsistent word lie treat tag equal link reality tag topic structural document thus encourage tag correspondence tag class limitation encourage proportion tag modeling furthermore high tag constraint pairwise performance generally partly tag individual component equivalent label topic enhance tag recommendation classification suggest tag document world find annotation recommendation image annotation recommendation tag multiclass svm image proportion tag tag tag tag train tag binary svms tag tag connect tag tag encode tag tag predict likelihood tag balance linearly good training annotation protocol top query tag use connected tag refine tag recommendation tag tag suggest include recall rate tag tag image set tag tag recommendation system give tag recommendation recommend tag tag test ability achieve recall tag precision lda recall h lda compare two tag recommendation rate lda show tag recommendation web page advantage annotation problem connect tag information major role rule false positive enhance precision still superior fig topic tag performance effectiveness smoothness pairwise document mrf bp inference parameter four lda mining many vision application unsupervise activity complicate multiple encode discover motion another historical interaction pattern work grant china grant modeling problem higher extract reliable tag topic structural dependency framework representation latent mrf belief propagation approximate hypergraph relation learn encourage label smoothness among image extensive confirm incorporation high state many text
case histogram unnormalize weight homogeneity monte comparison experimental program cm title language library histogram unweighted weighted solution formula ref histogram pt
accordingly preserve partition let index eq q projective xx constitute result construct let mapping take particular outer measure image restriction later borel constitute x element space finitely q conversely x satisfy hence regard theorem measurable measurable well first relate functional evaluation functional generate mapping evaluation functional set weak topology coincide map cx express large projective algebra deduce regard mapping trivially remains borel countable projective topology borel borel projective obtain rather manner ensure surely projective limit proposition gives formulate projective projective limit projective space p expectation element additive surely additive criterion term countable projective construction content additive along imply reduce countable subset derive result countable algebra generate ball sequence countable sequence additive finitely additive mapping projective ii surely mx mx form endow convergence imply mn whenever hence conversely assume surely sequence nan q convergence satisfie conclude finally obtain establish proof first projective proposition define probability conversely assume proposition f complete provide description problem list address reader measure theoretic obvious measurable partition axis hence encode family dirichlet product kolmogorov well assume dimensional event event consist simplex euclidean marginal live form sec projective measure dimensional space limit apply kolmogorov extension embed product properly formalize marginalization require construction illustrate marginalization merging event subspace kolmogorov natural formalize node scale use fill black body minimum width cm stroke fill body black minimum fill white fill occur equivalently x projective construction axis set useful implicitly algebra algebra algebra resolve particular illustrate set il axis shape direction situation illustrate assume shaped plane measurable obvious reason algebra kolmogorov closure countable set algebra axis parallel countable overall countable interest however event joint countable would countable arise measure assign event countable subset behavior measurable infinite suppose countable sequence hold even though along set aggregate substitute countable resolve acknowledgment thank associate valuable pointing corollary grateful helpful comment correction lemma corollary theorem bayesian construction dimensional marginal prominent dimensional dirichlet show difficulty construction probability distribution whose projective limit countable dirichlet projective finite distribution stick break overview construction exception construction key representation account surprising purpose projective limit modify prove put main dirichlet derivation sufficient stick specialize latter approach provide currently tool arbitrary prior make readily comparable prior notably process technical arise problem review detail product space kolmogorov adapt construction dimension label borel resolve event specific construct condition projective limit feasible requirement topological sufficiently accommodate measurable space useful without topological natural whether parametric existence regular probability validity de address generalization kolmogorov countable set substitute projective iii set space remainder apply going summarize sec brief overview construction relevant additive review notation relevant notion marginal topological topology underlie abstract call measure weak topology context correspond borel algebra partition disjoint probability ib jx vi x xx xshift circle fill b x x illustrate mapping image constitute intuition dirac roughly think analogue regard evaluation v keep mind topology topology euclidean space excellent exposition pa following construct suitable marginal law borel moment discrete sec example condition serve finitely sec contain probability turn distribution need p additive almost vx constitute random measure technical restriction mild purpose illustrate concrete separable banach banach metric space space domain topology compact cube distinction may dependent process process background construction three example marginal distribution let let dirichlet additionally problem ii generator provided resolve iii behavior know aware well follow specific dirichlet projective limit invoke tailor real dirichlet throughout mathematics dirichlet derive mix tree stick break remarkable construction list require arbitrary stick projective process trade impose restriction applicable represent phenomenon encounter theory example mean theorem factorial product space choose across subspace topological kolmogorov component process purely stochastically regard analogue factorial measure factorial measure general limit construct mathematical set projective small sense precise totally sequence whenever direct partially inclusion projective limit manner formalize define mapping space measurable regularity assume algebra generate underlie continuity direct di function f kf topological satisfy f f topology algebra projective limit space topology algebra make canonical measurable analogy f projective projective p analogous impose precisely coincide family projective guarantee f space countable direct projective projective stochastic refer theorem index ensure projective arise whenever projective theorem theory regard product
compute trace let sort decrease equivalent determine large uniquely present method typically order within second admm level reasonably large effectively b illustrate compare see differ produce see indeed bias determined ensure negligible implementation result n h h I relax previous subsection hyper write semidefinite matrix widely arguably iterate x iteration involve estimation generate select next paragraph denote produce result without first define residual variance diagonal refer applie may relate see formulation say denote resulting stand also formulation solve though involve quality multiply set x x become constrain unit determinant motivate solution identify explain variance uniform feasible convex optimal solution ascent alternate optimize compare aforementioned synthetic second historical price stock kind data model unit sample take factor variance orthonormal vector mf x n use plot represent availability variable difference outperform scenario largest well performance h synthetic similar effectively control residual tm plot moderate residual outperform tm variance grow elaborate important finance covariance risk guide experiment involve historical daily stock represent period dot daily compute price describe detail trade stock active set normalize log daily day application day day assess day begin day day evaluate daily day test slide try regularization time specifically select day estimate per parameter evaluate take ten ten day average define tm dominant natural historical observation superior allow involve lead difference alternative start simple residual identical let matrix move theorem suggest reason tend loading assign large large variance variation performance eigenvector load one factor load insensitive residual dominate eigenvector eigenvector imply preferable residual differ tm suffer similar tm tm incorporate tm produce desire strictly formally proposition give monotonically bias variance contrary accurately recover favorable incorporate ie estimate deal context assume among conventional provide computational experiment involve synthetic pre direction datum thing estimate guide subsequent interesting quality relate subspace interesting understand suitable finally body research robust variation pca pursuit sparse corrupt would explore connection work v rewrite associate multipli lagrangian denote kkt plug trace permutation g v v diagonal entry multiply side generally rearrange l finally plug complete sort denote note write n impose vanish imply j ik theorem eq fix scalar n therefore numerator limit rewrite law proposition sample inequality chi distribution l rewrite straightforwardly since eigenvector solution optimization f equation plug back yield result monotonically furthermore q therefore c r eigenvector differ symmetry arrive discriminant distinct root root great root great eigenvalue great eq q increase algebra mr mr sides r mr derivative describe experiment cross validation split whose candidate maximize likelihood select fed full tm choose rarely implementation em terminate I suppose algorithm evaluate equivalent uniform trading day stock return adjust close stock day stock day let great interval volatility stock week select tm range value proposition ex learn linear synthetic superiority exist theoretical explain bias feature algorithm requirement enjoy part due notable economic finance combination consider factor observation entail estimate loading residual seek explain approach learn principal variance pca efficiently maximize likelihood datum simplify ideally treat attention uniform consider number factor select portion datum baseline residual propose number approach maximize restrict trace model covariance serve model parsimonious validation similarly synthetic demonstrate estimate recent bias practically relevant assume stock demonstrate accurate residual aside aforementioned analysis efficient program sdp solve exist interior alternate direction multiplier long practically obstacle arise sdp formulation large solve efficiently typically multiplier pca wide extent essentially problem computational acceptable trace important difference however establish perfect asymptotic regime make trace penalty deal residual variance demonstrate treat semidefinite demand provide solve though research relate order graphical lasso detailed although share like fundamentally graphical model induce whereas result address bias algorithmic front develop solution build demand without estimate sample x think generate w residual small sample factor loading result model seek matrix leibler choose maximize maximum simply maximum sample accurately unless
adjacent possess social specie node explore far several simple heuristic node centrality maximize mutual next expect long history intelligence first couple generative discover network label gaussian field harmonic technique likely learn relationship community node label collective community differ information uncertainty case link end sign propagate class node generate correlate label simple could depend event independent classification tv undirecte modify pair wish undirected machine community assume community node node direct edge web classification likely classification order define integrate product particular bayesian edge hyperparameter dominate small beta allow user community dominate remain however entirely assume likelihood since graph take fix averaging overfitte paper determine include learn resource field resource node guess label explore mi node distribution difference q expect information entropy uncertain strongly correlate entropy sampling classification gibbs markov node explore allow collect conditional entropy write probability average label offer markov graph exponential real try marginal say markov condition explore explore stage quantity might explore classification define whose nod correctly could agreement gibbs correct classification know assume draw explore classification gibbs event agree numerator heat sampler except classification start chain class consist community community nod network commonly occur occur connect word appear point precede exclude direct simply noun focus attention uncertain early nothing perfect elliptical last algorithm explore explore stage use average markov chain performance gibb stage unlike mi aa specie point type specie namely value result stage use choose heat markov average type explore correctly remain perform somewhat mi include explore explore classified node early show confident wrong type variable aa correctly word wrong explore leave argue node perform specie large diverse specie type level account base specifically specie likely course regard mistake connect another extent consistent update specie specie iterate six every specie type topology suggest away occur four year world reflect hidden algorithm solely matter sophisticated probabilistic use active explore subgraph node centrality path go subgraph node choose heuristic heuristic variable leave high high heuristic early quickly mi aa heuristic perform process high high classify explore heuristic classify node surprisingly early bad mi explore costly one lexical algorithm good job explore relatively small certainly generalize probabilistic grateful mark web grateful institute foundation california usa ne topology label could choose word novel make even structure assume connect way category collective labeling biological attribute link view tend much correlate word internet business review define definition community group connect might english follow even divide know trying classify political social try infer focus discovery community community make assumption
pca mnist clear include explanation increase dimensionality generative glm properly overcome well regularization c c mnist mnist mnist scale letter glm summary computationally achieve order magnitude speedup mnist minute report combination section baseline learn k discriminative replace conventional rbf metric point specific linearly baseline optimize algorithm display average misclassification experiment result report table detail nonlinear combination metric poorly table metric attain good method understand metric normal heart baseline heart euclidean proper metric unsupervised unlabeled address iterate global metric apply significantly performance exploit learn nonlinear metric result euclidean metric mnist colored identity metric help well exhibit identity approach discriminative build upon semidefinite metric improve well combination compute combine ip space drop subscript clarity c linear term solution space I q eigen combination metric another combination probabilistic point estimate depend metric iterative list combine compute estimator kernel tp bandwidth parameter maximize gmm class assignment trick learn classify datum gmm eight method metric learn margin near method review learn metric review distribution modeling procedure classify metric finding much combine described speak define difference point near neighbor low purely base local weight density estimator compute tune outperform euclidean efficient sound glm glm c c avg norm heart glm scale shown generally reach train dataset mnist mnist letter euclidean scale count scale letter scale c c heart c c mnist scale letter metric represent normalization factor scale inverse bandwidth metric use desire formulation scheme constructing adjust accordingly metric combination simplicity us svm refer optimize combine coefficient solution depend include aspect ie simply local metric improvement metric promise kernel impractical heavy cost c normal heart baseline dataset heart identity method normal euclidean mean euclidean mean cluster metric metric extract mean euclidean truth class generative compute cluster use learn distance iterate long simple cluster measure metric essential compare distance demonstrate usage dataset metric well euclidean describe rand label rand return metric compute regularization overfitte generally validation ratio tuning cluster compute rand score validation tune learn use set rand performance rand average find mean improvement reveal department electrical pa science california ca lee department electrical engineering specify manner generative metric framework specifically learn optimize parametric base minimize training criterion combination achieve magnitude speedup distance datum learn approach discriminative metric serve block minimize combination extensive discriminative significantly competitive baseline train method magnitude metric distance interest machine aim improve training technique try point belong increase mahalanobis metric semidefinite use semi triplet nn interest discriminative technique metric aim try increase distance metric mahalanobis solution programming sdp positive semidefinite optimization learn computationally appeal asymptotic limit underlie metric near upon bias class conditional glm optimize simplicity desirable infinity classifier attain twice optimum asymptotic calculate training nn term metric glm technique knowledge empirically learn attain competitive training moreover glm metric learn solve sdp metric difficult distance distant unclear address issue metric global classifying view metric base kernel benefit generative combination highly kernel outperform compete method magnitude encourage technique metric need every significantly capable exploit generative adapt learn metric summarize average metric single reduce classifying empirically illustrate subsequent compose base base metric secondly extend intuition combine base gaussian kernel identity replace address important provide derive building benefit conduct extensive lead improvement performance furthermore computationally speedup magnitude organize review previous combine train present extensive follow derivation comprehensive approach briefly start metric margin near neighbor examine glm parametric attempt classification conventional mahalanobis metric follow terminology literature square mahalanobis transform arguably neighbor exploit identify structure metric significantly increase secondly glm conditional generative suggest attain competitive rigorously quantify relationship generative unclear generative adapt improve performance nearest classifier learn metric issue kernel classification consider metric define learn locally treat function intuition linearly kernel learn q coefficient kernel global metric combination metric simple combination empirical strategy note view apply transformation datum transform average computed identity near neighbor space p metric convex combination induce ip exploit close combine radial rbf covariance q bandwidth goal learn rbf kernel learn mkl convex metric semidefinite include reproduce hilbert represent close mkl choose rbf matrix difficulty properly base kernel problem formulation refine optimize specifically low use classifier vector machine benefit discriminative optimize combination preliminary indicate extensive reliably simple uniform present form appendix find appeal effective algorithm dominate calculation cost local metric add little overhead contrast learn optimization examine roughly moderate greatly outperform speed later competitive report linear comprehensive include
dependency lda small number non bad logistic method multi date largely machine drawback drop increase exhibit skew distribution world advantage respect number rare discriminative approach document labeling range probabilistic compare skewed text dependency decade publish document discuss limitation classification common dataset label power advantageous multi label corpus assign document corpus task dependency much set relatively instance label corpus occur relatively axis three multi label axis relationship log axis benchmark log scale plot point fall equivalent researcher restrict example popular yahoo yahoo fail however conduct yahoo construct dataset category yahoo leave classification yahoo dataset dataset mesh contrast research real contain thousand skewed frequency illustrate corpora thousand axis plot corpus vast label document example single document corpus stand yahoo yahoo health bottom occur frequency summarize dataset drastically world label rare previously notably multi drop dramatically real dataset skew illustrate discriminative poorly dataset full dataset classifier label positively document state vector machine classifier effectiveness neither flat hierarchical svms need skew yahoo many rare category svms substantial improve difference traditional dataset relate compare label document style large typical dataset label three yahoo health median per document document distinguish particular label little training illustration extreme additional assign document relevant reduce example feature label able leverage word token likely identify within likely away word word purpose remove wise since label relevant learn word construct document learn simultaneously separate training dirichlet allocation lda document mixture topic word primarily etc version learn label give corpus assignment rather learn time treat assign word frequent learn introduce label tend well associate associate word belong investigate version helpful type rare label pose purely svm york article show advantage lda rare text news article take new york human law game classifier learn lda learn contain etc rare discriminative game later rare predict nonetheless well job relevant probability benefit away effect far additional arise total label per accounting dependency label prediction classify document example assign document large like label ambiguity account word also motivation beneficial probabilistic dependency feature elaborate dependency label problematic past literature take correlation growth correlation consequently applicable probabilistic dependency type approach statistical task label document emphasis corpora model lda flat employ various prior novel dependency lda extend account label two variant vs variety specifically document ranking relevance yes ranking yes label contribution novel multi dependency improve performance simple accounting dependency competitive popular approach extensive label benchmark generative svms multi svms statistics svms svms prediction dependency outperform lda method remainder handle label text incorporate frequency inference perform extensive present task five corpora discussion vs statistic task adapt lda supervise lda label lda lda account correspondence restrict document document document lda illustrate certain qualitative advantage ability interpretable certain lda competitive svms differ firstly lda label include account label frequency additionally account lead improvement lda conduct large systematic include number label skew generative particularly method l relatively primarily yahoo sub see idea model compose document view early lda propose recently demonstrate probabilistic field compare test benefit able naturally unlabele semi crf particularly accounting label classify label pruning pruning upon ignore restricted dataset unlikely able account power hybrid generative discriminative label classification learn learn take document parent parent specify body prior elsewhere binary multi classification binary classification solve use suitable classifier employ task handle notably svms knn classifier vs since commonly discriminative multi propose work discriminative another build classifier label combination due flat generative assumption make lda tokens corpus multinomial capture dependency assume token sample topic document topic section describe depicted notation extend provide first describe multi inference description describe document model document multinomial type sampling word three model process label extension lda flat regard generate lda incorporate via wide label prediction lastly dependency lda label label token corpus topics lda dependency also unsupervise extract type corpus represent document flat document corpus document formally represent multinomial vocabulary document label hyper topic generate lda document observe assign token label label depict notation similarity flat present author lda author condition author l condition document learn l description bernoulli assignment e variable l lda reduce additional assign unlabeled document reduce despite generative label generative description lda employ distinguish flat frequency within corpus observe document label account difference law traditional flat generative account difference observe document wide multinomial generate multinomial distribution label think single vote hyperparameter label sample formally define word count document hyper weight label smoothing contribute label zero fully generative label sample hyperparameter multinomial always e zero infinity may observed label binary count bernoulli testing note bernoulli tend negative absence document label bernoulli slowly document turning label turned assign elsewhere flat model multinomial tokens j dd di graphical panel lda account frequency lda extend label label topic dependency prior dependency corpus frequency model induce topic dependency investigate past model relate dependency topic multinomial lda compute accord prior lda distribution topic j tt j labels di label w z graphical figure flat lda dependency estimate observe describe unobserve require multinomial distribution additionally test hyperparameter prior lda lda assignment document see label greatly serve upper low namely conditionally label furthermore estimation three dependency consistency evaluate one source factor difference attribute qualitative difference improvement could regard hyperparameter collapse sampler collapse gibbs sequentially update token collapse lda lda label long token time assign entire label document subscript token word integrate distribution term label distribution present indicate long chain single sample current assignment assignment word average chain several corpus document compute estimate document arm political house north nuclear flat lda collapse gibbs unsupervise lda assign across entire set time denote current token remove count topic integrate dependency lda topic label corpus chain run burn iteration assignment compute train ten chain mean chain set see learn corpus document present distribution document probability first proper inference lda unobserve proper inference document type token word token dependency lda involve inference label additional involve assignment token equation learn training assign document dirichlet document arise sample simplify dependency lda dependency lda description fully provide irrelevant flat label unobserve document exact token token token topic assignment must term word assignment token update document term elsewhere derivation equation label token assignment conditionally assignment assignment label topic rather eq label integrate document variable conditionally independent information propagate back assignment token label token sample token topic token pass efficiency act mean bottleneck increase token substantially compute reduce amount require careful storage still slow long give similar bad lda describe suggest require substantially similar proper explicitly level thus avoid information bottleneck create token also avoid costly step achieve directly pass treat substitute assignment document label conditioning motivate multinomial distribution topic compute label learn use parameter compute explicitly variable term lda approach expression step token label assignment token hyperparameter back token pass equation ignore actually influence assignment vector sufficiently assignment provide dataset even optimize sampling per fast inference computational lda base present note often incorporate label dependency read etc flat uninformative difference current token document reflect relative priori topic assignment addition dependency lda dirichlet reflect give infer mixture assignment relationship document label example four assign ten bold dependency correlation lda dependency improve rare learn training flat assign word probability label flat rank two include rare prior improve flat lda exclude evidence small rare label rank flat rank whereas rare high stay prior united relations dependency lda lda force attribute arm sale united labels force united international middle binary svms frequent rare label learn introduction key binary emphasis contain skewed annotate article annotate label assign manually york times indexing service document commonly benchmark yahoo avoid method employ lda represent word word document one document label prop ex health present statistic multi statistic power explain per document divide divide equivalently label divide assign label combination label occur average document set document combination label cardinality reflect degree truly label corpus cardinality frequently occur median reflect many exist sparsity clearly quite group relate label tell average unique label tell combination type deal dependency handle dependency unique three small document unique combination unique building combination reasonable hand mean nearly effective task metric performance lda base svm result show table svm condition lda advantage discriminative detail binary classifier comparison base scheme document count svms training vector svms parameter value parameter default weight penalty misclassification certain especially label desirable instance ratio instance hold set select close determined entire setting default numerous evaluation metric label broad focus prediction know prediction label consider make yes item irrelevant together choice task provide basis illustrate informative fair comparison lda possible binary prediction document prediction traditionally task increasingly grow etc literature adopt many metric literature base compare publish difficult multi version benchmark consensus evaluation prediction discussion early c one rank document predict rank notation bar item bar viewpoint set label rank predict rank broad document irrelevant document root general four bold curve alarm document document macro document document area area document combine macro document average document percentage document rank pt percentage rank irrelevant high rank relevant document relevant label percentage comparison roc curve statistic simplify publish rank pt binary macro average micro average traditionally label perspective averaged label however basis calibrate ranking etc approach harmonic label eq item summarize micro average macro compute individual matrix take micro confusion sum across confusion matrix confusion micro weight item instance weight item frequency must careful difference frequency increasingly skewed like ny macro poorly label vast power poor macro reasonably micro illustrate binary prediction see direct extension rank test predict rank sort transform set binary either assign top rank issue choose particularly selection comprise research selection emphasis thresholding rank cutoff calibrate cutoff predict break cutoff median cutoff expect instance frequency label train train document assign document learn svm note test document calibrate item pt break optimize item commonly refer break select information comparison theoretical calibrate tell high predict maximize average search close score micro fact optimize label generate false assign positive impact micro calibrate score prediction fix cutoff document prediction four evaluation rank label relevance document ranking prediction positive cutoff evaluation prediction show auc roc result ease publish calibrate absolute difference micro macro score evaluation oppose number label lda dependency perform well flat across outperform yahoo yahoo evenly case flat much dependency lda difference demonstrate achieve simple frequency tune svms non law outperform significant svm approach outperform tuned rating area svms svms prediction metric tune svms largely comprise thus overall svm prediction exclude ranking across benchmark svms outperform svms percentage label score much influence low rank see distinction performance power law dependency outperform svms skewed large cardinality lda outperform svms law mix metric yahoo dataset dependency outperform five ranking outperform measure label error measure evaluation dependency lda outperform svms health bad svms indicate relative without center see variance dataset plot label document train datum performance relative drop eventually flat tuned svms dependency svms lda drastically bad lda prediction term median label per label center per decrease e document improvement dependency flat document increase flat boost attribute inference test time dependency result intuition dependency lda metric document number document increase right center around zero dependency lda increase seven aspect ranking six partition cutoff base evaluation show suited numerous document perfect label include completeness calibrate represent proportional benchmark literature comparison lda base document lda outperform lda flat lda document document whereas gap relatively yahoo document even dataset automate dataset strict assignment tune consistently svms except equivalent notable dataset pose binary svms fix dataset intrinsic dealing answer introduction difficulty relate surprising outperform statistic improvement parameter conjecture difference dataset first svms difference actually secondly likely relative whereas test since include label evaluation split somewhat de performance rare assertion support performance tune svms perform svms evaluation metric overall intrinsic difficulty svms rare observe tuned difference relative lda outperformed lda outperform tuned svms measure macro macro equal regardless power dominate rare macro reflect svms macro measure model outperform support handle law svms generally dependency lda svms inferior health subset worse v method outperform base three amount datum datum despite dataset dominate lda label lda fair well interest dominate macro macro macro score previously discriminative publish label appendix additional reason vs binary sparse lda across score frequency macro score significance pair significant svms dependency svms frequency less five significantly level frequency svm dependency label lda frequent dependency lda svms frequency except prediction seven metric evident lda significantly lda scale depend dataset notably significant binary play however comparison across wish model discriminative vice versa purpose dependency tuned svms focus specific performance tune svms four prediction dataset task achieve fall qualitative yahoo multi amount dataset yahoo law dataset unlike dependency overall svms yahoo two v tune svms outperform lda relative performance illustrate evident else well suit seem suit base dependency lda outperform overall prediction quite yahoo health dominate label explanation fundamentally lda learn model direction label direction binary svms label across type whether label suited look lda experiment lda improve flat accounting label frequency improve flat lda prior accounting gain accounting dependency addition
figure draw claim meet establishe satisfied argue draw except fail return put mass run unimodal conditional output unimodal follow e ignore right least sample learn need right chernoff notice algorithm list number assign linear point represent maintain thus turn overhead run run explanation since atom remain justify form say differently unimodal change ok argument learn fail follow correctness either inside triangle inequality target return two guarantee close translate poisson probability plan cover must mean cover natural estimate translate poisson show highlight fact close variance bound figure output translate poisson value proof cm draw k produce eq treat obtain achieve guarantee allow expense omit routine boost reader time specify separately independent chebyshev variance correction check q excess e ii chebyshev imply proceed translate mean next guarantee output distribution heavy binomial follow inside cover translate poisson conclude use suppose remain distance translate remark priori whether inside simply produce routine competition candidate distribution draw select winner hypothesis quite chapter competition make draw return least return competition carry cm pair I mx p return either hypothesis immediate lemma proof suppose winner intuitively likely winner intuitively moderately unlikely winner mutually chernoff stop competition part competition winner otherwise competition winner conclude claim finite moreover algorithm choose output never tie output probability competition output triangle give show competition argue close competition prove pair cover notion competition tie cover recent improve choose pair distribution evaluate pdf take priori point binomial binomial return return absolute lemma describe part modification distance argument poisson learn replace choose appendix numerically within additive support pi tv p remark wish purely exactly hand hypothesis mistake define make unknown choose need much overhead particular running hypothesis instead proceed sample correctness immediate consequence choice lemma run run compute suffice need compute output learn poisson inspection bit hence cardinality spend produce merely involve subtract explicitly fall inside rest choose run theorem proper part modification proper learn output add step poisson translate proper proper distribution choose return return return constant cm sort list constant return put cover hypothesis sample procedure proper sparse explain beginning claim imply case statement output fail return put observe x sparse bernoulli ax proper construct run describe place event complexity run proper follow correctness happen close order translate xy routine search sparse cover distance fail close succeed lemma learn succeed form close learn succeed see routine indeed would k explicit j ks k sample complexity claim remain run competition carry efficient amount competition competition kb n k k kp w tuple tuple dynamic subsection peak efficiently weight bernoulli alternate asymptotic construct sum bernoulli distinct exponential algorithm give problem ok running time mode weight distinct unimodal sum weight algorithm hypothesis choose uniformly call relevant draw fraction constant error argument probability uniformly equal target easy exactly sufficiently must easy chernoff ever sample target entire learner ever go equally likely element note prove initial conference publication binomial conference concave distribution poisson binomial accuracy sample question subsequently poisson binomial mutually integer give binomial goal obvious come bit learn ideally output theorem except additional statement property involve construction reader contain note subset heavy permutation cover theorem cover cover subset cover distribution large cover small variation instead small also proceed argue cover review collection indicator collection filter stage respectively denote collection filter stage tv possibly heavy binomial form make cover heavy proceed cover define denote satisfie heavy concluding well convenience let define round expectation index additive expectation expectation indicator expectation variance k proceed separately symmetric index obtain proof stage filter expectation depend heavy stage sparse look distribution heavy binomial collection produce eq learn distribution briefly explain moderately familiar variation hypothesis though denote distance unimodal take bound target hypothesis expect pool hypothesis term remain carry bit begin true mode identify stage search function proceed appropriate analogous recall empirical come element point lie compute adjacent point lie explain paragraph paper minimize additive difference exceed find finally output convex cdf cdf simply collection conclude follow mass operation able input number produce poisson distribution assign run separately take follow motivated approximate within enough approximate particular involve satisfying bit complexity able claim indeed k algorithm te logarithmic q approximately combine q monotonic construct grid adaptively perform find known fraction exponent within error time e bit operation search te argue additive distance two grid additive thought give want distinguish output e te apply give hence run view show compute step recall say exponent term suffice error approximate expression try first digit multiply integer use bit approximation approximation bit constant exponent multiple fraction bit logarithm fraction within complete additive division clearly integer description om om rational evaluation rational division operation approximate truncate comprise arise combine claim claim remark conjecture claim remove mit ed ac uk cs unsupervise poisson binomial arbitrary expectation poisson generalization familiar surprisingly work basic far essentially class respect bit thus example quantity absolute draw positive result extension sum somewhat city week city pick copy course week pick many production analysis etc like detail snapshot pdf describe reader week answer yes sum random need thus binomial rich class poisson consider extension refer poisson binomial call discrete indeed arguably distribution nontrivial statistic tail important chernoff use survey control analysis survey understood literature natural essentially draw divergence use framework output surprising result bit output description follow property draw least output define bit string bit complexity indeed observation learn result namely sum sample weight sum sum weighted variable weight different draw use run complement distinct weight simple sum unknown draw broad study introduction via eq subsequently many proof range line extensive relevant approximate binomial distribution see approximate via simple fall approximation moment target attractive moment approximation perspective show unimodal unimodal understand computationally unimodal state continuous unimodal distribution argument easily highly one complexity assign mass could point estimate na sample could see chapter target sample still cover remove refined understanding refined aforementioned binomial outline approach section structure roughly speak say support sparse close translate heavy cover run unimodal make translate construct hypothesis translate whose distribution cover hypothesis output capture learn non translate translate poisson hypothesis distribution handle alternate approach unimodal algorithm show exist size apply describe cover size generic approach density work distribution high account versus stress proper algorithm many technical implement high careful detailed specialized sum sum many distinct direct distribution pdf density cdf denote conditional distribution sometimes mean often distinction support kolmogorov kx xy ranging kolmogorov denote think observation domain every sometimes vice versa poisson poisson represent go collection binomial lemma denote mutually
nonlinear cv method knn regression method project prohibitive large data approximation gram cholesky reduce method kernel reduce memory approximation gram x tr mf g n dimensional propose find project projection matrix onto repeat result expect call second class encode gram limitation problem include slice problem propose partitioning matrix eigenvector tb x rkhs denote xx mx n appendix note assume convergence eigenvector use verify variant estimator true choose kernel knn distance cv regularization b sir iteration work propose also require gradient well sometimes fail cause large dim heart breast uci repository visualization represent parameter choose cv knn intel core ghz sec result separate cross validation unstable mm mm mm evaluating method project heart breast cancer uci repository rate knn project slightly cancer hundred thousand fast computational sec breast cancer sec sec heart disease cm breast two difficult handwritten digit gray pixel although subspace separate reasonably classification knn estimated subspace compare cca baseline table give show error improve dim mm uci repository provide classification knn classifier effectiveness estimate subspace classification dimension save computational I uci repository c comparison knn show competitive dimensional dim cca variant dimension applicability little algebra sec degenerate true work focus unsupervise employ extension nonlinear feature important apply give interesting involve direction discuss xx literature terminology completeness assume c xx ab side xx c xx pn xx xx xx xx c xx eqs proof complete expression yx sufficient follow suffice rate derive assertion operator side eq yx yy yx xx yx yx xx n xx pn gx c xx hc yx yy xx yx fc xx assertion note eq side assertion thm purpose base reproduce space comparison exist applicability without assumption result involve modern handle dimension reduction preprocesse datum expensive expression focus concern purpose reduction effectively dimension effective although I extraction choice nonlinearity method ii nonlinear transform combination give classical cca independence wish span without parametric unlike inverse sir employ lie statistic slice computationally strong slice number slice class another kernel require careful apply dimensionality gradient regressor sec contain one estimate limitation high limitation method use characterize problem without require convex descent gram large novel reduction unlike exist estimate kernel virtue method careful solve call mean assume x uniquely rkh characteristic gaussian rbf measurable rkhs dense direct imply rkh rich advantage kernel schmidt discuss fact express general easily make kernel nonetheless empirical estimator appendix prove rigorously expression method derivative open euclidean space continuously
fast shown compare ht passive known exponent rate reflect price pay directly learn density helpful minimax consist pay class put extra denote joint triple class measure p aa infimum density htb n decision subscript control attain adjust p determine hellinger probability divergence inequality hellinger concentration let bound lebesgue condition exist number correspond lemma complete fx h fx qp h qp h qp h thus verify first since r fx fx h theorem leave reciprocal right information estimator reciprocal fisher unlabele slow small definition detector estimate probability knowledge distribution probability aim quantify label training show lipschitz probability detector estimate probability locality general maximum converge favorable detector base mle converge optimal parameter typical risk flat near insensitive error mle bayes error achievable matter unlabeled detector minimax hypothesis prior unknown disease know proceed pearson pearson one detector base bayes detector provide train binary hypothesis detection theory extend handle criterion view case machine learn knowledge density hypothesis density terminology detector special know approach develop mle signal testing detector minimize detector denote risk incur place produce difference quantifie express joint identically unlabeled estimate stand base great operator performance excess concerned label prior risk property extensively exponent nonparametric estimator attain van around excess margin exist rule converge super rate view special take matter deduce determined locally convergence proportional smooth smoothness actually mild condition limit flat near insensitive error mle rate minimax optimal fig depict case smoothness htb b b label convergence unlabele training discuss detector train density probability minimax take triple class conditional remove mle parametrize risk explicitly thus n general result pmf dirac assume reason assumption accuracy relate lemma describe lipschitz property satisfy tp q lemma constant parametrize margin exist case boundary reduce interpret eq fact derivative first neighbourhood smooth derivative satisfy near make leading fast corresponding domain show determine assumption consider derivative boundary reduce infinity illustrate minimax also set constant r satisfie detector increase increase increase decrease mle constant lemma hoeffding constant accord know fast bad show rate standard passive consider lie go infinity proof theorem rate degree assumption mathematical excess q measure vanish chernoff always guarantee lie finite often arise practice rate relatively speak likely label convergence rate attention meanwhile also
law law rescale projection denote rescale selection sequel compact call old smooth derivative interior small th old large gaussian covariate obtain observational study differ across contexts brief value df measure observation design dependent indicate probability partially assign subsection say convergence distribution set old inside coordinate converge rate kk f kk partially logistic I q rate whenever h old kk kk independent negative spatial intensity assign x dx hellinger say converge fast draw probability density satisfy g g kk r g kk vary intensity along model entire density model conditional let isotropic function b qx db b prior g n ix kk x n kk later refine posterior independent relate two existence receive prior relate exist set nn b condition map prove condition subsection extend ingredient calculation reproduce kernel hilbert associated rkhs closure v rkhs measure rkh hx dimensional ball banach equip supremum ball stand rkh vector old real exist w b w theorem universal b b nf f proof calculation one operate relatively take however support behave compact make relate rkhs rkhs x eq follow application cauchy hx q theorem notice w n clearly w qx qx qx qx qx q last probability show sd complete assertion set inequality replace therefore b b kb b q b nr b v n operate common way rescale law prior distribution parallel explore point view study extension present verify restrict finitely differentiable infinitely differentiable regularity condition rescale model nearly depend lead seem essentially define joint lebesgue measure thing joint view purely prove preserve interested useful give place absolutely hope issue joint fast projection shorter credible band amount joint formulation delta delta ball around invariance haar delta ball integration formula radial radial delta eq imply integration estimate mistake root integral approximated count taylor expansion exponent term taylor term case limit observe determinant replace solution kind integral know particle integral eq determinant result q final proposition know equip rescaling nonparametric posterior class rate dimension function could potentially obtain rate low amount loss general priori explore classification density density rescale process equip variable low exploration novel without convergence nonparametric density process widely analyse function spatio temporal nonparametric thorough likelihood process bayesian remarkable nonparametric common specification equip posterior
exist derive ar source process learn ar temporal correlation slow deal bayesian source space make operate effective low computational mahalanobis matrix datum local cost noiseless experiment superiority algorithm counter intuitive may motivate derive analysis finally discussion notation bold vector dimension evident principal diagonal square block principal diagonal column represent kronecker column trace deal temporal easy modify optimization base find bayesian initially regression rule hyperparameter perform implicitly exploit framework source hyperparameter capture need let ml transform block elaborate I block bayes obtain give block zero clearly associate zero associate block evidence maximization estimation refer solution hyperparameter bayesian framework framework temporal via way learn matrix hyperparameter assign source correlation good source totally importantly minimum property unique solution confirm hyperparameter maximization maximize yield effective hidden maximize simplify result thus challenge model consider temporal elaborate either environment basic nonzero hyperparameter contribution dictionary column identity identifiable nonzero worse arbitrary instead however ambiguity covariance obviously excellent learn slow speed sparsity develop source convenience operation attempt adopt snr show adopt quite broad condition consider term base note use rule rule ambiguity snr number nonzero row explain presence form invertible low medium case suggest add namely ensure definite element zero noisy modify name modify rule observe incorporate source load rule feed suitable rule load learn definite low note zero follow insight since generalize root theoretic essential modification order original analysis nonzero positive definite overfitte change conclusion equal probability irrespective still except set global deriving avoid overfitte explain recover source noiseless add mean instance attract minima discuss local function respect present result composition concave equation degenerate local minima loose meaningful minima result role local minima satisfy role whitening source source minimum role whiten us motivation modify state reweighte extensive computer conduct representative informative trial trial create uniformly draw lk index experiment source ar th source rescale noisy unit noise adjust measure indicate percentage total noiseless fail recognize index fail recognize index source index case mse use measure suggest experiment identity suggest small communication code matrix matlab accord suggestion communication reweighte iterative reweighte extension iterative reweighte algorithm count adjust terminate attain otherwise increment implement toolbox noisy exhaustive search practically pick give small failure could nearly noisy besides choose estimate experiment code benefit discount source vary source easily observe ar process sufficient temporal six different level multiple surprising observation excellent matter source environment behavior algorithm noiseless save case correlation performance b htb source noiseless dictionary source generate correlation level accurately much advantageous source compare highly number vector snr db ar ar large error trade increase algorithm performance large source point ar sufficient experiment maintain superiority experiment snr kinds process e coefficient feasible examine showing outperform performance gap increase replace averaging source random imply algorithm maintain superiority source ar outperform scenario noisy case derive sign temporal beneficial answer dictionary measurement varied experiment e reweighted snr estimate thus reduce nonzero prevent closely snr three exhaustive previously outperform noise imply want emphasize snr propose near performance algorithm use zero poor performance indicate advantageous impossible think correlation always temporal ar coefficient result expect surprising well achieve correlation phenomenon observe noiseless observe helpful appear close recovery plausible explanation interact combine determine correlation help limit dimension infinity location noiseless vector almost one show noiseless first row trial form dictionary measurement vector process common coefficient varied see behave curve interesting phenomenon closely seem provide almost counter intuitively close change condition number form e nonzero condition increase ill source behavior matrix gaussian matrix emphasize importance exploit temporal motivate ill issue correlation vary exploit work algorithm presence start consider temporal issue unclear trade modeling extend one classify source assign capture source one far source group grouping accurately capture area work play role whiten regularization indicate one way estimate ar process common snr add estimate achieve many form advantageous issue channel extend assume different noise parameter work author channel noise instead symmetric variance know clear inside framework framework load pursuit denoising suggest method modify l choose environment see algorithm adjust speed channel channel extend source support vary assume slowly interesting approximated concatenation several support change appeal work algorithm basic effective learning rule superior basic especially sparsity show poor practice solve problem block temporal derive base latter extension mahalanobis distance extensive superior desirable would like thank considerable help help experiment mr writing code mr md provide author helpful especially thank outline equivalent globally noiseless equivalent minimizer state noiseless adopt notation indicate globally minimum e achieve summary equal rank write l lemma optimize concave obviously local minimum minimum chapter extreme indicate convenience element index element il ii il let definite e kl finish proof zhang receive electrical university china ph degree department electrical engineering university california interest compress sense blind cognitive receive electrical institute technology
notation pair wavelet subspace orthonormal q last term vanish general geometric scaling translation lie term projection difference fine equation multi subspace edge connect say analyze approximation dimensional using wavelet result sec dimension embed continuous measure let chart asymptotic decay coefficient decay smoothness manifold quadratic smooth manifold high geometric wavelet seem benefit high asymptotic affected distortion measure depend vary location estimate thresholding would wavelet third constant may price replacing may achieve appendix generalize manifold intersection intersection manifold cloud affect neither eq think matrix discrete e small k rough course geometric sample dyadic wavelet construct dyadic cell center scale j j kk accord multiscale variation weight near weight molecular practice choose iterate st repeat iterate construction performance reader ii constructions iv construct dyadic definition display code code construct dyadic wavelet scale construct dyadic tree cell convenience htbp wavelet illustrate construction present manifold dimension construct achieve average wavelet geometric wavelet select scale approximation coefficient horizontal index arrange tree index coarse top scale block display wavelet index wavelet row wavelet wavelet rate error use compressed wavelet wavelet fig wavelet coefficient good reconstruction e scale threshold compression dyadic accordingly leave reduced coefficient magnitude corresponding manifold essentially say consider mnist image handwritten digits digits digit digits three reconstruction fig dimension leaf magnitude coefficient stop decay certain wavelet future work efficient coefficient sec index element right convention magnitude scale fit last closely part equivalently digit reconstruct scale dictionary image handwritten digit successively point dictionary quickly orientation distinguish feature face database extend illumination note background variation decide solely face discard display image reduce complexity keeping algorithm compress geometric keep leaf function magnitude see indicate lack reconstruct corresponding wavelet basis orthogonal lack orthogonality direction scale efficiently dictionary encoding construction geometric wavelet coarse fine seem situation analogous also direct construct wavelet construct construct scale encoding may perspective roughly vary fine scale projection well set choose local cm dyadic cell local wavelet construct dyadic center cm c kx k c remove wavelet basis complement projection htbp end band limit leave dominant sort coarse scale bottom fine geometric transform space smooth characterize energy dyadic I care fact smoothing frequency issue observe band gaussian comparable norm smoothness g unit space band family roughly block dyadic band expect geometry ellipsoid axis length dyadic dyadic approximate transform discuss technique dictionary encode need extra geometric wavelet goal encoding precision intermediate approximation version correction sec encode encoding include cost dictionary define element multiply coefficient nonzero precision local achieve child optimally encode cost example leaf optimally plane corresponding size way encode datum separately use parent approximate encode direction parent encoding produce iii combination pca direction scale lastly parent encode wavelet encoding encode need child node cost cost wavelet complex parent node top correspond child wavelet intersection store encode cost small correspondingly section prune practical realization define node quantify encoding bottom start encode pca minimal let parent determine encoding minimal remove child separate tree new parent discard tree examine wavelet basis accordingly repeat htbp dyadic mean wavelet node construct dyadic cell tree center tree compute basis encode cost achieve encode parent wavelet good child child form discard parent node accordingly let svd encoding cost various think sort fouri multi real science news comprise document model word dictionary set pruning threshold rate cost encode wavelet curve way svd error pca second svd coefficient correspondingly discard curve split intrinsic create dyadic tree kn j summing point stop precision reach strategy jk cost leaf project correspond geometric scaling term performance vertical axis axis increase computation show ambient noiseless noisy handling time computation curse ambient computation vary scale multi roll red point generate cloud hausdorff dot line hausdorff cloud roll scale map generation svd p p hausdorff similar distance ambient space look hausdorff function imply variability almost along similar experiment point shape noise term unable advantage intrinsic seem much construction take approximately sec technique measure support approximated multi construct report publication cloud distribution training belong fix scale drawing define way point plane database train project principal component encode run compare quality hausdorff randomly measure generate quantify capture variability generate multiple point hausdorff call hausdorff variability tradeoff hand requirement correctly fall greedy region correct bias geometry spirit publication refined application currently develop user interface interact geometric well set technique pruning construction cost slightly near minimal give approximation datum depend locally thereby approximation box one cast probabilistic subspace wavelet prove upper start every j dyadic respect dyadic calculation spirit coordinate write hessian calculation high pass pass hull curvature direction last bound formalize span top high tangent plane pass provide obvious assume interpolation curvature multiply quite mr thm exercise remark interesting exploit project example point dictionary either dictionary dictionary dictionary vice versa contrast nonlinear multiscale dictionary depend geometric resolution analyze space regime setting important various application gene array eeg manifold value datum investigation several old estimate intrinsic manifold construct application multi scale efficient precision particular b space organization geometric set signal research harmonic typically class term dictionary atom dictionary desirable highly concentrated require processing interpretation motivate construction fourier basis wavelet dictionary representation suitably define certain class simple originally prefer towards dictionary frame library cosine fusion transform usually organization dictionary trend motivated signal structure allow function construct optimize signal becomes typically complete give rapid possible construct sample harmonic approximation machine construction seek certain dictionary give current add size column fix minimize alternate refer reference therein construction dictionary svd bayesian involve heuristic intensive dictionary precision analyze mathematically challenging dictionary multi resolution analysis inspire multi dimension fashion analysis guarantee algorithm growth control depend datum wavelet vector space geometry wavelet share similarity wavelet transform etc crucial arbitrary nonlinear manifold albeit consider crucial construction may algorithms use wavelet paper organize geometric wavelet fashion sec introduce orthogonal variation two section efficiently cost sec distribution point future direction borel measure paper restrict theoretical section compact riemannian embed endow natural volume discrete metric small ambient typically unknown practice also additional assumption geometric resolution geometric cell linear efficiently encode
actually complex base base complex network concept partition detect shape present drawback identification provide division consider metric connection every distance accurate identification traditional chebyshev fu distance artificial expectation verify rate complex seem able clustering base potentially traditional complex topological connection power undirected matrix zero account present strength different cluster coefficient measurement allow reveal far purely addition modular cluster module brain function community develop community also partition basically group spectral g agglomerative choice depend complexity describe quality division term metric construct fraction connection modularity calculate value division network modular cluster identify understand region high community inter property object attribute length width group feature belong high object concept object connection quantify pair vertex vector similar object quantify adopt automatically necessary modularity complex provide many account execution database take organization modular measure natural similarity measure expect measure inverse distance value interval interval distance assume interval chebyshev values chebyshev fu fu assume limited divide former modularity repeatedly join pair merge great small decrease modularity division result high method lie walk community consider time execution method discuss result fast optimization account breast visualization project analysis cluster database range necessary attribute transform equal deviation call cluster small error among note limitation result chebyshev distance community obtain case equal provide specify database value modularity partition nevertheless proximity modularity cluster error case chebyshev metric analyze error without verify error case dendrogram chebyshev modularity c error first database result small produce distance community nevertheless produce modularity separation imply methodology cluster breast cancer database l comprehensive complex approach discuss section allow database cluster point separate well case cluster automatically maximum modularity traditional result complex chebyshev provide small b fu chebyshev clusters accurate well traditional algorithm go case chebyshev among mean database symmetric equally small rate circle figure present become tend take base mean complex tend figure mean complex based identify f chebyshev distance chebyshev distance obtain modularity determine density adopt measure chebyshev chebyshev distance cluster proximity measure represent adopt suggest improve new compare small account proximity measure chebyshev inverse chebyshev similarity feature vector community reveal small real world approach artificial application constitute promising possibility da algorithm theory graph distance partition spectral automatically overcome account algorithm complex quantifying find world database datum traditional chebyshev distance similarity identification rate intrinsic activity facilitate huge receive matter categorization human almost material due
discover cycle change cycle reduce schedule give experiment schedule gpu demonstrate key benefit exploratory work produce experiment smc sampler particular decide cycle perform mix relation quickly scenario produce particle allow practice mix desirable dominate perform general want smooth picture successive posterior fraction intermediate control recommend scheme convergence sampler sensible note aim distribution give run benefit smc exploratory plot function informative understand predictive fail full key aim report bayesian exploratory tool posterior describe statistic contain heavy run smc gpu sampler generate plot weight smc move true coefficient color one observe path mode jump away concavity marginal density mode decrease global mode red posterior global jump find iterate median plot cumulative quickly origin precision decrease scale scale marginal evidence q indicate fig concentration posterior tolerance correspond covariate appropriate important together plot highlight influential evidence strength change freedom fig moreover map much dispersion plot highlight plot plot four credible mean weighted function show marker weak association nan sparsity induce tail smc use uncertainty moreover calculate concentration away fig summary fig plot overview importance prior plot smoother investigate detail tail induce great rapid concentrate zero away retained fig plot fig evidence variable predictor comparison observe support coefficient fig summary plot snp turn adjacent snp marker leave ambiguity fig partly point association around marginal fig see red hold marker interval green considerable actual value association explore well present path aid understand genetic association summary coefficient range complex indexing prior provide scale likelihood computation demand would day worth time conventional cpu processing make core gpu producing improvement run benefit within working day acknowledgement acknowledge computational biology acknowledge trust centre figure solid line estimate mean blue red reduction shrinkage htp mode use htp green median red htp htp credible green median red htp htp htp plot credible median black mean red htp double htp htp htp htp green black red help nature association region wide association adopt exhibit attractive double tend pay attention obtain scale shrinkage precision heavily concentrate around prior coefficient distribute around map generate distribution prior computationally challenge amenable carlo smc scale parameter smc processing unit efficient inference generalize obtain genome challenge marker univariate test test marker marker highlight region genome motivation decompose span marker linkage lead marker regression priori marker responsible hence prior induce tool aid understand phenotype interesting signal consistent causal attention phenotype although likelihood via gain popularity seminal interpretation posteriori map prior penalty tend increase zero perspective little justification full bayesian exponential monte mcmc penalization coefficient towards evidence induce penalty increase mention statistic reweighte concerned publish article date analysis context recent map gamma sparsity sparse mixture adopt contain spike component broad classified relate irrelevant setting explore sparsity benefit allow visualize scale regression student absolute attractive analytic form interpretable prior task believe much explore form posterior path towards provide heavily origin distribute maximum stress model probability interesting predictor phenotype exploratory challenge suited carlo index graphic graphical processing first suggest context sparse likelihood double pareto prefer easier interpret logistic generalized context develop gpu run pay make hierarchical em take conjugacy inverse gamma different find q standard exist appropriate perform estimating asymptotically sparsity datum smoothly step method examine expectation asymptotically unbiased satisfie worth bayesian represent belief irrespective may receive recently deal solely exponential prior sparsity mixture list mention table compare prior know understand prior less amenable fast parallel gpu use freedom student researcher exponential ab form development use anonymous centre snps genome originally control identify cancer space snps column plot marker strong make multiple pseudo five coefficient realistic size generate bernoulli individual individual marker marker coefficient marker create explore posterior sequential smc prior dominate large limited impact
coordinate look mutually exclusive nonempty indicate single block index define quantify hold among block find partition multi identify variation theoretic early multi state system dynamic start start dynamically divergence transition specify generalization couple brain cc function partition provide uniform start bi h rest dynamic interaction identify modular information system great interaction every subset select stochastic favor unique due author propose normalize justification hoc work identify module principle penalization modular clear interpretation theory statistical assign maximize choice equivalently distribution kl term reach excess error minimum specify call state start prediction system possess perfectly previous factorize parameter condition impose depend quantify divergence take value previous point learn evaluate new infer mapping start state future risk attention component arise consequence minimal excess optimally interaction capture term train arise maintain value space parameter uncertainty simple approximated number p possible beyond distribution low partition provide predictive risk offer well small generally induce present principled weakly module amount trade emphasis complexity stochastic interaction group variable learn block search factorize parameter select decomposition generate multiscale infinite partition minimum form dynamic variety possibility exist product chain dirichlet priors b start index variable support dot bottom accumulate model graph total modularity cumulative risk system probability assume variable maintain amount variable value couple illustrate fig coupling modularity decrease modularity independent modularity grow without proportional flow partition choice pp uniform subset lowest factorize optimal decomposition start risk decomposition become causal organization lead decomposition compute copy previous show plot column calculated uniform starting choose whose independent state risk distribution show induce decomposition partition optimally start identical system dynamic risk decomposition architecture modular indistinguishable causal connection highlight interaction handle utilize semantic intervention recently theoretic direction distribution impose dynamic interest causal boolean frequently artificial life biology organization state mention start system change formally risk eq need start uniform state kl divergence b whole block reflect perturbation block dynamic partition consideration less statistical eq p partition assume risk causal interaction modularity organization model infer modularity predictive treatment connect theoretic provide identify module causal train dynamic test give rise measure identify causal framework modeling framework produce total quantifie predictive advantage modularity process note cognitive proceed suggest modular limited learn gain power term depend model form product model heavily parameterize think generalization module search decomposition probability depend fuzzy modular variable module overlap generally identify modularity biological modeling detection inference organization feedback identify understand modular study several many term statistical modeling amount simple trade simplicity predictive multiscale decomposition dynamical weakly couple module version dynamical gene network modularity broadly speak modular compose weakly numerous complex suggest effect perturbation great operational module arrange argue modular adapting change combination integrate arise result adaptive process fluctuation network pattern modularity acquire central vary scientific science reference formal approach apply discrete multivariate whether boolean dynamic time recent community static graph modularity organization dynamically interact argue analysis life light notion utilize domain modularity next provide outline modularity system
pyramid consideration solution ode practical example calculation apply age duration due case group year report claim health year tend incidence reason age pyramid statistical office peak decade peak age test institute diabetes center calculate duration disease incidence equation applicability datum ordinary differential deal disease duration disease calculation incidence formula mean age disease develop apply basic incidence prove helpful model mean respect disease consideration respectively transition intensity rate incidence rate intensity duration disease state suffer disease disease duration henceforth depend system equation patient eq play system derive ode relate change age rate close analytical solution special expression solution age incidence ode direct infer incidence rate addition incidence cross study incidence terminology duration disease total spend incidence age exceed first age age correspondingly age age specific ode completely rate remarkable age pyramid inherent consideration dependent function number people age simple rate birth rate depend age population office capture age replace incidence separately general population life office reference patient type incidence knot ht integrate initial via fourth r foundation statistical year age almost early incidence rate age almost twice high lead age year age
retain variance project onto pca average coarse search case equation interact eq step root interval error normalize energy c c c purpose potential rich reproduce extra double precision method line give percentage lose projection
class tree minimum principle space use store case compose associate encode induce represent total graph probability need ignore candidate network dag store parent include parent enumeration cardinality parent encode bit node encode length store description length depend encode training probability code depend length description dl px variable occurrence encoding count represent table parameter minimize rewrite hx entropy theoretic interpretation representation bit encode combine total length score statistic score proportional structure satisfy independence relation encode pg pd constant structure follow structure address difference penalty assignment namely datum parameter distribution pd nx ip decomposition p observe sequence te tt assign pd criterion use logarithm posterior pg pd ratio equation log structure posterior probability drop express assignment parameter amount gp configuration degree taylor approximate plug equation approximate diagonal get simple approximation term increase linearly approximate bic bic use without assess measure predict minimum except length high certain prior decompose x score base structure search whenever score structure part remain simple lead maxima constrain list follow simplest hill hill modifications modification modification accept hill list acyclic add remove graph choose high besides hill search anneal procedure exploit space structure example hill compute score check large candidate hill solve hill return turn parent key parent early mutual determine existence learn propose evaluate dependency mutual information xy px qx information discrepancy estimation qx qx number expensive easily probability utilize node define ix metric introduce j ix metric empty measure iteration incorporate structure candidate hill stop criterion usually terminate long increase update network candidate terminate remain unchanged score monotonically bound score guarantee stop candidate enter need sound identify true parent iteration acceptable approximation parent parent allow risk discover suboptimal unnecessary efficiency imply impose either search max hill explain parent problem impose algorithm capable hundred hill search usually optima list record loop minima solve local optima follow pick arc globally arcs arc search generate operation ordering iterate randomize order change conventional hill relax max parent author domain model optimal procedure modify structure benefit add parent single hill propose decomposable markov network iterate link set mutual hill list section imply property order would parent fundamental ordering scoring network order degree equal ordering order second current already imply order perform potentially disadvantage statistic ahead parent discrete simply count structure parent node thus chain carlo typically network structure reliably hill list define score good start order swap operation thus branch swap search order find prevent swap execute try parent node follow exhaustive programming dynamic feature pf unlike programming acknowledge feasible fitting usually add loss e importance value square algorithm encourage towards zero towards invariant test generally prediction logistic regression regularizer also loss quadratic minimizer close dependent regularizer regularization network coefficient drive play role equivalent equation see figure avoid density decay increase observation contribute fix setting non zero tend exact parsimonious property regularizer lead sparse graphical popularity year mainly choice list objective model n assignment subset give log partition aggregate feature order dot product prior z ise x minimize pn estimate graph nonzero weight neighbor gaussian author choice regularizer weight circumstance optimal choice certain full author direct graphical always multiple px least joint dag pd px distribution I precision precision prior I pp associate integrate sample th px ki p x map ki regression advance incorporate undirected graphical optimization kf f f mapping linear high th training category identification multiple branch system relation infer system use differential algorithm key illustrate steady g substantially approximated differential x tt gene network residual coefficient x w regularizers ij first decompose training constrain interaction explore graphical large covariance precision sx x efficiently nesterov description try minimize author propose find parent apply parent neighbor solve follow regularization log parent parameter hybrid approach since incorporate bayesian network incorporate selector encode bayesian encode algorithm author max hill algorithm bayesian algorithm share hill step learn discovery algorithm parent conditional max min variable minimum association parent step hill within constraint skeleton algorithm impose author parent node score parent neighbor discuss parent identify skeleton structure create mb mb mb replace candidate procedure potential parent application search mb sc exhaustive besides list similarity may euclidean boolean assume boolean gene try profile employ reconstruct network correlate measurement indirect interaction two thing unclear approach elimination justification discussion limited loop edge cycle category matrix interaction factorization max factorization reader refer corollary graphical modeling graphical intuitive structure insight inference sophisticated carry graphical field bioinformatics science analysis cope combinatorial space structure paper notation represent direct edge parent child call direct parent domain may attain probabilistic graphical condition formally localize identifie predict node imply theorem reverse direction node graphical essentially divide two group undirecte ht mrf call factor clique potential clique graph potential domain joint element non clique fully connect maximal clique maximal clique probability calculate normalize clique current clique variable maximal practice w function real training sample representation especially convenient evaluating difficulty partition ise node spin clique parameter represent external particle representing px variable normalize follow gaussian zero direct representation joint distribution network underlie compactly advantage network direct acyclic dag correspond random describe configuration x independence together probability via px px ht way critical specifie dependency parent clear size grow node take tree exploit node parent vertex split vertex condition root place softmax parent summarize contribution parent node common many associated choice density incorporate determine node sigmoid belief px j logistic sigmoid root take parent noisy popularity category well employ test conditional attempt constraint pc try pair possible size difficult reliably independence constraint lack objective try directly structure framework name bayesian determine existence independence pseudo slight modification undirected graphical vertex subset undirecte write adjacent separate adjacent adjacent direct path edge edge adjacent possible condition super graph become infeasible sparse besides reliability determination conditional reliable name
contradiction c determined rectangular coordinate system origin old rectangular coordinate dimensional less otherwise definite convex ac ax contradiction integer put denote p pt fx px py positive cardinality p jx I ir I jx x jx jx vx fx jx jx vx lemma dimension great easily combination naive linearization subspace precisely however linearization seem establish px bound degree aid scientific research technology second aid university establish dimensional ellipsoid set class employ asymptotic estimate dimension exact euclidean ball dimensional da section theory theory probability real integer component belong open great tx converse ta b b ta
variation model upper receive imply indeed difference account assess combine datum great posterior rv rv amplitude outer velocity contour rv period outer contour arise plot also euler turn noise also possibly cause say report rv appear uncertain indicate uncertainty show report ms ms euler ms ms euler ms ms claim six rv challenging combine rv rv combine set combination show five check latter factor order probable another parameter map set limit ms amount ms ms ms ms rv star star rv surveys advance recently fourth signal analyse combined rv model criterion however start check rv published rv publish additional rv bias bias combine set imply rv model bias corresponding period fourth day day see contour contour show amplitude parameter top bottom contour contour blue amplitude differ also contour publish less periodic datum parameter table rv day ms ms ms ms solution significantly solution estimate period day whereas day fact rv table ms rv latter estimate excess include likely roughly contain rv likely assess differ adequate bayesian rv make rv describe use uncertainty cause model magnitude suggest rv uncertainty force uncertainty parameter uncertainty considerably likely base turn table conclude uncertainty rv rv data criterion unfortunately possible turn rest note consistent four rv differ significantly solution period stability estimate period could coincide could stability investigate able system successful variation measurement model commonly whether statistical describe respect measurement whose dependence physical derive principle description numerous effect account system insufficient whether valuable extent assumption exact nature construct prior early use select formulae posterior model great probability describe model case star european david present simple whether describe mode derive criterion set usage radial adequate describe two datum set model reasonably radial update generality assumption need apply probability radial rv detect nearby nearby star rv survey rv update little bias individual determining analyse rv bias receive purpose method naturally commonly use jeffreys efficiently compare model whether great accurate measure bayes goodness set measurement goodness reliably however model assess contain accurate measurement reliably several adequate author determine source aware single discuss single describe importance signal limit sensitivity target survey model superposition trend corrupt analysis rv variation usually despite effort model magnitude dark rv excess explicitly statistical two measurement measurement way practice describe finally combination probable probable arrange system calculate magnitude determine odd determine hasting also several rv existence work play important assess nearby star commonly tool assess probability likelihood density parameter model interpretation assess confidence view probable strong adopt need solid ground probability especially probable possibility explain probability likelihood description independence measurement model model result marginal marginal probability therefore describe measurement bias description practice wants determine fact model bias measurement datum three number estimate e receive adaptive density sensitive robust enable rapid metropolis assume reasonably sample roughly period posterior density nonlinear rv however th member file chain indeed converge five integral digit chain likelihood sake uncertainty semi axis rv draw mass calculate axis density rv reasonably model well motion negligible practice though several rv understand adequate cause whose period noise cause surface refer aspect rv statistical lead analyse
base fractional brownian motion reason come mathematical underlie fundamental describe background stock trading inherent stock fundamental noise model involve fractional brownian motion study paper drift originally observation present differential equation fractional brownian fractional several drift propose fractional parameter differential involve brownian motion compare mixed standard non formulate strong estimate need behavior fractional derivative fractional brownian growth organize fractional establish concern strong sequential result strong consistency drift fractional brownian fu fu bf l notation statement concern introduce eq generalized eq b f l integral respect evident us stochastic wiener brownian parameter lebesgue integral lebesgue integral coefficient assumption hypothesis differentiable exist assumption exist satisfie satisfy specifically old start consider recall fact drift estimation hold together continuity space process mm integrable h martingale two eq observable condition c hc integral process functional convenient ei sd likelihood integrable sd result r tt construct preserve return assumption exist lebesgue observable assume generalize traditional estimate right provide shall investigate strong stopping time form version estimate technical maximal metric space separable element minimal center space exist exist constant follow increase suppose ab random assumption turn separability separability ex statement ready state asymptotic growth series h old take converge case immediately chapter directly efficient sd consistency evident evident non bound ds inequality yield example correspondingly linear version bound function exist non zero consider assume derivative integrable sufficient fractional derivative locally integrable arbitrary locally limit representation true condition omit random hold identically decrease derivative preserve zero estimate estimate strongly assumption addition consistent shall strongly alternatively strongly fractional interval likelihood however hold case present follow form accordance consistent coefficient bound dt strong assumption strongly check obviously dt bound must moment gaussian random calculate attain maximum maximum specifically follow apply formulae wiener integral fractional divide assumption vanish separate therefore boundedness relation behavior let denote several positive sake technical simplicity omit multiplier contain old q admit derivative se bound behavior standard satisfie eq consistency establish consistency case version sde evident construct let estimate se q proof fractional fractional parameter let way rewrite rewrite fu u follow combine ready fractional motion integral du opposite explicitly evaluated similarly eq derive similarly relation handle
survival splitting class early vs failure reduce bipartite survival outcome simplify though class model produce associate test bipartite method strong regularization naive bayes build univariate avoid multidimensional naive bayes curse sophisticated term strong approach smooth method cox well large study type computation real life exclude set size artificial keeping advantage datum build predictor problem build weight vote several implementation observation two survival calculate predictor smooth calculate outcome scoring implement experiment class survival censor censor exclude cosine transform make evaluate spaced default naive density error g smoothed use procedure stand function polynomial two weight formula outcome class tie ci step post improve dataset threshold selection use optimize make update calculate relatively filter tuning marginal class eq frequency ratio posterior conditionally equal prior point association influence otherwise kernel procedure fix ensure smoothness unlike advanced risk modeling method parameter tune survival sir david cox risk hazard cumulative individual failure cox hazard ph hazard dependent hazard rather score r package popular cox ph cox include feature exceed method build zero change aic aic training miss patient free survival diagnosis three characteristic overall among nominal compound fu moderately thing model recurrence death treatment option fu tumor survival patient advance cancer score patient perform usual activity characterize loss dataset record dataset patient feature bc breast record primary description disease al cell contain expression associate microarray cox apply data expert relevant along include patient author aggregate signature signature gene modeling aggregated patient value feature dataset split train split method test split index cox ph regression record miss neighbor imputation cox mean ci bold font features measure small smooth good low advantage prominent high process version original value include signature equally aggregation aggregation penalization use cox ph time almost identical ph model lead selection advantage indicate superiority regularization confirm advantage conduct control aspect instance series experiment artificial gradually dataset experiment conduct list miss near neighbor draw training set size ci number high set small dataset smooth experiment confirm smooth rank quality half year confirm cox ph regression good survival available select artificial logarithm distribute assumption depend variable uniform half assign censor censor censor time indicate observation occur censor formula every reality make less clear record size multiple sample show method train cox ph regression add achieve cox ph tendency method overfitte good accuracy smooth dataset datum serve justify survival study easy factor affect outcome noisy high address analysis bipartite ranking multidimensional avoid curse smooth marginal smooth proved comparison survival cox test method life dataset systematically instance sample artificial gradually increase indeed number make valuable study rank motivated risk bipartite application another study traditional associated sir david cox dependent hazard event survival cox proportional hazard regression popular survival hazard unknown individual result hazard cm bar e mail medical modeling dimensionality simplify algorithm aggregation predictor
atom sampler synthetic bernoulli patch value th stick break prior factor collect iteration variance process roughly fast second present integration two process region analyze poisson process use fall interval poisson alternate calculate replace result break stick break r program david nsf p foundation google breaking construction specifically beta derive truncate beta tight develop process part nonparametric prior collection binary obtain ibp focus dictionary motivate ibp chinese step stick ibp stick breaking finite limit ibp process mean show provide paper derivation addition tight literature section poisson provide immediate beta vary concentration mcmc stick efficient b process review beta process evy review break construction beta evy weight lie discuss generalization atomic measure value contrary measure measure beta parameter bernoulli process show ibp ibp clustering beta follow ibp condition least sampling measure example draw beta poisson generate count poisson pairwise underlie q base goal notion stick break general discrete stick breaking play thank largely seminal stick follow representation previously atomic sequentially receive draw stick break atom stick keep illustrate reduce construction q approximation process show stick distribution derivation derive beta poisson prove beta differ beta process collection section generalization beta basic poisson lemma process lemma concern variable second superposition poisson underlie countable collection independent process superposition I fix let I ic superposition countable lemma dd calculate summation group group use calculate atom weight I df df stick breaking process role truncation mcmc evy measure measure form solution calculate poisson decompose follow eq complete stick underlie process dd beta stick break superposition poisson truncation construction center contour bind plot corollary process arise representation characterize beta process discard underlie corresponding measure closeness truncation bernoulli set parameter could globally share closeness process measure truncate beta slight modification round say less minus bind account atom truncation construct process mean probability distribute truncation additional integral increment give definition constant base poisson limit present partition poisson transition kernel superposition modify construct round atom draw atom draw weight use break break union new process idea stick poisson atom consideration likelihood integration computationally sampler derive quantity specifically collection bernoulli atom care exponential atom multinomial atom atom belong
demonstrate constructing allow hide infinitely typical approach compare alternate compute likelihood generating model ratio likelihood may even hypothesis otherwise choose derive regardless standard algebraic therefore influence therefore exceed interestingly observation bound trivially need rule latent possibly occur number sequence rewrite must put test line weight observation exchangeable k pa simplify domain constructing simplex form hull weight coin requirement normalization exchangeability refer extreme point form extreme boundary region convexity distribution positivity seek example exact hull half polytope suffice description convex instead distribution determining find outside explanation hand conclude example tt outside demonstrate hyperplane dot suffice produce course experimental visualize test problem large develop tool find algebraic finite polynomial inequality conditional parametric inequality equality constraint positivity observable write measurable like represent call hull set construct relaxation converge ultimately alternate representation contain formulation implicitly translation also contain convex hull hull finding comprise amount relaxation set q converse sum square body semi completeness describe polynomial deal simple relaxation degree sum polynomial guarantee negative write semidefinite amount problem semidefinite technique demand positivity bound thing easier set cone default polynomial form sum x g I also cone exclude provide impractical guarantee convergence hyperplane maximize format translate semidefinite matlab convert sdp code large solve difficult construct rigorous example correspond call weight coin hand increase produce constructive specific polynomial statistic direct acyclic reflect rigorous conditional independence equivalently specific distribution term although graph allow intervention rarely inequalitie test quantum specific realization inequality introduction formalism briefly imagine detector pass summarize basically party measurement outcome measurement local variable formula nontrivial reproduce expect party reach hand large evidence favor achievable measure quantum particle extend space observe polytope np whether distribution look tight relaxation individual behavior highly correlate effect influence actor neighbor individual similar suppose friend time begin would certainly influence friend typical attempt covariate either substitute case come relevant take latent actor action various attribute e edge possibly unlike consideration asymmetric give correlation write restrict possible difficulty transition change essentially freedom static demand crucial stationarity transition look stationary coin unlikely independently influence coin intuition precise see mapping observe distribution structure combination e directional variable value parameter take outcome principle consist outcome would like pick mapping q hull visualize statistic correlation vice versa state symmetric fix latent take hull region fig outer dot want e hull polynomial omit brevity hull line construction test explanation correlation alternate reason alternate produce latent second could produce correlation impossible loose nevertheless deduce strength usefulness start real online news com edge semi know model started pick copy slice change apart consist indicator possible outcome distribution ia hoeffding give increase bind suffice quickly increase
analogy mechanic annealing generate stationary state initial increase put mass global crucially prove schedule use schedule guarantee reach surely anneal common geometric schedule schedule anneal sample anneal instance construct sample readily anneal level use accord proceed sequentially modal way straightforward mcmc distribution statistical mechanic learning field many recently transition importance metropolis hastings monte paper mcmc bayesian three describe follow importance employ sampling distribution essential ess measure correspond mcmc sampler name subsequent aim derive efficiency illustrate concluding remark distribution parameter respectively distribute markov generate intermediate sequence subsection adaptively sample obtain anneal motivate rigorously generate size correlate desirable rarely property mcmc stationary importance use independent eq importance reasonably accurate condition analytical distribution markov chain form describe part dirac form metropolis understand example walk metropolis hasting proper transition simple illustration consider panel interest j panel hasting j mention density continuous overcome take j j motivate j n j n n iv define us markov chain generate j ji indeed chain enter accord n n calculate acceptance involve replacement ergodic markov continuous proposal calculate discussion height mm state proposal total generate state generate accept stationary anneal chain aim application e normalize sample usually readily suitable distribution previous sample posterior description aim choose anneal affect aim advance require often available adaptive sampling degeneracy ess ess imply sample ess j prescribed characterize eq level produce ess small reason select aim anneal adaptive annealing give rise height threshold ess anneal level level n I stationary aim j increment anneal total distribution anneal generate chain algorithm description markov generate annealing density choice aim absolutely monte carlo generate sample accurate estimate like highlight difference role hence thereby sample sample become aim increase generate markov chain refer critical safe posterior use assessment numerous diagnostic comparative none one finite mcmc assess run chain requirement understand easy multi exist theoretical unimodal affect speed anneal intermediate inaccurate aim intermediate importance prescribe threshold behavior regardless speed evaluation important relate high speed need follow ergodicity ergodic take place anneal chain notion call integer uniformly satisfied kolmogorov nd equality center ergodic always fulfil hold give enough next reasonable gaussian aim produce practical case recognize acceptance jump neither picture property high acceptance implement aim bound expect associated annealing probability proper transition transition bind equal accord accept candidate step side nothing else acceptance expect candidate basically says never exceed acceptance rejected automatically reject repeat dimension modal dimension model modal mixture denote vector refer display posterior cluster overlap reflect space level posterior I chain anneal b black annealing figure independent find grow aim hasting fair comparison metropolis likelihood I chain aim trajectory markov aim successfully mode interested mean vector component aim variation average display posterior factor run interesting nearly I demonstrate truncated density sample markov sampling posterior five examine challenging detail implementation run outperform capable generate mode five scenario outperform table dc sample level proposal whose spread monotonically high local large local sample neighborhood far look local scaling run observe peak global suggest small anneal indeed natural concentrated attention table anneal monte variation sample aim decrease affect total proposal improvement correlate variation illustrate feed forward neural network model approximation goal function compact base example feed forward activation input unit hide connection connection weight unit properly sum unit cube architecture input unit component introduce prediction output principle produce subject probability prediction initial assume posterior framework obtain integrate nuisance mean aim per threshold ess aim total anneal sample approximate hand intermediate approximation plot visualize approximation plot paper scheme sampling distribution aim well importance simulated annealing behind draw intermediate posterior approximation independent hasting algorithm sampler generate draw name ergodic derive show condition often fulfil parameter recommendation value theoretically demonstrate three modal acknowledgement science foundation award california institute technology finding conclusion recommendation national foundation remark proposition question computing sciences science institute bayesian many posterior monte carlo applicable posterior distribution popular sampling annealing call uniformly markov recommendation demonstrate three example markov express quantity interest respect monte expectation estimate average often encounter multi explicitly strategy importance markov briefly review method play nearly old readily computable sample estimate converge almost surely number condition hold prior stand error especially nearly although make preferred major instead especially know multiplicative constant depend call importance yield poor estimate efficiently importance prior different therefore inefficient complex challenging minimize variance early optimality normalize integral independent posterior distribution physic use
put mass consider conjugate chi restriction ensure wide support beta assign belief highly origin score close fine straightforward recursion analog hierarchical predictive writing let predictive likelihood write predictive run value produce implementation maximization bfgs pr produce recursion methodology factor order score upper pr choice nature induce visit replace number add stability parameter estimation experience permutation negligible reduce permutation maximize discovery represent argue local false rate fundamental pr readily false example choice investigate simulation benchmark result fourier take ensure find nz du tail nz nz asymmetric portion mass concentrate origin nz away zero six form time pr regularize average permutation table summarize simulation accurate omit pr specifically practically namely smooth zero time roughly second rate rate nan figure oracle oracle fdr message test across sparsity similar non test relatively false rate spike large theoretically bayes bayes observation may interesting dataset careful versus microarray consist differentially score fit clearly fit fdr thresholding report express score standard z report hope comparison oracle method pick reasonable bit oracle false plot black black solid gray fdr value gene fdr gene fdr oracle fdr restriction component nan purely verification kind seem could insight entirely formulation accept notion produce z magnitude j selection zero ability still perform case yield model argue abstraction indeed test score magnitude origin give order interesting ordering wider issue conclusion focus work behave nan score indeed identify impose strong theoretically nan suppose base assumed uniform purely verification could insight biological entirely nan justify formulation make parameter identifiable likely encourage flexible situation concrete average identifiable wide abstraction score whose magnitude shift order wide nan breast study exist fail discovery point dimensional set classification gene collection classification statistical shrinkage whether interesting reveal limited resource force interesting employ exhaustive calibration infeasible multiple vector exploration intensive multiple testing software software find website acknowledgment author grateful associate anonymous suggestion discussion portion department mathematical university see take e degree satisfy therefore recall say generality imply rearrange modulus unbounded bound contradiction symmetry equality relation easily complete variation predictive pr pr development consideration mix detail find focus text unconstraine e kernel pz u nz gradient compute return identifiable consideration empirical nan case use computationally recursion nonparametric mixing lead estimate false simulation demonstrate handle tail turn realistic conclusion predictive recursion testing arise abstract score define derive treatment control characterization rejection insufficient perform major development shift treat independent treating exchangeable testing model share separate elegant testing problem assume mixture describe distribution z reason adequate choice various draw throughput particular responsible approach offer substantial advantage technique currently formulation application typically link phenotype exist two take encouraging close conservative detect majority interesting microarray study breast study method produce cover tail score leave nan zero gene make number tail study breast group representation likely magnitude belief score wide range exist fair interesting method study mention find classification similar simulation score range base group efficiency recent stochastic recursion estimation mixing dominate address use mix marginal strong dirichlet calculation group model specification parameter plug present section non nan example artificial microarray oracle likewise breast produce fit tail consistent sophisticated technique nan lebesgue version identifiable general identifiability nan identifiability shift tail correspond nan incorporate similar belief component strong tail histogram report comparison separate tail important drive force
derive pairwise associate weight image optimally align optimally transformation transformation dimensional shape orthogonal linear construct principal map dimensionality heat heat field transformation hilbert particular metric distance application meaningful transformation organization connection manifold way manifold point cloud dimensional specify vector heart mapping construction way embedding laplacian cloud sample dimensional manifold operator function result convergence except geometry achieve basic connection laplacian explain report diffusion nystr role play heat heat relationship diffusion geodesic distance nearby application reference alignment extension topological though datum cloud high dimensional manifold manifold cloud consists view orthogonal cloud local plane pca fix keep convention except confusion arise neighbor satisfactory choose boundary curvature due slightly define data neighbor purpose disadvantage proof pca monotonic diagonal define vector purpose emphasis nearby singular advance neighbor exactly b estimate number zero singular value curvature singular practice dimension value account enough percentage threshold estimate singular singular integer dimension manifold round sum median intrinsic q minimize sum estimating compare step facilitate notation write svd column orthonormal know define vector column establish eigenvector corresponding interval never actually form requirement perform intrinsic use try simple able alignment suppose nearby parameter later manifold boundary tangent space space exactly matrix differ orthogonal equivalently operator copy subspace usually exactly necessarily orthogonal close orthogonal local basis optimally align orthogonal refer optimal orthogonal basis alignment matrix basis align nearby align graph vertex correspond point basis align iff weight construct matrix either multiply zero w regard length vector tangent average non linear single transition sum end use complete eigenvalue diffusion mapping eigenvector usual dot diffusion weight start affinity consider connect sum path transformation length orthogonal multiplying transformation order result analogous operator geometry path point curvature thus add may happen affinity affinity affinity consider matrix affinity square path measure connect also transformation agreement differ eigenvalue eigenvalue decrease tv li nd ti affinity finite dimensional hilbert manifold embed invariant dot diffusion dot product v li li ti inner space mapping consequence well large truncate equivalently ti ti non manifold would real truncate diffusion slightly different diffusion mapping matrix eigenvector another vector schmidt embed datum hilbert proper vertex define diffusion ti ti ti unit angle embed point mapping suggest distance ti ti mapping obtained suppose define symmetric degree nd tv sd tv next diffusion discrete walk diffusion manifold graph converge pointwise operator smooth function laplace operator point graph laplacian eigenfunction pointwise convergence sample manifold laplacian converge high error case converge laplace state potential depend operator additional terminology thus specify way generalize map map heat operator heat laplacian diffusion eigenfunction th vector laplacian formulation far riemannian manifold assume dim riemannian metric canonical j k x accord support connection laplacian lx di lx dd lx I hold slow compact however dim riemannian embed induced way choose dx x I choice appear consequence term discrete connection homogeneous satisfy remark boundary approximate heat dim induce laplacian geodesic square integrable vector diffusion eigenvector detail obtain wide purpose numerically embed bar numerical rotation calculate theory agreement bar diffusion distance distance manifold embed mapping compute vertex correspond truncated diffusion eigenvector case embed embed numerically compute due sampling truncate embed embed similarly numerical h square transformation result usage want eigen connection embed truncate embed embed dim sample equally spaced point interval diffusion embed dimension embed truncate embed figure h sample spaced grid fix truncate calculate truncate diffusion calculate eigenvalue embed smooth xx xx xy arrive real nystr extend local alignment mapping approximate alignment find subset project embed represent orthonormal decompose suppose step orthonormal basis approximate plane local use among point inside ball center use eigen field finish early graph converge generator heat heat connection laplacian adjoint tangent bundle accumulation distance heat smooth analytic vector mapping hilbert pair need vector choice property map compact manifold dim basis field laplacian vector diffusion continuous note compact smooth xx xy xy tc xy xy x mx ny tx tv tv tv embed diffusion theorem diffusion distance diffusion behave geodesic distance smooth dim close similarly expansion coordinate denote possesse close bundle tangent bundle also equation thus eigenfunction operator rewrite q heat kernel laplace operator laplacian trivial bundle also heat put fact besides robust reference alignment object dimensional periodic shape describe problem dimensional arise comprehensive problem reference arise graphic structure image ct determine structure channel protein determination ray ray since far attempt ray ice thin typically disjoint promise structure typically whose pixel image proportional line path image highly impractical projection ct shot maximal ice partial homogeneity set hereafter random rotation structure collection noisy rotation orientation orthogonal transformation three rr tr describe orientation exclude intensity plane integral along path image potential fix laboratory coordinate know ray column orthonormal plane angle clean rotation rotation basis plane rotation plane rotation extremely group noisy raw within single double within result average angular average raw clean projection white noise simulate choose visible denote angle plane average noisy alignment snr clean frank invariant mean procedure identify similar invariant define euclidean optimally align rotation center rotation operator image angle prior distance radial angular rotation result center true pixel refinement procedure challenging assume image worth note proximity average cross distance proximity practice distance although emphasize effect find average share angle perhaps plane ideally ball center neighboring alignment angle happen match plane lead spurious neighbor average image poor image invariant neighbor know direction image angle indicate identification neighbor angle outlier snr outlier direction neighbor whose angle indicate angle way perform much na I near neighbor average distance share reference distance angle utilize mean algorithm snr experimental averaging far improve detection information plane rotation angle base powerful noisy snr near diffusion pair neighbor angle degree histogram angle figure panel neighbor identify invariant distance identify angle number neighbor apart algorithm show robust introduce diffusion map algorithmic consistency transformation along connect affinity inner point hilbert distance point distance orthogonal transformation scalar procedure seek alignment also rotation diffusion correspond orthogonal rotation organization structure graphic shape framework become collection manifold orthonormal tangent space mild manifold prove transformation procedure approximate careful pca prove manifold sample uniformly lie heart framework connection graph prove operator sampling prove asymptotic heat generalization diffusion determine latter double basis tangent space example vector mapping topological cloud extract use modification vector order product high act form topological harmonic form laplacian topological therefore modify approximate laplacian instead laplacian basis use tangent may parameter effect vary problem multiscale resolve recommend incorporation mapping difficulty face deal locate necessary limiting estimation space recent method improve result heart diffusion whose block matrix tool analyze eigenvector eigenvalue may also matrix haar orthogonal presence example matrix mention early mapping framework alignment process transformation focus utilize point probably ask transformation utilize remark diffusion mapping framework group compact less obvious arise extend diffusion compact group partially support award dms nsf award fa award gm foundation wu discussion regard express university institute stanford university air force appendix mathematical background reader familiar operator surface rate directional derivative eq extend field way notation directional denote direction vector field look first generalize embed dimensional surface smooth curve affine span collection tangent denote embed tangent plane tangent please plane affine figure tangent vector differentiable map abuse usually understand embed rigorous field please field derivative difficulty make belong easily change bit curve guarantee theory face make sense obvious differential play essential role fix field parametrize q ordinary along curve denote back address derivative say want direction comparison sense live notion derivative point tangent plane always mean know field vector axis definition generalize field detail operator heart equation property setup roughly speak manifold surface theorem state throughout smooth dim compact embed metric induce canonical denote dx np step expect large hold automatically tensor embed denote ease notation sequel denote connection whenever divide form approximate embed tangent plane prove approximation crucial proving geometry x high orthonormal dim subspace orthonormal minimizer column p well near seem bit radius indicate observe improvement relevance deviation error pca near boundary without choose manifold choose remark expense slow orthonormal alignment procedure connect boundary crucial proving proof lx orthonormal accord third approximation section tangent bundle parallel theory lx lx fourth expand laplacian operator plus term first third vanish vector field sufficiently smooth potential vanish laplacian geometry field x put theorems theorem theorem theorem get I ip ip lx I upon divide vanish specifically dominant surely proof please fix identification p taylor expand exponential map x tv xt eq obtain conclude next embed q evaluating give pd taylor follow calculation small taylor similarly put get please fix sufficiently choose enough neighborhood expansion together fix elsewhere properly translate pca coincide leave singular rewrite denote geodesic radius identically number q evaluate moment substitute apply taylor lemma integral odd power vanish symmetry sphere consideration become move establish estimation purpose establish bind define calculation similarly provide quite say inside identity size due curvature finite statement note rewrite identity symmetric orthonormal eigenvalue order dd matching note generate sampling column dim span point without full integration domain jk odd power vanish expansion case bernstein inequality provide
gaussians mean could explain single address change sample program depend specify program transformation frame replacement program abstraction abstraction remove abstraction replacement abstraction abstraction abstraction abstraction abstraction program abstraction abstraction abstraction body remove assign body instead value pass remove abstraction abstraction variable abstraction program abstraction instance abstraction body abstraction abstraction abstraction make name abstraction abstraction body abstraction abstraction abstraction application list abstraction abstraction abstraction map abstraction body pair body program abstraction abstraction application location deep nan primitive else abstraction program abstraction abstraction abstraction abstraction abstraction list function abstraction change frame remove abstraction abstraction empty abstraction abstraction change variable position define recursive argument application argument remove argument argument variable position abstraction abstraction change transform abstraction program remove application transform compactly represent identify induce structure place noisy two call replace abstraction abstraction variable instance replacement abstraction program shown begin color color uniform choice identify instance replace abstraction change color node size program begin v node datum color choice application big simplification affect program search improvement fit datum principled apply less generate remove v color color color argument replace choice indistinguishable value look program abstraction create argument abstraction advantage potential redundancy merging argument transform frame abstraction instance match abstraction abstraction variable program abstraction possible match find program abstraction match pair pair equal car cdr cdr match replacement abstraction variable match abstraction instance match similar application object number explain first choose abstraction choose abstraction whether abstraction match variable remove argument similar argument indistinguishable unlike program start v color datum apply program match variable true begin gaussian v uniform program size color gaussian compactly represent transform call pass function program terminate induce recursive illustration capture abstraction valid variable recursive call filter abstraction application abstraction recursive call abstraction valid terminate terminate abstraction call nan valid recursive call terminate call valid instance flip recursive call uniform choice recursive call terminate abstraction non define abstraction name hash abstraction name list abstraction hash abstraction abstraction checking status name hash ref abstraction name define set name new status hash abstraction new status abstraction abstraction status terminate status abstraction f status abstraction begin abstraction name program abstraction name abstraction terminate base branching list abstraction abstraction map base base call illustrate transform abstraction transform f remove change definition former outer body remove get flip datum start demonstrate compactly infer process call represent program flip color color color generative explanatory size color use gain noisy use instead construct frame third noisy replacement instead return sample return call replacement abstraction instance map instance gaussian induce noisy use program three third show node three incorporation three structure color color node merging generate node make stochastic noise color node f probabilistic implication solve take run want generate remove within problem step set program noisy occur program dimensional recursion program generative one induce simply call overall merge procedure posterior define depth transformation depth top transformation sort transformation program search depth lp program depth program well take sort program perform recursively program transformation depth filter program frame depth transformation transform program filter depth program apply map depth program reduce preserve program transformation us program semantic preserve transformation program semantic preserve apply transformation semantic preserve transformation change program transformation semantic semantic semantic change program define transformation semantic preserve program program transform transform program program program semantic program program compute frame sort program program program semantic semantic program log program semantics semantics program program posterior prior program program semantic posterior list new illustrate example domain colored example program sample program pattern show pattern recover representative merging result differ color gaussian repeat ten incorporation learn create width color f program ten mostly mostly flip merge abstraction depth begin f uniform choice tree true flip repeat width recursively call stop build begin f f f flip f observation reach trade prior recursive adjust occur merge demonstrate width define color color size begin color flip uniform demonstrate parameterize recursion place program pattern branch run width set trade likelihood introduce program branch would amount program compression define node define flip branch branch define color program program function single color node size pass branch end begin v flip uniform size uniform merge induce model central translate program program identify repeat compression goal include efficient computing furthermore consideration systematic machine algorithm world capture rich limited human engineering pursuit plus minus report approach program pattern choose language program extension abstraction generalization extent merge induction explanatory use abstraction form find key nest list might series tree green branch branch either pattern intelligence human rise form rich language capture program probabilistic program play prominent modern learning free lead pattern able capture feasibility learning class much limited develop tractable investigation take explore might proceed expressive identify abstract program probabilistic language represent program program programming program parameterize recursive searching merging simple colored algebraic extend guide use probability begin transform explore transform program explicit move formalize transformation learn probabilistic idea color e status contain detailed code illustrative example document progress complete inspire high aim illustrative example merge search accurately merge transformation posterior give datum building incorporation set generate hypothesis program transformation model generalization successfully artificial represented gram extend merge rich program represent context parameterize use transformation compression describe implement program give form use program type lambda abstraction operator probabilistic program objective program program program estimate em move move abstraction transformation improve search strategy objective function move algebraic point translate derivation sequence datum nest expression attribute list node tree color color size visual way represent interesting merge tree var flip primitive primitive uniform color apply program merge different list etc merge data incorporation incorporation going create expression evaluate algebraic combine uniformly list tree tree expression rest datum easily convert expression tree question leave incorporation structured feature program true data color color node size gaussian node size node like illustrate generate program pattern improve meaningful branch body branch body branch branch branch body branch branch size flip flip branch branch datum combination flow lambda program branch body connect branch branch connect small node color create identically color reflect capture branch program initial incorporation structured form program represent create symbol function name frame abstraction body make name abstraction body define name abstraction name body abstraction name body abstraction name abstraction define abstraction body fourth program consist list program body define keep track motivate idea computation transformation affect program program log semantic preserve list semantic program program third likelihood fourth program preserve define program incremental forward choice sampling observe evaluate tree smc core core smc argument repeat symbol find symbol member repeat sample mcmc use parameter compute node child score data parameterized attribute parameterized attribute parameterized list attribute original variance attribute attribute parameterized inf gaussian variance third incorporation program pattern prior abstraction abstraction aim syntactic pattern program call newly create remove act computation merging transformation potentially lead well generalization property successful implement lambda anti matching pair program occurrence program program program nan program cp size cp compress compressed program abstraction body program define list unique map anti abstraction abstraction transformation simple size node node b true f node v behavior transformation semantic equal transformation partially match contain common example common create potential abstraction transformation partial pair partial match expression anti tree expression list interior primitive element number tree partial match expression find subtree representation represent anti common match anti proceed recursively primitive primitive return list return frame anti begin define variable var primitive primitive primitive add add else pattern reverse anti list match root match match since match match list size match variable match find list primitive final lambda create partial match attempt abstraction abstraction body replace match replacement abstraction frame
configuration algorithm krige surrogate function nest reliability analysis evaluation function optimizer extensively numerical paper article rectangular load moment respect axis load refer neutral additional material elastic perfectly stress originally whose original paper gaussian small coefficient respect objective formulate follow failure choose optimal limited satisfy constraint minimum probabilistic denote form index cost line row reference coefficient variation failure carlo second approach slightly lead turn failure dependent example self quantification third reliability krige surrogate limit krige constant regression refine per refinement iteration mm reference krige comparative reliability illustrate index behave acceptable occurrence acceptable mode structure due load failure cd exceed yield performance read stress member ab load read euler load force member ab read probabilistic variation along kn gaussian mm gaussian find rectangular cross member reliability requirement respect result ab cd ab ab cd ab ab reliability form reliability check reveal reliability index implement approach plug krige surrogate confirm krige save provide error reliability oppose reliability h aim develop reliability design engineering consume expensive evaluate probability krige strategy counterpart indeed convergence evaluation quantify sequentially onto final numerical efficiency property krige surrogate build call simply surrogate one thus performance also worth refinement make add thus availability platform krige efficiency amount real latter investigation publish author grant present design g nonlinear finite use starting failure quantify estimation failure advance krige surrogate error surface failure sequentially krige surrogate reliability analysis refinement reliability reliability sensitivity analyse adaptive surrogate estimation finally classical order problem krige surrogate reliability thus nest literature three structural mechanic mechanic aim find cost often lie boundary account uncertainty realistic despite attractive field limit mostly simplify assumption intensive attempt efficient bring field sophisticated real word safe within evaluation quantify minimize induced development devote formulation literature review argument surrogate resolution section introduce krige specific emphasis put refinement krige surrogate nest reliability loop model describe read objective bounding call soft constraint consist prevent negative infinite constraint evaluate additional system failure oppose involve evaluate box finite probabilistic minimum acceptable probability failure term integral define word joint later gradient design might consider sufficiently argue full eventually account randomness extensively thank analytical simulation c straightforward consist reliability analysis loop conceptual often lack since require consume range performance weakly reliability nest able formulation often refer nest despite concept reliability transform reliability inner reliability quantile argue much nested resort reliability optimization towards refinement criterion note reduction overall run demonstrate applicable grow increase availability resource pc double loop sequentially reliability optimization perform assumption problem simple efficient closely relate notion partial certainly soon deterministic build design simulation design uncertainty design probability closely reliability explore possibly refined minimize whereas approach consume task surrogate surrogate build refine evaluation various surrogate reliability surface polynomial expansion machine krige classical reliability theory literature surrogate krige krige surrogate address mathematical call space simulator present consume krige spread depend prediction purely lack discussion distinction source reader refer property krige surrogate essence krige start uk process read term part require mean autocorrelation function read accord autocorrelation autocorrelation generally autocorrelation know common zero polynomial ordinary together dramatically output fidelity choice may low fidelity analytical build simplify construction krige subsection aim gaussian correlation mean krige indeed ii mi symmetry relation prove krige autocorrelation common statistical krige methodology field turn methodology well suit purpose methodology maximize respect one optimality numerically tractable within toolbox application consist weight improper pdf volume though explain hereafter note pdf sample mean well highly concentrated property cluster failure extreme failure uncertainty standard unfortunately failure section spread useful surrogate accurate reliability stop refinement reliability define application criterion refinement stop propose criterion additional replace respective accordance simulation reliability refinement g generate candidate improper technique sampling reduce use newly point update krige autocorrelation update reliability g perform onto failure order measure tolerance k new I refine illustrate apply nonlinear limit surface show contour population slice give pdf several mode section black krige black bound line interpretation blue point adaptive strategy previously krige surrogate augment reliability krige indeed build krige reliability would refine krige call augment reliability standardize n suffer argue cause performance present augment reliability keep equal vector indeed augment account design consideration read pdf assume illustration provide augment reliability space axis simple augment make limit surface potentially optimization precisely accurate choice onto space marginal vector able compute univariate assumption contour region tensor confidence interval margin order quantile interval involve margin parameter margin quantile level follow problem function margin domain rectangular derive margin analytically numerical solve mean gradient property quantile respect location improper volume refinement krige double loop mean proceed optimization nest reliability reliability perform reduction krige surrogate simulation sensitivity detailed advantage failure indeed point state failure joint pdf analytically probabilistic distribution copula trick unbiased read follow sample estimation estimation additional run simply estimation concept easily subset simulation detail toolbox matlab sample refinement improper pseudo refine optimize refine optimize refine j j ji refine c region bound
trajectory origin consider quantum markovian quantum study planning describe spin p warm support grant aid science reference reveal recall approach uncertain model situation kind heat heat several concern govern temperature might quantum paper term stochastic quantum markov quantum neural us extensive asymmetric build basic network early pointed energy treat spin researcher matter pick material remarkable theoretical mathematically concept storage capacity phase retrieval spin phase utilize replica introduce prevent build compose namely spin boundary control capacity temperature heat storage capacity energy function monotonic consistent signal analysis enable derive couple self mention theoretical neuron heat artificial might kind conventional model add classical hamiltonian shall quantum especially investigate structure diagram replica method static equilibrium understand already theory evaluate recall powerful point equation take account correlation asynchronous dynamic dynamical replica utilize sub self noise derive quantity theoretical recall quantum systematically candidate dynamical deal shall quantum investigate differential respect overlap paper organize follow origin artificial quantum clearly quantum carlo recalling dynamic quantum deterministic build pattern master monte static apply case namely pattern via asymmetric conventional last summary divide namely model put noise quantum system hamiltonian identical hamiltonian network eigenvector classical mainly consider dependence however even numerically hard reliable become huge realistic brain use quantum computer recall dynamic quantum algebraic calculation due operator hamiltonian namely order classical system slice simulate attempt flow master regard equilibrium neuron strength classical go symmetric hamiltonian transition specify k denote neuron neuron obtain immediately imply classical slice master spin flip operator pick overlap build states sum substitute introduce notation get around complicated expectation k flow average equation local field realization sub k notice appear nothing effective neuron neuron state along axis differential form substitute tm tm side equation integral subsection carry overlap slice call inverse study numerically validity approximation simulation successfully validity recently argue path immediately mind
decompose eq estimate access second measurement q simulation incremental determine cycle cycle average trial size size incremental set incremental cycle algorithm fair gradient exchange metropolis gradient average give respect minimizer biased away factor second derivation eliminate result due replace curve incremental db gain factor steady value steady state mse large long regularization fig deal localization problem diffusion strategy know component cost multiply eliminate node minimize individually instead seek arise communication diffusion manner node access case available base estimate target neighbor simulate stay size incremental average exchange metropolis gradient close mse expression diffusion step become performance diffusion mse furthermore localization local gradient next algorithm scenario along incremental step continuous track decay step tracking diffusion optimize adaptation distribute interaction track state localization incremental cyclic side sum iterate q show guarantee right converge side eq require recall definition respect arrive block block block stack vector top block x nm mx x symmetric note nx diagonal inequality generality maximum define large magnitude q obtain theorem radius matrix stability matrix examine satisfie eq norm stability mn diagonal also block diagonal substitute within require obviously guarantee condition edu global individual allow information real help effect noise continuous error steady apply diffusion study incremental method require cyclic link diffusion network enable biological category sum individual component spatially distribute maximizer biological network interaction resource allocation online latter underlie manner network technique optimization manner notable incremental consensus incremental cyclic cyclic manner determine cover np cyclic along path share cyclic stop consensus vanish consensus optimizer steady state however environment prevent step early motivated introduce adaptation use global problem process process continuously diffusion encounter resource cognitive pattern recognition optimization wide class function approach tend algorithm size condition datum necessary adaptation result exhibit organize sec use taylor expansion optimize alternative strategy sec square statistical perturbation gradient sec application problem localization finally conclude paper notation vector regressor take simplicity letter denote quantity regular font letter denote diagonal entry stack symmetric difference collaborative manner minimize individual function differentiable minimizer function network work attain track chemical physical frequent context biological group location location common problem process individual study challenge nevertheless converge taylor approximate optimize alternative descent within neighborhood spatially motivate approach node represent value assign neighbor leave coefficient new combination nonnegative convex actually guarantee strongly treatment ahead approximate taylor subsequent second result substituting expression ignore approximately expression relate original function second side wish hessian node expression lead explain kk share reasonable initially summation adjust early term function approximation correction step proceed development practice say observe vary index approximation common view eigenvalue replace show stage scalar embed rewrite scalar nevertheless localize constitute point development minimize move namely step size adapt close get optimal mse sufficiently size lead add estimate update correction add one intermediate use arise examine optimizer neighbor would replace help bring influence arise generally estimate therefore replace incremental widely literature describe q note nonnegative coefficient arrive combine diffusion structure originally square satisfying intermediate involve receive neighbor use step intermediate generate perform similar order motivate alternative adapt diffusion originally square error square ahead strategy albeit vanish ensure towards choose sharing optimization node agreement size local word coefficient weight doubly performance every ensure across address strategy treat diffusion three mode interact choice derive version exchange enforce doubly decay satisfy one error introduce q subtracting give relation twice hessian symmetric component apply introduce across symbol kronecker product introduce assumption cost gradient follow exist ensure function strongly minimizer exist denote past definite x similar subgradient method norm uniformly bound restrictive allow unbounded condition allow grow instead noise regression u case easily state note random absolute appear necessary gradient quadratic cost common square linear instantaneous assume illustration purpose arrive strategy originally extend solution square sequel square towards steady state steady performance rate answer largely influence constraint examine diffusion variance weight evolve evolution reasonable profile form expression characterize relation evolve hand denote actually truly recursion nevertheless relation recursion regard however via turn challenge reason alone filter analyze filter adjust argument enable state node ahead expression jensen derive norm eq upon weight symmetric positive definite matrix k establish note yet n meaning lead combination hand inequality sum obtain introduce error write notation denote component entry vector nonnegative combine inequality certain condition mean bound use subsection state small size stability satisfy condition entry appendix examine conclude validity establish strategy stochastic gradient absence tend vanish interesting free impose provide size stability expression since norm entry worst verify enough nonnegative factor define eq sufficiently size denominator substitute step step
individual recall hoeffding use martingale martingale easy define great lemma proof martingale eq combine great inequality entire pick dependent simultaneously despite great compare inequality corollary corollary imply n hence corollary match corollary derive minor general mention imply bn bn martingale corollary least tight hoeffding simultaneously project tight corollary hoeffding application value drift tight q tight hoeffde whenever actually practice manner obtain bernstein bernstein opposite case estimate inequality bernstein pac average kl bayes inequality grid vanish inequality martingale difference variable variable enable reduce study simple study application analog hoeffding hoeffding logarithmic case drift walk region tight hoeffding bernstein inequality importantly present concentration average multiple simultaneously inequality control standard one important tool evolve process useful domain analysis importance line convexity imply first note expectation side order last concentration bernoulli information bernoulli variable factorial restrict tight since function convex corollary inequality variable markov take normalizing proof theorem variational root back theory physics relation generate respect bound use theorem respectively change measurable distribution eq since exceed reason finite jensen nh h r convexity divergence hold great inside nh nh n normalize minimize distribution simultaneously proof value many possibility instead relevant range note bound cover grid ni relevant need value strictly take condition pick exceed substitute factor know obtain relevant range minimize pick weighted complete substitution individually simultaneously intermediate technical condition martingale fact apply reverse conditioning result author valuable improve presentation european community european publication reflect author computer science institute systems science college research reinforcement supervise unsupervised processing bioinformatic ph mathematics universit e thesis solve seven international distinguished mathematic universit work verification system pac computer universit di interest theoretic author games taylor ph mathematics subsequently complete sc advance publish college development inspire theory boost show analysis co introduction support book analysis publish receive mathematic symbolic university technology learn california accept area theory symbolic computation research project european union interest include exploration lemma college uk mail universit mail universit di mail mail ac present set weighted learning set importance reinforcement interactive encounter comparison difference shift expectation independent derive tight analog hoeffding bernstein pac bayesian fundamental modeling study hoeffding present inequality shift function analog hoeffding index martingale represent martingale denote everything clarity possible average simultaneously evolve illustration inequality infinite individual high averaging law depend sample inequality importance weighted widely estimate draw proper reweighte control expectation base unweighted form order observe average law round application technique reinforcement average generate hoeffding bernstein bound martingale sequence shift lemma variable lemma p analogous theory probability combination analog great explicit bn similar obtain hoeffding however certain less tight hoeffding tight bernstein provide control average multiple evolve result
value work posterior plot produce stable estimate replication seven quantile typical left panel plot link right plot fit visually smooth fit value significantly autocorrelation plot autocorrelation appendix nearly cause numerical numerical place evaluate ht plot jj conduct replication table error next exponentially superiority setup except error set illustration error follow estimation value estimation illustration generate fit case index general limitation far satisfactory median increase demonstrate setup tn table save compare show still ht ht finally tc pose economic statistical tc datum occur wind north early quantile analyze influence wind fit quantile extreme three tc intensity tc intensity index qualitatively reliable level especially column histogram figure imply equation performance simulation suggest fitting obtain sim unstable quantile algorithm superiority modern carefully partially collapse simulation quite encouraging exponential possible explicitly incorporate mean function normal thus consideration zero exponential flexible possibility model mainly spurious underlie come likelihood particular may accurately accurately estimate reader incoherent inference quantile intersect proceed separately nothing prevent overlap term computational ordinary pc fit dataset setup minute burden slow speed mcmc algorithm variational desirable scope index pose serious challenge thank associate three anonymous helpful manuscript education detail n e mathematically distribution full inverse nearly singular integrated avoid inverse matrix parameter consider q conditional modify metropolis manually tune manual simplify transform run partially collapse sense specifically obtain partially collapse modify sampler intermediate marginalization improve example corollary division university school business usa distribution mechanism develop approach single quantile nonparametric vector popularity monte carlo careful consideration conditional partially collapse frequentist gaussian chain quantile sim provide estimation curse nonparametric covariate compare offer nice fitting base spline literature sim study velocity acceleration study trying identify incidence stable estimation inspire work area frequentist long bayesian spline process gp gps subsequently sample work limitation supplement description response relationship response tail robust tail article single quantile quantile response implicitly quantile dimension univariate variate treatment quantile adopt laplace alternative support hand constraint identifiable norm mathematically impose keep parameter advantage specify note matter heuristic sign norm prior choice put component sometimes interest recent identically prior attractive far generalize normal g generalization variable generalization straightforward
ia jj array equation norm I matrix order pdf distributional linear moment marginal independence well array variate eigenvector provide optimality estimator merely check variate consider paper variate assumption rule variate covariance state problem put restriction last matrix attain array variate flip square assume n l ia calculate root st array observation q array diagonal array let known estimator consistent hand assume kronecker delta develop size considerably assumption covariance kronecker estimate obtain could ccc ccc ccc ccc plot compare piece general array slice array assumption variate estimate eigenvector eigenvector kronecker small reduction dimension obtain ordinary n assume repeat result summarize matrix center delta covariance regularization expression tumor compare tumor normal tumor definite model theorem hold testing covariance inverse propose rank calculate reject figure array htbp compare true eigenvalue right compare true kronecker delta estimate use scenario reasonable variance covariance identity diagonal kronecker delta finally figure true kronecker htbp htbp htbp observation classify tumor misclassification covariance estimate normal tumor summarize finding misclassification practice addition permutation reduction covariance parsimonious model propose covariance like issue important glasso package implement shrinkage conjunction covariance individually sparse selection sparse glasso glasso estimation glasso shrinkage glasso aid model expression dataset order variance expression tumor color high little among discrimination remark cluster extraction paper introduce call essentially assume kronecker finally implication high collection capability lead dna micro array million easily less characteristic bioinformatics field variable usual sample covariance method classification estimating essentially covariance realization kronecker delta kronecker corresponding matrix
quickly nearly demonstrate increase sequence marginal converge set payoff follow feasible saddle point restrictive carry admm relate method decade field reader refer spirit fundamentally adversarial definition robust department electrical stanford university email stanford optimize structured uncertainty problem game nature subset variable try assign maximize von minimax randomize strategy player several structural property minimax assignment choose satisfy dimensionality introduce pass solve minimax generalize max belief propagation whereby strategy player call strategy optimize opposite game capture situation agent limit pool remarkably applicability economic theory learn theory estimation game try design good statistical nature choose close engineering reduce maximize appropriate course inherently importantly real design nominal try bad player theory control present take complementary explicitly index let pure index parameter robust course hard indeed adversary special case approach follow factorization shall objective sum local whereby node control naturally abuse terminology shall objective robustness graphical model effective relationship think instance subtle disease medical diagnostic system normally model family probability g logit expect distribution justification prediction robust robustness implicitly never carefully investigate apart graphical simplification strategy introduce established game device play try expect utility try consequence linear program von existence saddle point strategy pair form nash equilibrium order nature indeed remarkably bad expect utility pure achieve pure hence key general formalism pass minimax finally work ise ise alphabet unnormalize inaccurate challenge find uncertain strategy depend model variable distribute two finding model tree classical check figure summarize result different run increase perform give consider nominal response experiment compare strategy strategy strategy optimum strategy outperform long adversary nominal value control node neighbor denote analogously control discuss condition distribution small since belong lp graph writing take play role problem generality field mrf marginal pay product thereby loss generality graph distribution degree specify mrf completely marginal minimax definition belong fact simplex necessarily point possess separation tractable relax marginal denote polytope q exact alternate multiplier solve optimization problem transform admm guarantee case admm short presentation admm reader q indicate norm admm try solve optimization closely augment indeed despite
ad full present iteration phase avoid division little matrix algorithm sample cost strongly influence computation need well run rough computational empirical iteration chooses add objective derivative choose coefficient compute e value alignment continuously parametrize base parametrize good early kernel restriction solve encounter program matlab simplex center k kernel real see list method cr continuous multiple kernel search follow list cr da kernel provide follow combination kernel improve achieve single family dimensional performance illustrate h design interval label show experiment dirichlet kernel investigate classifier think good single good frequency misclassification frequency rate kernel frequency figure cr misclassification close frequency show indeed discover frequency sake parameter space dirichlet kernel define insufficient easy discretization dimensional method serious space parameterization multidimensional require inferior wish always obtain frequency design second vector seven measure repeat experiment misclassification alignment running version nd search bandwidth bandwidth average modify parameter kernel bandwidth constitute find material large number since able cope drastically small idea scalability nd large tune increase multidimensional running nd htbp finite base kernel matches generate frequency kernel intend value without come kernel kernel appear parameterization kernel continuous table uci repository rank contrary letter uci letter experiment take du expect fastest cr stage method da cr around dc cr fast rest continuous iterative algorithm solver whereas terminate algorithm require fast alignment novel method learning span kernel ascent forward stagewise correlation kernel method show benefit compare successfully deal dimensional kernel space experiment start might performance think design competitive interesting future whether exist provably although directly natural dictionary search however thorough investigation work secondary alignment large necessarily misclassification completely implication help construct synthetic potential avoid continuously parameterize example multiple significantly alone datum competitive multiple show multi parameter growth kernel notice limit directional derivative rule calculation cyclic uniqueness notice denominator solve root root contradict value work maximizer detail iteration objective function sign local minima mnist uci letter experiment specify iteration function kernel select uci need find cr employ matlab multiple multi ca particular run space expensive equal element kernel weight soft validation tune validation value decide experiment I factor choice bias though similar cr also validation set together result positively run everything training classifier validation nd expensive alignment readily multi method see transform performance alignment kernel alignment breast cancer diabetes heart lemma lemma pt ari department compute science ab base select step demonstrate art variety world explicitly demonstrate importance dictionary kernel base set however amenable demonstrate positive continuously parametrize implement ascent stagewise alignment technique several strength case multiple learning parameter heavily influence method function idea consider kernel learn linear belong parameterize continuous e extremely gaussian set available literature kernel paper reference priori problematic contain moderate say since number moderate discretization accuracy furthermore independent might explore avoid continuously future gradient directly main alternate jointly convex excellent avoid forward stagewise robust implement expect suffer minima belong group stage kernel two stage kernel fact alignment metric objective technical implement compute objective solve nonlinear optimization problem solve overall excellent completely boost greedy cf chapter new exist efficient closed determine add serve purpose explore potential advantage limitation technique reliable despite potential implement successfully simple combine kernel issue weight fact recently finally method cost stage method dataset previous method experiment new competitive less reveal insight method performance kernel give rise overall use metric bad like necessarily well primary overcome use combine seem mainly comes discover fact method search avoid conclusion dictionary important dictionary dictionary certainly already discretization infeasible dictionary competitive art kernel considerably control construct example nontrivial family achieve method gain correlate predictor set give rise center surprising derive favorable method method learn discretization publish force multiple virtue lag illustration difficulty parameterization deal show observe considerable gain algorithm outperform accuracy comparable require conclusion arise study discretization discretization exist finite kernel hilbert kernel interesting idea run method find spirit stagewise alignment align desire output empirically learn technique example combination continuous powerful priori performance world suggest dimensional kernel g exponentially multi kernel learn differ whether simultaneously happen far concerned start restrict search stay algorithm restrict lp qualitatively different statistical restrict rule tune limited option explore finite kernel kernel reproduce kernel learn happen happen single stage drawback work usually choose dramatically case continuously parameterize family derive natural kernel might prefer final however picture surrogate stage parameterized base study family kernel optimize require kernel positive semidefinite combination show semidefinite sdp semidefinite program computationally expensive simplified show sdp solution compute standard meet claim generalized learning kernel parametrize kernel choose number propose iterative optimize add general belong dc technique example function rkh formulate problem learn large group learn base kernel kernel parametrize control practice base value learn predictor base kernel usually base continuous alignment similar case alignment learn alignment outperform shift intercept alignment dataset let positive semidefinite apply alignment alignment use program alignment non negative kernel learn maximize label kernel ridge learn kernel alignment center semidefinite px semidefinite center kernel alignment dataset kernel emphasize linearly separable linear perfectly expect target point long alignment vary class conclude center definition alignment combination
existence uniqueness ensure uniqueness definite definite unique show optimality showing prove lemma sum implement centroid carefully computing distance centroid vs fig illustrate expect eigenvector log close slow centroid matter application application riemannian metric riemannian metric retrieval explore instead minimize seek prefer geometric solve make median differentiable ignore obtain necessary iterate nonlinear correct mention author contraction thompson claim deduce uniqueness contraction fortunately valid choice metric large generalize eigenvalue section projective ia iv iv projective prop analyze calculation prop third prop iv strict contraction start must attain minimum definite solution complete strict contraction generate iterate stay compact construction satisfie whereby empirical behavior fp different collection compare fp gradient fp turn remarkably manifold implement intel riemannian collection wishart study positive definite qualitative similarity hermitian definite notably divergence define divergence derive main present property metric hilbert develop mean median useful hope encourage investigate new acknowledgment grateful department university visit remark conference positive definite matrix attribute strict function draw view cone definite divergence sequence result connect riemannian divergence distance several property riemannian demand nonconvex use theorems matrix bregman divergence stein divergence jensen bregman geometric curvature hermitian definite manifold study manifold curvature possess ii closure form close convex cone view enjoy convex nonlinear algebra manifold view role see manifold drawing motivation view cone definite matrix connect divergence riemannian distance importantly divergence common riemannian demand build view hilbert usually transpose hermitian denote usual x f riemannian riemannian q logarithm counterpart formally name metric definition already divergence alternative simplicity think image exact paper highlight crucial speedup times compute latter toolbox time take matrix compute implement state art average fast alternative popular euclidean compute make cholesky slow much limit reader aware cm aspect pt hard connect riemannian distance connection cm question relate positive scalar cm necessary solve result global optimality proof establish previous jointly present substantially outline difference space support appear corollary similarity riemannian table well concerned theorem jointly prop remarkable contraction corollary corollary lipschitz inequality study q claim fix thompson metric formally viewpoint differentiable equality side inequality behave certain choice chen hermitian let arbitrary nonnegative typically asymmetric view dissimilarity q unnormalize occur two multivariate gaussian divergence theory applicability divergence symmetric attractive perhaps representation formula use advantage reader alternatively jensen bregman multivariate eigenvalue aa equality observe iii iv v prop enjoy convexity concavity equality ii convex divergence hessian prove rich distance result fail multiplicative harmonic prove metric prove chen b yield thm rely available alone nonempty inequality subset n help e imply inequality prove suffice number observe tx lemma another ce nc define converse deeply decrease cone thm claim let follow matrices ii definite theorem positive kernel problem use list quick l ref ref b ba ba c nb ca ta ba b ba ba ba b ba x kk u easily connect gm gm give numerous attractive among variational important surprisingly enjoy characterization even ba bx bx translate whose saddle thus go observe principle realize actually show prove geometric prop includes result special gm concavity e thm jointly continuous suffice eq monotonic determinant multiplicative combine identity establish fail quick suggest attain strictly share contraction term depend elegant metric divergence determinant reverse geodesic positive curve satisfie albeit weak result observe illustration theorem exhibit strong contraction curve straight empirically fairly gm arbitrary geodesic prop theorem help claim power monotonicity let scalar satisfy prove auxiliary prop vector decrease write relation usual follow general monotonicity mean scalar equivalently recall monotonic monotonic equivalently inequality obtain symmetric immediate prop basic analogue important property derive solve nonlinear similar monotonically decrease equivalently eq see operator prove shorthand see semidefinite follow plot show contraction display eigenvalue translate shape curve powerful compression connect thompson key
bring together york explore potential respective great success book advance mine advanced technique vast amount understand universe attempt window vast field like spend contribute year easy well website thank thank go space study york usa nd co york sciences strong research traditionally people science mathematic believe mathematics university york invert process aware scientific also science book fourth article make I narrow great field science discuss book agree field look data set discuss look expensive several progress people contribute hard science forget discuss possible innovation author field history support thesis title book west thank come york york ex institute studies http institute introduction pe universe gauss mining challenge analysis source machine understand universe high exploratory partition science understanding time series scheduling observation inference ari flow dynamical reconstructing
define perturbation closed curve large trace norm operator norm small norm unstable convex surrogate complexity property reweighte correlation net scientific domain statistic interpretable admit look sparsity knowledge many combinatorial inference induce lead framework year body lasso optimally correlation prediction estimation support however datum exhibit block diagonal sparse involve case toeplitz lasso although lot lasso instability elastic net add elastic blind group predictor correlation line focus ideal intervention net modify behave precisely follow propose trace nuclear norm show reweighte around relate include correlation synthetic trace lasso regime row note whose outside support norm singular value maximum value matrix norm equal consider predict assume training n widely predict avoid especially ridge square estimating drawback thus behave np norm estimator pursuit later reference therein unstable select correlate residual hand together elastic estimator adaptive predictor adaptively norm square introduce comparison group sum norm effect group norm introduce regularization depend normalize normalizing reweighte norm norm predictor support lasso adapt setting dimension span equal surrogate lead trace lasso property norm canonical orthogonal hand correlate behave like uncorrelated predictor behave statistical towards lasso convexity highly always minimum respect penalize eq appendix consist show flat loss trace section penalty trace allow newly special norm matrix line choice therefore q homogeneity triangle consequence linearity state norm special introduce overlap group lasso normalize family norm trace lasso unit see unit see bind upper norm elastic norm elastic fact single particular covariate strongly covariate uncorrelated behave like pairwise net important quantity unfortunately norm dual norm dual time estimate extend index optimize subgradient inefficient subgradient subgradient iteratively formulation infimum attain use optimization minimization infimum attain reweighte equal iteration warm lot converge start empty decrease eq start op use prop obtain second rewrite abuse second term interesting fact term pairwise net synthetic classical draw identity block eight experiment toeplitz design observe elastic lasso perform model almost ridge elastic net adaptively variable couple selection regularization trace approach
cx bx b cx function maximum system connect length elastic external noisy force model newton velocity mass sde write calculation write function specific parameter might two evolve observe trajectory learn reconstruct interval structure actual length observation effort devote develop complexity model devote learn use model pseudo paper gaussian sde upper bound prove match autoregressive however develop complexity substantial address question sde line yield quantitative scaling sample generality parameter random unknown distribution subscript omit interested property define value variance identity tx regard denote scenario mutual depend system account tx tx bound theorem follow tx var var fx fx simplify apply sde understood process interaction matrix first provide dense transpose matrix support matrix per row support large enough sde dense shall fundamentally regime sign sign tx together sde task square dense indeed upper sde vector denote jacobian non stationary small tx ax assumption existence solution sde energy prove throughout sde admit useful establish linear sde symmetric interaction stationary sde var ax tr conditional estimator expression form substitute proof show trying support require average unless bind follow uniformly regular generate independently define small maximum eigenvalue guarantee calculation law notice unless entry limit second compute low jensen step law define distribution since claim denominator multiply bind numerator denominator write stationary q arbitrary rescaling factor give probability matrix symmetric pa pa pa evaluating prove non family var x xx f ix ix e individual component therefore partially nsf award nsf dms fa fellowship later proof recall define define symmetric gx g g dividing
cost integrated indicate topology significantly irrespective connectivity aspect regard qualitative reflect expand discuss remark form cost topological differ express topology exhibit integrate mask subtle computationally integral mc require another integration population require comparable need possess vertex fmri however hold recommend selection denominator ng thus cost reflect several relie rank may topology tie rank value cost level create sparse network arise however non occurrence counting tie differ tie rank artificial limitation integrate several omit take topology unweighte arguably cumulative importance edge edge thresholded graph retain remain especially metric monotonic one also address cost control monotonic ignore contribute topological integrate metric combine weight respect use thresholding exhibit undesirable tend cost topology provide well dash cost specific plain cost average subject line suggest effect metric conclusion apply topological main prove general set independent topological usefulness integration difference topology unweighted weighted short main proposition general equivalence topological measure flat space cost development sophisticated effect consider correlation easily higher preferable emphasize build large integrate topological put network whose cost nature structural necessary integration justify usefulness control interest weight network thresholde artificial clear mean graph architecture ensemble thresholded fellowship institute health research research centre health mh south foundation trust college south would also anonymous valuable input integrate first integral method advantage provide interest advanced order mc theory first expectation convenience drop express straightforwardly possible omit order converge almost strong number provide integrable follow special mc permit refer mc density variate especially integrate integrate sample topological nature specify underlie mc topological care substantially however need rely context imply tie rank recommend resolve tie rank assign order tie rank random introduce amount tie high restrict simplified division purpose standardize percentile rank result rank rank order formal indicator similarity function cost verify unweighted verify equation discrete integration notation monotonic modify rank integrated suffice simplify requirement tt however complete contradiction conclusion suffice least weight short satisfy path short connection vertex denote invert entire clearly contradict prove claim p conjecture proposition run head weight separate topology e ed college institute health research centre health south college institute department school correspondence concern send sciences box p college se uk statistically principled population topology inherently unweighted evaluate limitation compare population efficiency comparison network differ key integrate controlling eliminate monotonic weight unweighted cost difference difference constitute valid global simply recommend report weighted cost integrate topological analysis provide cost topological finally limitation integrate subtle topological integration global last decade biological physical adopt approach interest originally seminal world free network idea adopt experimental research classify topological brain network different cognitive behavioral task common brain difference control patient author evaluate property task brain study spatial modality eeg fmri compare subject level spatial resolution question arise rigorous network network arise connectivity strength network topology topological edge draw comparison therefore secondly issue division weight unweighted graph difference topological depend whether unweighted likely unweighted concentrated graph theory biological originally nature interest consider relation element unweighte regular weighted correlation likelihood different population readily identical systematic manner straightforward network correlation interested correlation thresholding matrix approach uniqueness since language discrete mathematic correlation real value one concept originally unweighted consider weight firstly evaluate use unweighted secondly integrate possible network unweighted cost edge analog probably author explore integrate topological respect substantially differ relate cost topological coefficient derive integrate topological successful manner although way therefore literature network formally integration compare topological quantitative manner qualitative adopt network generative regard os enyi model common unknown refer type topological enyi lattice qualitative rely category contrast wish perspective whereby specific topological os enyi different level global therefore qualitative approach topological quantitative topological generative latent topology appear well suited definition weighted become therefore situation conclude weight share integration useful cost section basic two family topological measure network integrate quantity contribution outline describe fmri investigate us monte finding science close conduct weighted report arise exposition comprehensive complex find follow measure refer quantify metric mathematical definition topology term unweighted graph network order pair node connection I refer element edge cardinality denote graph describe topology short undirected denote triple index entry necessarily satisfy matrix draw unweighte one range development paper apply include synchronization likelihood simplicity lie interval standardized weight strength association indicate straightforwardly practice pearson standardize weight lead major since transform positive subset may positively secondly link positive negative take spurious close likely weight explicitly direction standardized correlation matrix sequel refer specify discuss limitation cost integration graph topological propose interest refer I metric former quantify transfer globally capacity information locally characteristic global local although metric wide range analog useful measure retain interpretation cc applicable wide range specifically efficiency metric irrespective graph neighbor family topology efficiency one remarkable local information unweighted summation short path node subgraph neighbor interpret information transfer value transfer similarly efficiency interpret efficiency subgraph imply unweighted distinguished edge set denote quantity distinction notation expectation probability technical general apply topological level interest quantify unweighted generalize term density unweighted quantify relative connection proportion match implicitly definition eq element represent cost generalize unweighted matrix order value association formalize natural choice entry formally standardize weight starting unweighted concept connectivity depend upon choose standardized respect different measure consideration fully two type weight cost equivalent currently quantify topological aspect weighted I integrate metric define family quantifying translate unweighted measure include weighted version rely unweighted short subgraph pair letter stand edge weight set path take notational introduce normalize association value use choose shortest regardless exist path every index topology integrate transformation level identical create vary compose regular topology corresponding association value build layer topology every regular layer integrate graph approximately contrast substantially integrate range hybrid topology globally efficient gray fill corner ex background scale every text node anchor west anchor west node anchor anchor west west anchor west anchor west anchor west west west anchor anchor west anchor west west anchor west anchor west anchor west anchor west anchor anchor west world transform regular hybrid exhibit lattice layer regular entry arrange facilitate integrate cost potentially substantial topological however population provide pool find statistically significant comparison network full yield identical group naturally opposite direction seem subset integrate full cost fixing suffer firstly generally justify practical secondly consider cost level network conclusion cost subset report nonetheless difference topology irrespective mathematically strength may reformulate difference proportional v w standardized interval trivially efficiency cost level cost attain integrate nan cost cost crucially difference cost return case exact two proportional integrated integrate respect point proportional thus identical derive hold invariance cost true monotonic metric unweighted argument formally undirecte monotonic weight version satisfie q proxy solely depend diagonal ranking set independent monotonic transformation argument topological I originally set unweighted integrate hold level function preserve original ordering tie rank adjust advantage topological topology roughly irrespective successful topology connectivity singular advantage measure invariant ensure comprise interval cost integrated topological measure demonstrate limitation potentially mask example addition integrate topological metric network rank cause induce topological structure rank allocate cost level discuss limitation connectivity characteristic directly topological efficiency equation serious limitation potentially type show setting weight efficiency whose proved contradiction situation real data difference zero nonetheless add weighted relationship theoretical therefore highlight various topology recommend report cost topological illustrate publish sup sup sup sup inf inf inf inf back back wavelet hz frequency band fdr correction base location correspond centroid inferior orientation axis analyze task base imaging fmri mc procedure cost integrate brain cognitive solely give description use basis fmri working subject letter second ask whether letter trial previously nan subject automatic template series wavelet wavelet hz main network fmri paradigm condition repeat wavelet concatenation simulate find final analysis subject functional correlation specific coefficient construction network illustrate univariate edge subject test thresholded discovery connectivity subject experience cognitive load description support mc mc propose cost convergence medium sized derive working memory task figure minus report formula dash grey variability twice mc indicate study compare integral computation calculation reasonably good quantity mc error constitute negligible mc derive computation could uncertainty associate interest
unsupervised machine valuable well unsupervised method far rate convergence assume image unknown pixel pixel propose procedure vary intensity test threshold picture choose consistently detect size one establish uncertainty present paper distinction formulate type literature knowledge research wavelet relation stronger hold detection object generality detection give test black object develop object vary intensity nonparametric block uncertainty relation mathematical analysis theory review proof relation detector device start notion infinite sense site connect site neighbor site q say assumption critical statement infinite even finite difference size site see say site precise cluster infer whether site configuration distinction locate near critical infinite lattice white modify cover detection intensity construction basic case graph discretize topology neighboring paper consider signal denote indicator function give identically variance measure detailed detection mean e previous construct test compute explicit mild proof denote finite triangular consist site mean arbitrarily resolution think write eq depend way define pixel side view inconsistent pre let suitably weak condition square site n follow type arbitrary detection actual error remarkable type object suppose constant type error exceed type cluster pixel intensity exists sequel realistic signal precisely give non degenerate respect define q device intensity properly display assume sufficient incoming explain appendix test perform read versus analogous fashion hypothesis strength bound detector device explain intensity detector likewise scale threshold sub detector device consideration attempt eq nan probability black cp ps ii let essentially situation lemma n suitably function theorem may contain length correct scale right indicator completely hold pick strength overcome scale positive intuitively clear ratio horizon prove like error explicit valid noise give q thus proper estimate triangular lattice depend black contain origin please asymptotic statement instance sharp sequel depend triangular lattice appendix upper eq eq intrinsic clear something statistical procedure therefore something signal distinction explicit condition literature gaussian wavelet strong wide threshold close lattice statement lattice large lattice nan reject relate strength ratio assume detect sufficiently large sufficiently matter principle lattice give automatically effective type ii error word derive necessary ii study interval fulfil however justify time lattice exceed exist might continuous sufficiently detect exceed circumstance principle minimal detect provide size bound detector device intensity signal rf signal super account property able potentially threshold begin statistic terminate nan reject reach repeat cluster time overall asymptotically slow original unknown color carefully rejection indeed perform single test occur threshold analog type tend exponentially tend fast pay able get plan paper acknowledgment read manuscript collect going prove basic introduction event increase indicator whenever state add completeness decrease subgraph sigma configuration occur q inequality let q ff disjoint site finite increase q statement denote event sequel write measure ss marginals sa p pa e ps ps p finally q depend closely connect lie size connect expect note write q connect adjacent path connect connect meaning site path site switch change disjoint finally dp pa p nx ny p ny ny ny p ny integrate differential detail see mention proof finally material matching finite probability site mark black mark white black boundary mean polynomial number extend infinite consider per power series size clearly infinite graph self polynomial triangular match probability triangular lattice instance actually special polynomial thus construct match equation similar discussion device device detector device able signal intensity detector device contain cutoff intensity cutoff lose signal happen view notion bound detector device equivalently reformulate still conditioning event course yield important
term cope interaction logistic exploit motivate attractive regression output linear coefficient reasonable value remove may stationarity adaptive entail considerable burden iii raise instability curse modeling recover equation unique readily find infinitely many hardness program basis pursuit motivate l l uniquely ls continuous least invertible grow translate numerically l ls determined may resort regularize readily store invert computing regression th call trick compute also neither estimator lasso weight improved price require ridge isometry albeit regression generalize model regularize logistic probit regression successive lasso problem polynomial expansion generalize polynomial certain improve coefficient level toward aforementioned constitute generic solver tailor estimator method coordinate scheme outline next core iteratively w scalar ii component minimization th entry sample iterate simply define eq update constitute coordinate alg guarantee apart computation alg coordinate unlike counterpart offer memory enable track vary l ls solve factor invariant variation exploit priori suffer instability overcome limitation follow polynomial recursive lasso call instant burden reference therein time spirit batch single form alg present hereafter cyclic convergence regression set alternative setup approximate expand basis formulate solver polynomial cast homogeneous n p cf objective insensitive though kernel concern primal domain section specify whether problem capable identify polynomial expansion study though tool call rip involve restrict isometry isometry initially identifiability noiseless uniquely recover unique analysis extend constrain bn bn bn rip describe hold properly early rip establish identifiability compressive setup definition suggest derive one ensemble rip bernoulli possess constant identification toeplitz latter bind versus scale former attribute dependency rip involved system filtering input independently practically frequently limit gene beyond consider possesse rip exist condition form combination possess rip favorable matrix unity norm unity quantity coherence rip appropriate normalization least enough desirable vanish requirement inherently inner positive desired soon study rip equivalently intercept dc toeplitz parts toeplitz q quadratic submatrix readily verify satisfied transition raise rip insight scenario actually substitute upon expansion hold translate adjust carry answer pose modification solve modify resemble replacement wiener polynomial wiener module white distribute analysis appendix define form c suffice scale bind agree filtering whereas due involved dependency describe sparse input recover comprise noiseless filter setup row statistically independent rip polynomial tight deal expansion e measurable endowed set tt kronecker delta involve entry rip rip universal rip basis trick apply basis upon stack properly replace one thus translate sparse rip rip possess straightforwardly independent input probability observation scale compare scale increase matrix explain structural polynomial paradigm admit product input typical multilinear setup comprise coefficient inequality rough bind sample phenotype additive alphabet latter alphabet analytical entry mean rip exploit appear linearly mass multilinear entry fact characterization readily aware relate difference input alphabet oppose one rip multilinear draw multilinear positive constant whenever n often phenotype scale previous section probabilistic bound identifiability representation evaluate applicability sparsity real indicate attain accurate impulse describe system model corrupt counterpart plot follow estimator ridge scale agnostic outperform condition offer accurate even assess setup mse carlo recursive conventional recursive system draw carry moreover fig aware closely study effect phenotype trait height seed follow gene effect marker determine trait pair trait since population regressor determine trait analysis multilinear paradigm synthetic detailed population evenly marker phenotype linearly express intercept effect leading marker note marker ii effect regularize center intercept fit parameter tune fold pe unseen data v estimate small pe determining estimator mse provide keep estimator software take min final see attain pe pe column ridge yield model yield avoid coefficient entail real north american outline height population consist phenotype marker cm genome marker cm tr allele value main involve leave fail handle large weight pe attain parsimonious spurious peak infer magnitude offer effect ii effect main marker iii marker main exploiting estimate abundance allow parsimonious expansion estimator need meet analytical solve efficiently find computational load enable analytical rip show high order interestingly generalize generalize regression aforementione numerically verify develop adaptive exact outperform simulate extend high utilize considerably genome tool surely essentially around dependent useful sum dependent random partition exhaustive exclusive subset respective minimum intra independent also uniformly minimum may yield correspond always construct way dependent degree vertex degree assign every vertex differ partition intra key guarantee dependency sum necessary realization possesse rip thus rip event rhs exploit bind triangular yield rhs entry exhibit component linear bilinear nonlinear system notation indicate th inner product start upon hoeffding multiply account n yield bilinear diagonal diagonal sum generally two split partial sum example entry express sm collective return entry splitting trick contribution regard immediate hoeffding whereas cb already nm equivalently depend sum bound include trick yield n exploit splitting trick sum k mx lb simplify presentation slightly consider sum three use degree dependency associate product entry together write k ij kx mx k bb entry associate dependency bound diagonal imply arrive translate whenever yield acknowledgment thank update ir ii r ridge ridge lr corollary support international fellowship european nsf grant award mn play identification range value meet method compress offer cs common sparse initially pose build identifiability principle sparse sufficient measurement rip commonly meet generalize result counterpart verified phenotype lasso isometry appear frequently engineering speech name nonlinearity smooth offer memory impulse expansion mapping filter approximate expansion apart extensive character recognition recently valuable phenotype relationship wide association jointly investigate albeit nonlinear input linear via bottleneck grow raise computational record reliable dimensionality issue however admit expansion exploit nonlinearity nonetheless polynomial expansion attribute parsimonious underlie parameterized sparsity expansion formulate motivate polynomial expansion drop contribution adopt sparsity
passive large present achieve excess smoothness involve question relate implement extend result answer logarithmic coincide derive argument every one iterative transform goal work ph numerous discussion grateful anonymous carefully read claim choose enough mean partition dyadic since cube exist dyadic cube q attain point eq concluding simplify brevity additional notation recall excess union event eq q imply section section lemma partially support arc nsf grant dms mathematics ga active investigation provide bound generalization achievable underlie obtain rate selective confidence band couple design denote goal predict practice remain often happen obtain associate label pool almost suggest classifier active devote modify property previously passive learner obtain design difficult collect situation learner outperform constant convenient characterize investigation discover generalization zero fast well passive however available noise decision boundary computationally adaptive field standard polynomial covering derive regularity decision assumption restrictive h old bind excess active label budget upper namely certain h old satisfie show plug learn regression excess second propose attain possesse within organized next introduce specifie make qualitative work statement govern observation sample sample usually measure respect measurable excess indicator omit minimal infimum measurable function restrict attention belong old continuously taylor distributions property bt cause impose two behind deal regular grid unit partition consider nest dyadic partition unit cube regular eq definition nature minimax excess sup sensitive local assumption another useful characteristic regular certain subset fit say belong derivation piecewise use define dyadic cube note view span haar nest desirable selection projection decision mind sign design support similar sigma follow appearance assumption algorithm nonparametric subject mention impossible adaptive confidence band subject follow satisfying allow away decision boundary lipschitz convex satisfy proposition appendix finally satisfied brief since definition appear naturally emphasize known restriction estimate desire base plug piecewise mention passive plug classifier attain iteratively improve construct shrink confidence band piecewise procedure h estimator use active clearly band classification potentially serve modified concentrate domain tight band control require priori noise adaptive proceed main recall risk active framework rate goal output excess converge fast build upon minimax constructing similar present reader infimum estimator kullback go back let cube partition follow belong center always integer infinitely distribution marginal independent mr define support union dyadic spread tend respect smoothness requirement check satisfy noise consider sphere next step well mm stand distance follow theorem nx briefly denote admit choose way set n eq finally lower since noise sign tractable show attain rate factor cover interesting decision proposition order complete address remark regularity assumption budget replace define simple resolution iid follow around bind consistent procedure play appropriately choose ready active detailed budget lb nn km lb k dx lb every sample divide k ks ks towards general iid absolute enough bind hand ready satisfy remark fast rate passive main b depend act claim active set finish satisfies q easily risk inferior show prove final finish projection onto
strategy process candidate determinant follow counterpart approximately strategy process covariance give collection candidate costly greedy term gp hence sample simplification dependency choose reduction consistent entropy collection candidate maximum lipschitz maker limit costly maker point call iterative function quantity simplification observation basically formulate use uncertainty model difference noisy noise entire second solve note good detail evaluate numerically describe replace candidate htp objective respective specific formulation constitute sum state substituting approximate search weighting describe meaningful objective order transform normalization individual dependent estimate function observation utilize value specific respective df provide combine thus provide weighted address objective minimize convert constraint many real life normalization counterpart constrain optimization scalar respectively provide objective practice emphasis estimation learn maximize method disadvantage prohibitive problem bind simply identify make method along scheme part balance vs search implicitly however variety function time decision make framework set noisy past observation change capture noise dynamic capture stationarity change algorithmic summary specific provide bound objective variant variance function error next action maximize objective state next proposition observe multiple issue space visualization purpose hence choose linearly I estimate consist show peak figure point eq htbp htbp htbp quantity well correlation depict result bound objective give sum process price rectangular region grid interval estimate weighted sum utilize space sample minimum price use depict optimum although optimum value still satisfactory consider price million percent depict htbp iteration third setup brain show rectangular peak location show location point htbp six function respective real location htbp datum make limited article theory search recently substantial historical valuable inference measure version subsection focus communication machine book decision make play statistical learning complementary discussion topic foundation gaussian favorable book treatment gps include understand nonconvex method gp focus amount hand model iterative search particle random nature distinguish apply latter quantification connection shannon utilize state theoretic problem likewise box account information criterion discuss subsequent active gp heuristic rejection foundation adopt experimental observe priori reality methodology work indicate already domain actual adopt present method replace specifically kolmogorov complexity priori probable accordance hence describe simple gp incorporate kernel theorem different consider problem obtain costly scheme obtain entropy shannon theory provide metric aspect optimization framework allow vs trade vs obtain amount point illustrative grid dimension e resort carlo case address issue include anneal resolution candidate biological analogy maker biological priori meta gp refined environment domain preserve resource much evolutionary fast member explore meta refine meta evolutionary assume competition framework present address decision account content datum theory relationship optimization framework future exploitation adaptive sampling relationship evolutionary make game theory acknowledgement support thank discussion subject make decision maker posteriori nonconvex except costly observation present information maximization quantify information acquire entropy nonconvex maximize model adopt approach art result maker preference world decision whether obtain information consume characteristic infeasible option leave maker develop strategy collect miss concrete maximize lipschitz except small may start observation may make decision location unknown paper capture pose observation multi optimization aspect structure manner enable aspect explicitly incorporate concept scheme build field machine specifically concept quantify acquire theory modeling system adopt capture aspect objective formulation heavily aspect application domain vision handle datum make play play important develop strategy time advantage scalability gaussian process frequently encounter seem method mine mid due decentralized resource decision complex system wireless local decision security relate decision effort action relate security management costly operate part large system limited concrete definition motivate helpful aspect generality nonempty domain fully adopt provide reasonable unknown possibly simplify assumption lipschitz main distinguishing limitation observation cost information assume objective nonempty strategy q may domain adopt approach beneficial allow usage incoming alternatively cost relax formulate scalar iteratively location conceptually strategy unless rarely complicated strategy collect information derive gradient heuristic explore step direction randomly jump also flexibility entire objective belong interpret maker parameter separate slow capability search fundamentally impose explicit priori learns manner available computation search opposite end utilize extent almost regardless valid wide field security economic management system human close spectrum advantageous search former end predictive unobserved balance usually achieve balance belief function fitting focus process within framework develop multiple reason preference elegant combine secondly gp well immediately model drawback namely heavy really amount available comprehensive treatment gp therefore decision make set observation gaussian formally completely characterize since noise matrix use infinite real accordance gp point outside training x scalar characterize gaussian define distribution objective furthermore uncertainty provide subsection subsection discuss possibility maker address quantify obtain precise answer principled manner example search shannon information readily framework measure discrete variable quantifie quantify final entropy conceptual shannon good obtain information constant action less threshold uniform case simplify derivative clearly roughly ignore quantization boundary effect middle prior provide process repeatedly quantization learn search strategy make limit present address problem
common poisson random let independent poisson multivariate poisson large focus bivariate process bivariate case characterize independence independence innovation distribute notice diagonal still still joint expression derive matrix expression poisson innovation normal distribution infinity numerical optimization maximum previous run simulation study behavior example two parameter two parameter interior size smooth function set size table standard converge go figure table though valid size sample use approach count propose account autocorrelation occur create serial within contiguous statistical exhaustive risk across require reference north american split east west chinese south american decide together west east secondly past entry magnitude database period mention website mid affect cutoff test st th keep period investigate first autocorrelation autocorrelation count measure degree order range hour propose serial autocorrelation independent thus pair show ratio poisson compare show sampling follow combination along quantile proportion significant thus provide useful add really table dependence contiguous serial correlation important independent expect mechanic one count sample influence interval hour occur find first degree autocorrelation autocorrelation finally occur second panel percentage importantly contiguous similar model model combination large part contiguous percentage far apart often sufficient table combination statistically combination fit contiguous different area frequency significant line show unconditional number moment west table part table daily three day poisson generate day observe day west autocorrelation autocorrelation thus average number day quantity west potential management west compare diagonal context compare model situation west e pick significant analysis one directly value influence hour frequency west converse west count causality poisson hour hour hour hour hour hour hour hour hour hour hour hour hour hour mean hour hour hour vs hour hour hour hour hour hour diag vs west vs hour hour hour hour hour hour hour hour diag mean frequency hour hour context predict large medium likely observe five propose independent poisson fit diagonal independent statistically sampling model finally diagonal autocorrelation explain future size medium case weak frequency hour table explain temporal decay rate hour frequency last unconditional medium hour frequency causality significance set illustrate impact hour medium autocorrelation period indeed cross account expectation much section cross finally approximately consistent name hour hour american east chinese south american denote number medium magnitude period name mean north american west chinese south medium management application price total area city important total area question section area propose see significant south american south american locate limit observe west assume south american occur half bivariate bivariate occur table leave focus day especially tail first increase auto count increase week whereas probability long occur k n take lot effect probability model tail conclude scenario confirm effect tail useful short term risk pricing derivative market model c model american day day day day day day west american day day day day day day k west south american model south american day day propose south american day day west south american vs matrix large distance contiguous perhaps day take account period overall medium theorem corollary proposition universit du pr email de universit du pr email finance represent occurrence integer occurrence extension cross autocorrelation fit various bivariate count many contiguous cross autocorrelation seem count multivariate point apply reason series model excellent autocorrelation autocorrelation investigate operator application treat car order independent derive multivariate attempt random variable notable poisson application paper autocorrelation policie cat cat cat offer periodic component various location upon poisson cox process self release review markov hmm toward location location see g purpose autocorrelation location count another site area main investigate management management consideration view example cat influence dynamic lack count prediction occur well causality count vector random component process diagonal chain probability satisfy associate positive assumption irreducible recurrent strictly eigenvalue td variate autoregressive matrix variate finite
estimator possible scope involve quadrature adaptive grid base capability smc overcome limitation smc base ei value dataset gaussian unknown soon smc however quickly algorithm function reference always lag behind new algorithm kind study evaluation ref ei ei ref ei ei proposition real past evaluation difficulty use article decide find maxima use ei deal implementation idea ei view define sake generally mean ei distribution conditional algebra ei expectation easily see follow impact applicability deal integral deal issue dirac use method therein sequential carlo smc care follow small population evaluation region x objective particle nx ng nx reflect past consider exceed initialize sample instrumental pick construct nx I c ic j nx
involve particular partial linearization incorporate fista several synthetic strength compare relative induce norm among fista good fista appear alternative fista variation different discussion chen matlab completeness convexity iteration apply hold let let get q sum instead hence n yield stop criterion dual outer iteration gradient iteration iteration optimality primal feasibility vanish e residual fx line fx k residual satisfy every inner residual derivation satisfied fista optimality condition feasibility vanish clearly z k x residual consider iteration primal feasibility q feasibility primal residual residual lx cx l cx l v cx lagrangian evaluate augment lagrangian outer readily outer c cpu avg e fista fista e fista fista e e fista p c c cpu avg fista fista fista fista e cpu avg dct fista dct fista fista fista p fista dct fista p fista e e fista c set sub dct fista dct e fista dct fista e dct fista fista dct fista p fista dct fista fista cpu avg dct fista fista e dct fista fista dct fista fista fista e fista fista e dct fista fista c c data set cpu avg fista fista fista fista dct fista e e fista dct fista p fista e dct fista dct fista fista avg sub fista fista section problem space regularize induce incorporate pose considerable challenge non sparsity induce norm unify augment lagrangian variant efficiently build block linearization splitting accelerate algorithm alternate fista method norm sparsity alternate direction augment lagrangian dimensional back classical lasso particularly appeal gradient take sparsity show assume define feature allow array application image extension adaptive paper minimize loss regularization eq index element possibly overlap pre without penalty cast overlap close approximation nesterov solve thresholde fista upon practical convergence liu fista proximal operator penalty solve descent gradient et al use unified tackle base linearization alm fista solve equivalent problem contribution augment lagrangian block alm splitting linearization fista serve key fista tune every evaluate rich breast cancer video unify base variable splitting lagrangian method solve problem obtain repeat component group group structure sparse equal time include appear loss term equivalent elastic reformulate merging penalty net loss solve augment lagrangian algorithm augment lagrange multipli update parameter amount place view x cx l guarantee exactly instead minimizer abstract subroutine theorem inexact version mx lf converge condition augment lagrangian subproblem solve cx l update call outer subroutine index penalty regularization without well direction overlap approximately minimize minimize update lagrange multipli procedure serve multiplier accommodate example splitting fx iy solve simultaneously simultaneous multiplier partial parallel proximal algorithm solve subproblem soft td jx side many set highly determined stay store subsequent large mention cholesky system conjugate comment section fista linearization alm apply sparse regularizer alm lagrange multiplier define linearization lagrange multiplier alternating implement fx fx k variant fx eq hold matter violate minimize ensure convergence execute likewise ready lipschitz satisfy optimal regular iteration among first hold easy satisfy entirely equivalent linearization variant lagrange multiplier redundant subproblem hence thresholding subproblem solving give optimality gaussian yield system section cholesky factorization amount comparable straightforward accelerate partial present algorithm apply alm split solve write even gradient compute cx alm need backtrack line fx adopt scheme initially decrease number reach minimum prevent hence leave factorize splitting potentially linearization lipschitz side allow alm line become intuitively job load hardness augment lagrangian define variant set property hence implement subroutine linearization k kx equivalent lipschitz iterate easy equivalent exactly difference outer lagrange multipli stay inner delay lagrange multipli load balance section obvious full linearization gradient minimize take large size due constant present version fista th outer termination stop iteration primal relative inner table expression directly gradient residual th outer outer inner objective gradient residual fista k c k k z k c x z z except fista fista hold solution return become increasingly evaluate quantity lagrangian experiment primal two basically terminate iteration subproblem high inner criterion slightly alternative stopping loop context fista require th iterate update outer heuristic primal residual subproblem subproblem adopt kind keep appropriate work dynamic scheme primal residual yield small hence outer iteration fista propose arrange adjacent overlap three output vary increment b version scalability base direction fista perform slightly show obvious fista base large cholesky fista system solve plot right fista fista due number dct approach dictionary cosine dct contiguou five non increment considerably hard heavily exhibit run c algorithm leave whose already fairly figure tolerance gap keep cpu versus fista outperform regularization fista perform growth fista trend outer show fista large factorization fista cholesky factorization fista subproblems performance fista fista model serve update fista significantly improve scheme scheme work turn case fista suboptimal fix report well algorithm numerical compares stay fista job gap fista obvious usefulness test world breast cancer tumor measurement goal good experiment survival patient pathway naturally group regularization summarize table fast tune scheme speed solution keep constant return instead identically fista fista behave fista number inner error different leave rmse compute low rmse yield well magnitude return group pattern rather background main segment foreground image frame frame camera pixel form ax noise combination video frame foreground lasso regularization patch still group consist element alternative net yield well linearization fista p lagrangian also solve solve mask truth foreground ground truth maximum special
kernel rkh characterization observe reproduce machine onto span w inner linear fourier adopt fouri lebesgue satisfy tell integrable prescribe rkhs lemma paper evaluation shall characterization infimum make convention proved fact get kf h lead another feature moreover nontrivial adjoint argument particular choice countable prove translation reproduce invariant characterization invariant translation exactly borel measure inclusion relation translation invariant kernel q borel topological space absolutely subset function denote function norm later hilbert borel translation kernel happen everywhere clear pay borel continuous lebesgue integrable measure essentially equal everywhere radial kernel variate area area define borel define reproduce borel vanishing equip argument rkhs borel k dd absolutely lebesgue measure inclusion six invariant kernel learn apply mathematic apply norm transform identify page combine fouri function spline spline inverse give kernel denote singleton exponential application kind hold np h b p relation rkhs separate let hold use one define right infinity h unbounded origin neighborhood origin therein thus estimate yield go inclusion rkh form constant relation hold break prove step clear possess almost consequence handle way h b nb p prove eq everywhere method ii combine explicit e g e r equation laplace delta singular measure borel absolutely common vary nj yield tell corollary hold equality achieve h eq claim lebesgue dominate origin corollary h monotone equation h continuous zero ix upper tend convergence everywhere positive hence conclude measure absolutely continuous choice result argument straightforward calculation hilbert schmidt represent recently construct multiscale wavelet introduce hilbert schmidt nonnegative denote hilbert give belong schmidt associate nontrivial space similarly exist constant arbitrary imply denote delta equation suppose cn bb give example inclusion hilbert schmidt schmidt necessity contradiction nonnegative clear schmidt kernel assumption proved sophisticated mathematical radius reproduce fall proposition result explicitly sequence distinct nonnegative example include periodic see eq b n n nz nr b assumption especially finite kernel inclusion let reproducing hold eq contain matrix denote hadamard product still definite kernel notational shall definite k g complete kernel restriction sequence remain kernel kernel respectively j get remark remove last let limit reason proposition ready result analytic nonnegative coefficient origin nj nj nk ne proposition able prove k investigate inclusion relation small positive constant refinement relaxation refinement accommodate reproduce investigation characterization let hold simplicity satisfy f f f k word conversely since lemma imply finite introduce clear move applicable schmidt example borel absolutely introduce kernel characterize inclusion measure respect nonnegative denote exist constant schmidt dense refinement kernel increase chance polynomial kernel kernel positive proposition refinement weak refinement china science program air force office grant reproduce science hilbert maps correspond reproduce full table invariant example operation reproduce briefly inclusion equivalence refinement reproduce
disease early throughput genomic genome wide disease genome study dna copy gene throughput disease gene annotation phenotype example know tool david others annotation gene poorly often association phenotype availability molecular disease high genomic phenotype mine clinical phenotype molecular protein network provide relation tend interact utilize module disease study disease gene gene query similarly set select quantify relevance phenotype phenotype coherence top rank phenotype phenotype target query connectivity connect phenotype discrepancy phenotype rank target phenotype query coherent upper phenotype phenotype phenotype association phenotype gene utilize phenotype gene network gene retrieve phenotype assumption rank relevance gene set coherent association disease phenotype disease phenotype framework phenotype gene laplacian fig laplacian result gene seed walk propagation relevance phenotype similarly laplacian random fig ranking phenotype rank gene phenotype highly quantify coherence real problem phenotype unknown design phenotype combinatorial ridge close form select disease phenotype variant phenotype good match gene gene phenotype rich reliable phenotype share gene purpose gene phenotype phenotype relevance phenotype phenotype walk simple exploit network phenotype gene map phenotype know association utilize propagation heterogeneous combine phenotype explore gene module phenotype phenotype interpret good formulate query phenotype discovery consist network phenotype gene retrieve phenotype association phenotype query denote membership query gene phenotype target phenotype coherence laplacian utilize global gene laplacian normalize gd g laplacian connect receive ensure gene neighboring consistency query global relevance query contribution penalty close eq empirically iterative eq phenotype phenotype form solution score infinite laplacian modular graph use scoring count direct set measuring query information fully coherence coherence whether query phenotype association disease association relevance graph phenotype phenotype whether give connecting phenotype similar propose coherence regression couple close relaxed simple score phenotype reconstruct neighbor cost equation norm small take standard ridge real vector simple post one phenotype significantly score phenotype full solve require matrix inversion thus j j enumeration enumeration disease phenotype phenotype query phenotype score phenotype use pearson disagreement relevance strategy conceptual simplicity incur calculation phenotype find multiple full inside loop line overall complexity want retrieve phenotype explore configuration exponential leave one task predict recently discover disease gene association validate finding copy network phenotype represent disease phenotype edge phenotype text clinical record calculate association connect node version may experiment association disease phenotype gene association connect disease phenotype experiment use copy expression two protein undirected functional dna contain around million weight reduce cutoff weight generate million weighted evaluation label propagation shortest sp association average walk label propagation phenotype choose balancing method cross query disease phenotype phenotype performance receiver operate characteristic curve interested phenotype near area roc evaluation select rank phenotype coherence top gene phenotype rank large cost small penalty quantify connectivity disease phenotype know gene association select fold rank query gene phenotype rank disease phenotype compute phenotype phenotype plot threshold score assume evaluate well disease gene association association study design disease phenotype phenotype since gene experiment phenotype newly disease disease retrieve report sp new phenotype network necessary accurate far compare disease third mark denote multiple trait category disease trait rank cell high diabetes diabetes disease predict disease phenotype disease individual disease influence implication pathway annotation case collect disease phenotype discover disease trait phenotype disease trait phenotype subsequently disease trait example predict phenotype breast gene novel rank disease gene gene also include gene similarly disease phenotype rank breast cancer phenotype top gene phenotype association phenotype set disease gene disease trait disease phenotype disease trait match multiple phenotype report match phenotype well rank disease trait among within top notable cancer breast cancer disease phenotype gene within accurately breast phenotype gene rank rank disease around fold phenotype fold connection phenotype rank disease phenotype phenotype breast cancer gene directly rank disease phenotype gene directly interact rank association phenotype result association suggest gene disease phenotype network disease association statistical significance connectivity fold association obtain interesting also disease know association disease phenotype poses reveal rank cross validation cancer interestingly network compare disease share disease study disease gene reveal disease suggest incorporate topological association successfully discover association case disease phenotype dna copy copy identify target disease phenotype disease copy change region collect dna copy change copy study dna copy measurement snp copy change detect tool default detect number region query gene phenotype rank disease top rank rank target disease state copy gene previously target human reveal disease association find htb include human cancer copy trait rank rank cancer cancer cancer tumor phenotype frequently gene differently gene experiment algorithm predict disease differentially express gene gene expression collect cancer gene profile array differentially express differentially express disease quantify differential propagation predict disease differentially moderately encourage neighbor differentially association disease microarray gene second disease trait cancer algorithm disease association gene retrieve phenotype rank association fig disease gene currently gene c pt identify display pt discussion set genome throughput disease detect association phenotype significance molecular analysis fail find disease experiment phenotype report improve phenotype gene association gene well association global comparison disease gene algorithm utilize hide gene network label explore gene phenotype quantification relevance score association evaluate association bias utilize datum handling association shortest ridge association phenotype enumeration encourage limitation heavily gene association target disease phenotype introduce thus useful disease characterize well understand disease closely phenotype
medium support get corollary remark ex address minimize optimization hard approximately singular value certain calculate completion rank matrix smooth generally equation speak base instead infinite index respectively sparsity minimize minimize application reduction since infinite cast find singular matrix analyze robust approximation side benefit bind completion efficacy movie datum equation various context trace surrogate closely low singular extensively mainly context trace encourage rank produce study several guarantee rely assume incoherence entry trace involve program large problem minimization selection mp omp formal guarantee norm however choose choose rank omp algorithm sp come elegant rely e isometry assume smoothness solve psd turn wolfe well important change though tackle enforce neither fully difference analogous wolfe fully greedy selection discuss involve approximately improve approximation assume v scalar simplify real du finitely many u v input tolerance initialize I v tb tu u mapping eq easy convex fr problem optimization equation arbitrary forward start find maximize magnitude partial assume obtain maximal though element even lead singular expensive maximization implement element see analysis value increase note approximation whose contain set v equation write write whose svd tell unconstrained optimization minimize minimize practice maintain pseudo code runtime runtime function describe later runtime choose section lead improve still increase section find direction possible verify describe simple start find candidate matrix small let value rank increase decrease analysis tell sufficiently restrict replacement rank guarantee increase guarantee rank constraint guarantee regularization vector term equivalent norm add pose trick singular singular step reasonable effort step expensive completion runtime diagonal word change solve variable competitive perform solution competitive theorem smooth eq zero constraint low trace competitive convex strongly implication several problem example movie find agrees square minimize low rank easy otherwise element row element write u take make least runtime completion parameter j rank predict unknown high low assume frobenius da sensitive outlier replace frobenius norm constraint propose minimize norm require work smoothed replace absolute huber otherwise smoothness huber bound therefore entrie huber discrepancy lemma matrix completion dataset rating movie integer first try decrease singular matrix inspire proof decrease scalar like one yield examine balance
detect pick happen say practical issue purely guarantee behaviour white goal look colour maximizer maximizer q noise possibility design question j nh formally maximal work noise statistical terminology randomize crucial within pixel case moderate easy strong size obvious algorithm relax square triangle shape formulate picture find depth thresholded black large report object black cluster search deterministic description rigorous complexity classic book picture probability nc dependence implicitly mean think image detection pixel point algorithm remark simply detect object sort thresholding avoid although work possibly heavy tailed example heavy detection thresholded detection thresholded picture maximal cluster safe much efficient calculate moderate applicable cover contain square size consistently neuron pass detect neuron acknowledgment van discussion proof devote proof also convenience theorem suitable predefine take operation step chapter standard also save position pixel false denote white pixel outside black additional lie marked cluster thresholding prove open origin conclude book terminology increase large large contain obviously symmetry affect increase decrease ci j p p j n ci j p n k immediately detection follow detection first follow site lattice path leave maximal vertex rectangle slight modification pp object picture square picture square site take cluster big proposition phone phone correspond propose method detection image detect shape appropriate prove result algorithmic quick noisy indeed look noisy question ask primary question field cancer automate detection picture intensive procedure particular picture surprisingly vast engineering skip crucial impose nonsmooth object object noisy shape advantageous detection one practitioner require reconstruction detect usually method rigorous performance free drawback though study subject technique present detection nonparametric number asymptotically consistent pixel stop human stop could serious limitation case affect scope object vast majority detect background colour intensity object thick colour background moreover background case paper simple rescale colour section new section main testing computer work interested colour colour colour background noisy black black background white belong meaningful object within pixel noisy pixel observe begin formulate model observe correspond stress level way continuous moreover easily arbitrary noise statement let black omit since identically white colour image q standardized colour noisy image look colour colour account intensity grey pixel pixel black colour obtain grey value white pixel colour previous procedure would call analyse picture paper let exist satisfy satisfied small pick need research formally definition step transform picture pixel picture call vector moment pixel picture furthermore colour graph black white far edge follow pixel
high profile operational loss supervision discussion management find survey conduct supervision capital risk market risk capital working adopt change international adopt operational capital requirement broad seven event business business failure process management risk fall risk treat separately result bank include term consider nature assess model understanding modelling framework capital internal deal quantification financial bank around bank model statistical tailed infinite family stable loss stability combine ii requirement agreement aside capital implement framework requirement occur expect loss rare bank methodology operational area finance broad approach bank capital advanced model bank adopt comprehensive internal risk quantification flexible believe environment quantitative criterion sufficiently account impact lda frequency simply fitting parametric mathematical compound matter situation expression compound class stable develop paper compound affect setting operational company growth forecast year economic factored scaling threshold record choice distribution binomial binomial generalize pareto stable family important distribute heavy tailed scenario tail extreme providing process likely financial introduce give stable family enough incorporate light tailed mean loss incorporate expert opinion scenario approach adopt model select combine compound compound capital compound fit business process business wide correlation typical frequency framework adopt lda ii requirement business year horizon section framework present cell business compound year event count typically year nature rare particular infinite propose classical available modern processing calculate algorithm recursion long history quantile function heavy recent development class admit analytic solution bank risk calculate requirement seven eight business depend drop index risk unless utilize loss business convolution density sum calculate convolution thus q nz convolution analytic integral form consider generalization close stable ensure stable possess useful skewness year cell four parameter parameterization appendix tail decay determining skewness exponent heavy light cauchy tractable family stable admit closed evaluated point typically characteristic discussion intractable simulation efficient function basic require lda binomial poisson doubly poisson bank positive considering analytic es case lda asymptotic comment capital aggregation model contain dominate time increment month distribution model variance less arrival loss increment arrival process poisson clearly year loss loss extend model doubly derive binomial year process extend doubly mn doubly binomial standard beta convolution lemma mix compound consider bayes conjugacy know hence doubly negative cumulative distribution close conditional application mix compound derive conjugacy property know hence recently poisson begin poisson process exact loss poisson cumulative close eq convolution random weight compound conjugacy poisson q consideration study capital binomial doubly develop analytic sum representation determine sum theorem demonstrate paper compound asymptotic truncation finite tail compound median allow term lda compound searching value integer specify n w r p n nr l sum compound express analytically eq determined demonstrate truncation discussion section table derive frequency setting frequency addition matlab intel ghz cpu source code edu file c cn cp binomial beta gamma ga frequency beta binomial beta poisson ga loss compound correct simulation study simulate develop via obtain year obtain expression point cover represent capital model compound doubly beta model derive frequency doubly theorem frequency low poisson high frequency gamma compound doubly gamma derive frequency frequency carlo widely generation compound demonstrate accuracy derive estimate excess estimate example demonstrate truncation binomial doubly binomial binomial doubly poisson versus estimate cdf square cdf mean demonstrate squared index simulation monte equivalent ccc carlo low sec min beta negative min min poisson min min binomial min beta sec min poisson min times versus monte estimate next result provide distributional combine lda distribution present closed form trivially result lda lda compound I jt analytically comprise mix specify q apply density cumulative analytic doubly lda model ability heavy typical aggregated compound model loss first throughout year intensity result compound scenario consider increment increment capture binomial success change derive estimate simulate exact low high frequency scenario term develop truncation
heavily depend let procedure poorly effect come da procedure path matrix scale likelihood recognize poor behave look validation organize section show weak part local degeneracy weak element posterior von asymptotic sequence law chain stationarity guess distribution impractical good mind say always work inefficient good bad degeneracy exclusive degeneracy degeneracy poor behavior sometimes kind convergence mcmc weak call interpret large iteration semi jump embed markov law almost hand identically general take lemma n follow ergodic tends claim measure assume always strictly procedure component perform well modal switching degeneracy weak consistency assume close bad illustrate obvious relation write integrable take scale q consistency local degeneracy localization da ergodic see also convergence process weak order invariant rewrite convergence probability procedure direction asymptotic high small asymptotic particular latter local result da independent hasting compare da procedure mixture elsewhere briefly procedure iterate f hence obtain proposal close kullback leibler posterior n consistency da consider mixture illustrate difference da procedure fairly unlike trajectory da behave weak consistency effect difference quickly size da da suggest check effect different value order da htbp check mcmc le suggest comment difference may article present suggest order verification chain diffusion probably da procedure probit numerator denominator propose grateful suggestion ng gx rt side q argument I negligible follow contiguous vice versa pd n probability prove numerator nf numerator nc bernstein von come distance bernstein assumption dx da diffusion limit become sequence respectively moreover nx x read remain term expansion n p next show p previous n n n n n n h nc n equation come term side n nr n n n write give convergence study markov reference therein apply ix da tend proposition continuity representation may space theorem assumption assumption remark assumption commonly year result convergence convergence
use validation set well benefit output binary node filter use class decision node arrange bottom testing proceed leaf computation logarithmic filter similar study reward context know contextual bandit include health care task context challenge past proportion take either former leverage overcome policy evaluation value good consistent doubly low variance well policy doubly become environment receive feedback action internet whether user ad receive ad health refer patient version assume kind address contextual bandit dm use incorrect data model reward model reward restrictive hand usually possible quite suffer especially differ evaluate technique dr estimation overcome doubly robust statistical estimation incomplete dm correct unbiased drawing inference age family ask survey census statistic condition weight estimate form form predict outcome available doubly estimation accurate predictor doubly bandit analysis model deviation doubly style doubly robust beyond study evaluation doubly estimation therein internet effect feature focus evaluation address doubly asymptotic various assumption make paper idea language reward estimator bad variance offset tree think offset case estimator describe sophisticated contextual new reveal choose concatenation precede reward observe collect estimate maximum lead well classification challenge previous reward datum limitation dm condition good refined instead approximate action proportion old new function argument since understand policy direct variance issue become gets take doubly take advantage previously study learn informally baseline correction estimator accurate perform estimate truth dr deviation multiplicative express expected value clutter fix derive identical magnitude contrast direct variance policy reward suffice significantly however direct leading estimating evidence dr feedback benchmark average internet armed contextual bandit allow compare evaluation dataset letter datum draw iid class cx kl policy label loss select reveal component form summarize adopt repository letter dr rmse dr dm letter page dr ft offset investigate accurate policy context constitute policy test obtain dm dr estimate dm dr estimate loss ridge result bias rmse squared predict dr estimate accurately suffer apparent enjoy effect come policy dm focus dr apply dr time partially decision ridge partially run policy optimization advantage great evaluation dr includes offset specifically optimization implementation version rather weak dr version offset tree tree finally one descent induction different choice number internet website record million associate feature age gender calculate summarize unique random true however situation impossible scheme special formulate arm formally partially label sample define adopt denote univariate deviation th randomly dataset subsample
henceforth simply describe ball ball respect avoid confusion conventional definition uniqueness radius available radius set interior minimum continuous compact nonconvex present minimizer acceptable axiom compress sense rely provide certain definition state asymmetric previously format rip rip isometry large vector symmetric minimization accurately accuracy estimate sp guarantee solve least rip proceed form rip compressive measurement rip estimate coherent kk obeys eq parameter approximate ideal feasible f term error ideally residual merely step small desirable restrictive choose control increase decrease imply measurement increase iterative rip I algorithm reduce function determine error nearly sparse power law choice suppose hold obey x thus apply inequality inequality note q use yield desire case inequality please pl prove define projection onto thereby recursively statement theorem descent sparse square norm onto ball statistical requiring small method surprisingly leave unclear intermediate x x projection ask design develop provide building method may find future sophisticated nesterov method constrain learn f property projection every proof contradiction ix ix I lemma therefore simplicity assume real onto coordinate another onto x j I jx jx jx fact gradient constraint uniquely define entry q partial derivative desire pp pt statement hold regard root root respectively illustrate thereby since case zero peak strictly fact p exactly claim yield fact peak suppose obvious contradiction w otherwise identical merely algebra always sign contradiction rely isometry entire guarantee iterative thresholding suggest group increase accuracy compress restrict isometry project problem signal observation govern equation linear simply collect achieve merely singleton least describe imaging absence independent ideal extended norm convenience throughout scalability functional merely merely count theoretical minimization however alternative framework regression solve square eq tractable include thresholde pursuit compressive pursuit solver solution approximately iterative outline henceforth limit recognize soft originally pursuit also solver consider shrinkage iteration iterate sufficiently contribution comprehensive regime unlike feasible satisfy therefore relie fact long extra condition convergence name therein ignore exhibit sublinear optimization guarantee decomposable norm include linear possess objective
home sale work stock price market unclear market collective web daily volume stock correlate daily volume stock particular query volume many peak trade day carry dataset web engine enable user show dynamic collective finding contribute early financial www introduction activity leave digital transaction mobile reality field science physic several social phenomena direction concern epidemic people usa activity keyword actual care paper address issue obtain early financial market issue financial ultimately phenomena signal complexity anomalous collective great interest policy intervention appropriate start however context market show volume shift correlate price yahoo engine list exchange assess hereafter relate daily trading hereafter mean cross correlation mean causality analyze search activity collective hereafter consist stock market financial stock price query correlate transaction stock lag week week query volume correlate week trading volume yahoo engine volume stock aggregate volume suggest trade stock pointing investigate market daily scale even volume proxy market address forecast market novel analysis traffic activity market frequency volume volume stock trade company top plot trading volume company research motion limit material simple inspection reveal time series tend peak fig cross trading query pearson q trading volume range cross correlation positive broken line volume trade volume day beyond lag day trading vanish cross trading volume clean cross completeness support clean table stock spurious origin volume volume trading volume confirm finding also vision market influence activity contrast appear much short seem resolution result coefficient present query volume trading volume appear large coefficient opposite assess significance set company randomly pair trading time company average permutation range small present explain term general remove event trading time series verify table stock report support value cross stock observe stock consider drop distribution underlie investigate series tail fig support discussion validity correlation responsible correlation stock percentage explain turning towards volume trade volume well trading volume volatility appear volatility exponent range therefore volume trading volume fig effect respect correlation absolute return result branch volatility even trade origin due volatility cross correlation volatility autocorrelation support determine provide trading volume find volume cause want causality argue cause interpretation causality link wrong result direction trading claim volume observe volume tail investigation fig support picture come focus yahoo yahoo expect user stock interest user market display important news corresponding page among link various window interestingly month whole robust along certain restrict high correlation trading volume amazon com netflix month year user check portfolio stock suggest financial expert search uniform way addition trade price drop drop show appear evidence query trade user bring surprising picture volume user store yahoo engine assess stock trade stock focus trading volume price volume compare trade stock computing delay existence web traffic trading stock confirm analysis user query stock month suggest market crowd finding explain market proxy furthermore query reflect composition find assumption portfolio composition know empirical market confirm take bring incorrect disease portfolio balanced could market straightforwardly epidemic financial market fact latter differently ordinary news financial agent market short disease reason early carefully effectively detect sign also believe promise currently kind semantic material overview contribution previously say user yahoo event place market basic market activity stock correspondence activity stock trading volume variation volume relate search query volume volume compute conduct analysis perform take consideration contain stock string company query volume trade permutation causality several analysis assess significance correlation statistical user work user issue query finance refine extract typical look one look stock analyze compare volume volume national association trading market united precisely analyze market amongst index contain financial include outside united list daily financial yahoo finance focus attention volume yahoo engine span year mid action user interaction engine page return decide volume extract aggregate daily text contain stock yahoo text company name incorporate log moment engine aggregate volume different annotate represent query activity count daily distinct company daily query distinct user query volume yahoo finance volume trading volume compare every trading volume stock separate query company company extract source volume series interval range mid mid contain stock financial available trading thus uniformity working day work stock series volume series trading volume company coefficient delay choose maximum lag week five correspond cross coefficient consideration coefficient case lag range consider worth individual observe stock support information worth stock stock stock hold stock also table table lag remove time trade interesting correlation seem event news second company name observe weak average cross correlation query spurious query origin query nonetheless stock represent string natural language life completely study level query company stock support information filter query noisy spurious query restrict report correlation lag volume volume query relate company clean log correlation volume trading volume show one volume correlation user volume trading lag statistical permutation test randomization permutation validate nan permutation outcome true among determine value least extreme reject verify significance company volume cross query volume trade high company company merely consequence market activity market activity pair trading volume cross permutation ensemble series volume company pair series trade company pair randomly compare macro real dataset company pair company get value test find separately understand deep actually one scenario company company trade company company form every company empirical value statistic sort similarly volume company trade company correlation company trade company query volume real dataset include p reject value case stock query volume global finance relate search traffic activity general worth present trading volume volatility trading volume volatility volatility source volume trading autocorrelation back volatility price proxy query trading define price day stock price return build series series positive return rt rt rt rt length similarly involve trading volume stock cross volume company break report correlation basic price return reflect stock exhibit variation web search concern stock show correlation volume volatility break significantly query trade solid branch case volatility observe autocorrelation expect origin correlation query trading return negative volume reason consecutive clean table one get causality whether test cause reject apply causality volume trade volume trading volume company cause trading volume company stock whether causality opposite direction include however query volume noisy extract perform manual result table summarize outcome specify volume volume compare company cause column provide fourth reject report observe opposite strong time cause trading company oppose user row examine reject much opposite direction reject already observe experiment weak consider user volume case trading cause volume autoregressive trading volume volume volume cause trading also interpret follow query volume help predict trade reverse argue principle term regression instead deal query
without receive feedback action play otherwise action probability play result correspond confidence action play figure action exponentially call tb game horizon confidence horizon v v v g tv building worker routine routine discuss round update receive change subtree store node tv v tv recursive play tree v either exp optimize corollary tuning tuning book show figure exp transformation second action outcome v pl v v w h er exist outcome throughout play measurable universal give er become let stay index tt jj j far least bound martingale associate denote play time history recall construction algorithm explicit th unbiased random x hence bernstein inequality achieve desire bound depth either exp satisfy exp induction hypothesis true depth action action furthermore word loss contain event know last last switch step even inequality outcome assume step exploit hypothesis sum root call child eq action euclidean non trivial loss game monitor game exist minimax regret present work adversarial ensure dominate action small outcome close finite dominate action simplex cell set action space polytope denote induce interior small pair cell cell decomposition cell hyperplane I characterize follow action dominate kind cell ic dominate game mean action cell decomposition correspond non dominate clearly cell coincide pc p pc pc f choose lie two fix combination dominate cell compact convexity contain closure well length also p eq choose p v part adjust necessary continue lemma quantify kullback divergence define divergence divergence eq depend proof lemma notice sum coordinate modify generality coordinate write logarithmic second taylor expansion around universal substituting let third term dp always lemma action satisfy avoid indexing action later randomization time choose outcome monitoring outcome outcome appendix imply ki ta imply continue collect averaging get lemma proof carry state action close partial outcome pair outcome n action put ensure game hold remain obviously take corresponding trivial present outcome condition lower bind er constant minimax let important minimax degenerate degenerate action find degenerate generality dominate reduction section far non get generate outcome outcome later replace variable internal outcome depend depend algebra depending prove inequality rearrange substitute statement subtract lead lower bind low independently give choice ensure separation clearly trivial b choose monitor minimax question degenerate degeneracy category game minimax nonetheless conjecture include important open result generalize game outcome finite monitor restrict opponent two outcome result hardness regret condition believe generalized situation play respective generalizing result algorithm exploit two yet outcome bind exploit assumption opponent ab bc lead proper continuous minimum action whose large action action non dominate dominate loss vector vector da action lemma compact imply outcome regret know thus tc depend c c c I ic I sp ic ic regret take outcome randomization clearly omit prove deterministic denote history history section since determine general feedback action game action product sequence last divergence notation none action degenerate play action th x tx al de ec game mathematical framework decision repeatedly suffer receive feedback signal minimize progress game classify game finite regret either computable four toward classify monitor online monitoring constitute framework decision feedback arise make expert multi dynamic pricing pool convex partial monitoring game opponent receive feedback signal neither outcome reveal learner player learner assumption opponent opponent learner keep loss opponent learner suffer loss ask cumulative competitive viewpoint take cumulative compare I round growth sublinear per round approach design focus monitor optimal view measure hard motivation behind determine minimax achieve large game action set action outcome full game determine lie time therein game loss lie bad regret least monitor game loss action loss lie inf know exp know optimal strategy regret bad learner bad recently design possibly action game game step classify minimax monitoring game exclude later minimax fall call four game respectively give efficiently characterization four class admit efficient regret factor design et trivial game choose algorithm partial monitoring game number outcome notation denote outcome outcome denote alphabet real learner opponent game proceed round choose opponent outcome receive nothing else reveal learner particular remain simplify assume opponent deterministic learner equivalently allow learner feedback outcome internal random learner incur learner constant ta eq supremum outcome infimum also identify probability dot product characterization preliminary partial let one action dominate dominate pi outcome neither hold outcome game action bind monitoring game action bandit example armed bandit game feedback instantaneous advance classical bandit loss constrain game condition type action game hard classical bandit game regardless loss vary action action dominate game minimax fact want sequential test suffer loss formalize monitoring game cm q outcome correspond correspond dominate minimax regret result also notice picture translation picture bandit efficient situation sequentially spam email output additionally request label email incorrectly request suffer email request feedback loss formalize game c correspond request third spam request outcome spam chain neighbor game reveal outcome q action dominate minimax game regardless choose round guarantee regret cm non action thus degenerate game cm
embed plane lattice various embedding site triangular lattice occur statement directly proposition site big least constant hold prove acknowledgment like thank van helpful phone phone novel statistical algorithm noise wavelet system noisy digital pixel propose efficient technique quick detection noisy detection one look image question diverse cancer automate picture make intensive procedure skip immediately impose object permit nonsmooth noisy mild object interior detect aforementioned automated cancer material procedure work well precisely image method advantageous object use practitioner medical perform reconstruction image object rigorous analysis error differ substantially subject wavelet technique nonparametric necessarily asymptotically pixel algorithm drive stop human stop appropriate white focus serious limitation essentially vast differ background colour colour object paper applicable rescale colour organize statistical detail crucial theorem testing devote main numerical computer discretize setup colour colour background mathematically assume black pixel belong meaningful image pixel noisy pixel pixel paper always begin analysis assume array value stress nan fully necessary mean restriction quantitative original corresponding free dependency throughout degeneracy symmetric since symmetric ij p p ij complete ready describe thresholding pixel colour colour pixel grey observe grey begin think pixel colour grey value noise pixel colour transform picture mathematical main fix exist condition pick vary transform set pixel picture call pixel colour edge add edge two connect vertex crucial definition lead testing procedure property view square subset view black white picture lattice triangular thresholding theory formation crucial explain split vertex belong original left background vertex subgraph observe cluster go detail theory introduction mention relatively difference theory quantitative behaviour efficient randomize detect phase simultaneously appropriate mild noise triangular lattice statement proposition explain lattice applicable nonsmooth nonparametric observe design question ij nh formally probability noise time crucial kind picture small observe precisely object moderate easy strong contain least pixel furthermore purpose relax triangle shape character finite ready false run step find picture first thresholded picture large find size cluster define define search graph procedure deterministic classic book chapter symmetric e picture exceed n nc mean order think work image pixel detection square use consider size neither sort thresholding indeed noise tail tailed thresholded value false detection fine fig neuron advantageous gaussian independently pixel image picture version picture run simulate neuron detect picture pixel value grey picture thresholded study consistently image nan true picture algorithm thresholded consider moderate size algorithm consistently detect pass neuron run shall site triangular cluster contain origin triangular satisfy eq otherwise need
know assume illustrates data four bayesian mixture mix prior cluster later make mixture choice model mixture generative define assignment assignment assume observation conditionally return rt example cluster rt interested assignment grouping switch care label care preserve ignore rule assignment fix assignment denominator sum group become salient limiting discuss contour finite specify cognitive advance cluster allow previously unseen assign call partition independently discover rational categorization crp derive name imagine table imagine sequence restaurant customer table probability restaurant table customer customer partition show generative crp customer circle crp crp mixture model assignment customer sequentially assign class customer number table intuitively large customer crp invariance order seem counter customer assignment accord chain term come equation group place customer customer assign customer cardinality first customer contribute customer contribute probability customer third contribute probability write rewrite equation per notice identify customer assign exactly group simplify denominator write joint equation reveal exchangeable group group configuration depend order customer crp model table though exhibit finite active customer draw observe concentration control table customer treat return rt crp us place cognitive process set process specify mean transform force notice skewed outlier examine infer assignment cognitive process addition crp away parameter partition group predictive crp predictive distribution crp previously unseen example time force decision experiment color gibbs factor influence way model intelligence application latent usual component small factor much contribute see representation primary mode popular observe property latent recent review specify preference kind g intelligence assume factor intelligence loading influence noise extend factor loading product two binary mask variable indicate continuous sometimes spike spike zero bayesian approach infer factor mask ica fit likelihood classic application intelligence argue intelligence intelligence test participant highly test highly several pattern arise unitary intelligence resolve factor intelligence convenient reality intelligence avoid specify instead allow row correspond intelligence infinite correspond generative ibp customer circle ibp customer select factor note ibp rather chinese process component ibp infinite similarly customer dimension infinite arrange line sample customer sample never customer encode customer ibp ibp crp unbounded crp one latent ibp tend ibp key crp ibp crp ibp illustrate return intelligence intelligence intractable approximate one g give one typically estimate crp ibp analysis collect reasoning task participant histogram factor posterior sample indicate combination around although general intelligence investigation right display first load ibp demonstrate example histogram factor infer generate burn relationship loading infer ibp describe factor ibp generative analyze give latent generate datum thus computational many close way treatment beyond scope use link package implement markov chain equilibrium draw chain eventually sample markov construct thank exchangeability property crp particularly amenable consider use distribution term excellent survey crp gibbs factor inference different figure two drawback assess idea posterior family search member close recover unless typically crp ibp operate crp ibp factor operate available alternative selection parameterize broad array bayesian variational provide direct space model find convenient modeling number component search efficiently explore widely limitation worth past year idea diffusion idea volume concern across group similar virtue group provide cognitive characteristic degree coupling set extension beta share individual document word share across return rt imagine subject infer underlie cognitive suppose cognitive share different proportion precisely kind hdp limitation dependency capture markov class hide sequential employ class infinite hmm hierarchical dirichlet alternative linear dynamical autoregressive moving evolve linear linear dynamical explore nonparametric switching mode hdp another type dependency arise dataset disease location also occur nearby way capture dp variable spatially couple generalization dp segmentation allow nearby pixel pass devise specification dependency covariate people age risk disease recently author ideas ibp factor discover hidden output variable discrete supervise output non linear function place supervised proceed function gaussian inference process analogous mean parametric predictive develop prior input linear mixture relate marginally linearly within output model gaussian process output input build flexible grow review model potential scientific noting address choose general may preferable finite capacity factor treat tool work model argue human categorization adaptively number observation domain ibp human argue human decompose feature strongly object treat part find change include segmentation theory acquisition recent volume introduction review statistic acknowledgement grateful ed comment version manuscript manuscript suggestion suggestion fellowship nsf nsf google crp ibp combinatorial way understand construction review perspective crp ibp factor dirichlet dp parameterize distribution denote appeal measurable measurable partition dp average play high value mass concentrate process infinite atom atom independently property connect restaurant consider draw examine repeat note repeat draw exhibit define partition distribution chinese restaurant process parameter share independent dp assume exchangeability de say exchangeable conditionally collection prior generate reasoning equation theorem dp add step dp mixture crp mixture good break stick beta stick break dp via provide constructive stick break segment proportion remainder stick give draw representation corresponding admit atom note dp weight sum
close quantile value nominal significance report quantile become nominal especially cauchy affect distribute conclusion table detect consider ordinary noting performance especially case competitive c c cauchy mm c cc represent true denote total combination threshold parameter multivariate normal mean matrix experimental try three dimension non gold combination discussion list variable linear combination combination cc cauchy rely normality addition table deviation small diagnostic simulation apply propose tumor diabetes tumor set tumor patient continuous percentage except code scale accordingly diabetes consist subject subject variable body mass ratio progress diabetes addition diabete also investigate apply avoid variation variable table case cc diabetes set tumor ct great association tumor tumor datum set among set diabetes table cc variable little bit large set linear marker tumor diabetes diabetes diabetes c ct tumor data diabetes diabetes diabetes diabetes tumor diabetes diabetes diabetes tumor index ct cc type index age diabetes datum index ratio diabetes diabetes diabetes index paper first evaluate potential diagnostic individual variable gold measure share threshold independent roc curve variable threshold existence index useful practical maximize measure normality valid realize lar lar selection scheme conduct binary standard available variable unknown use density estimation propose variable computational large combination moreover ordinal suitable cutting elsewhere random pair suppose assign control give sample index gold distribution cut end instead explicitly define weight critical define width smoothed lemma distribute converge find page identically random identically gold marker condition triangle due tend generality tend let density complete theorem acknowledgement partially support science roc theorem remark operate roc useful tool diagnostic power gold gold become less useful evaluate diagnostic finding diagnostic auc index good potential diagnostic scheme addition descent maximize new even huge estimate property performance roc area gold roc tool diagnostic study moreover combination diagnostic gold gold standard evaluation classifier roc curve choice threshold informative index continuous organization national health suggest cut finding diagnosis cutting point interest new advance criterion evaluate necessary may changed conclusion evaluation diagnostic standard gold prefer motivate standard affected cut gold outcome gold lot report roc curve still gold maximum roc problem bayesian gold roc gold construct gold however approach issue especially variable biological genetic study auc propose gold study multivariate lar normality violate nice lar selection study performance range cut property next evaluate diagnostic individual variable measure find example summary detail appendix novel auc type gold standard briefly roc relate real variable example disease subject primary difficult look surrogate association develop association gold threshold subject member auc random respectively subject disease non disease threshold hence threshold diagnostic association independent cutting point gold monotonic property roc auc diagnostic integration cut weight support weight cut cut point treat suppose conditional rewrite distribution dimensional continuous linear multivariate normal z iy independent identifiable therefore combination linear coefficient standard identically gold generality centralize datum subtract tp z I regression clear regularity dominate convergence consistent estimate omit variable calculation matrix calculation numerically unstable sample relatively overcome obstacle small shrinkage lar lar regression lar propose combination lar adequate rely normality lar omit lar instead start linear combination variable follow extension sigmoid sufficiently window identically distribute continuous gold marginal conditional detail smooth summarize theorem suppose identically distribute vector constant hold need choice depend part curse method maximize positive anchor vector dimensional component anchor generality coefficient anchor identifiable base pt calculate derivative coefficient dl pl find
already powerful learn rich set expert vc expert pac factor expert reflect divergence flexible allow treatment expert policy expert experience treat shape property contextual reinforcement follow survey paper discuss result apply derive instantaneous per bind bandit problem derive instantaneous concentration result martingale sequence martingale difference variance define pac bernstein reference independent satisfy apply definition reward play round action independently respectively reward action round high expect good action define martingale cumulative martingale follow round q probability simultaneously translate logarithmic theorem improvement due bound apply generalization important state art alternative previous art technique rich infinite contextual bandit pac enable involve mutual bandit plain expert dimension complexity analysis prove co cluster unsupervised appendix theorem lemma bernstein bernstein inequality martingale second statistical physics bayesian ready take tt union fact take technical due new left bind instead negative decrease without positive apply hand modify tight tight identity verify q want get theorem enough requirement ok great substituting choice thank support network european publication reflect coherent exploration improve precede combine pac bernstein combination study multiple simultaneously evolve exploitation reinforcement selection empirical datum fit question good knowledge trade simultaneous consideration illustrate follow imagine pool additional know bandit imagine contextual consider increase simultaneously exploration trade simultaneously extend pac feedback sequentially pac decade make development method pac lie successful model pac pac bayesian prior bayesian derive generalization classifier margin beyond supervise domain surprisingly limited domain potential advantage apply bayesian reinforcement recently researcher suggest use regularizer reinforcement incorporate bellman justify provide guarantee pac batch reinforcement learn reinforcement difficulty pac limited reward action take estimation hypothesis evaluate sample size hypothesis limited action action bad
minimum across vertex hyperplane place piecewise average model smooth robust highlight minima design minima response surface method approximate programming involve unconstraine additive approximation robust regression broad resource portfolio management article novel nonparametric show consistency approximate show frequentist regression sampling need test stochastic semi scale well moderate limit could extension stochastic extremely tool sequential iteration fit search approximate set many resource wide perhaps combine preference product tend likely covariate age education function great economic reinforcement grant es national institute result cover need time cover place measure place space net constant g I numerator prove lemma norm main one jump chain balance satisfied hyperplane hyperplane addition hyperplane hyperplane hyperplane jump hyperplane initialize draw posterior draw sensitive space hyperplane order efficiently region create region describe create create proposal region subscript indice j define component proportional obvious hyperplane high start hyperplane adaptively add follow hyperplane along produce collection along linear combination covariate interval covariate assume choose partition mixture define observation fairly evenly section corollary proposition economic research reinforcement function constraint sequential state decision propose nonparametric characterize unknown hyperplane induce prior consistency norm reversible jump posterior computation function py n nf easily modify allow concave regression negative economic operation research economic production stochastic optimization know advantage estimate first place condition derivative substantially solver multivariate little piecewise support hyperplane solution infeasible observation generate regression weighted kernel subject solution find sequential programming convexity condition dataset propose convex partitioning split fit like hyperplane scale covariate guarantee univariate poor minima hyperplane intersect location sensitive hyperplane hyperplane could averaging produce smooth rely implicit hessian translate increase dimension discretize covariate normalize slope point place prior polynomial restrict spline prior place restriction basis dimension closely regression convexity enforce projection create set project back prior guarantee hyperplane define reversible markov carlo convex dense subspace determine space numerical produce term outperform objective potential suit objective support hyperplane support point semi hessian place restrict meet function automatically meet specifically hyperplane arbitrarily maintain straightforward follow factor hyperplane give prior spline bar place parametric prior number parameter reversible jump markov chain bar introduce knot within dominate detail consistency assign converge one small neighborhood grow rate size contract maintain despite property multivariate frequentist framework show show recent literature relate estimation show consistency mle transform estimator restrict asymptotic attention monotone spline extend prior let true throughout rest paper denote quantity counterpart eq examine actually convergence dimensionality full bound compact generality assume first compact restrictive unlikely occur bind cover plausible design compact range plausible choice atomic p x f ensure sufficiently hypercube let lebesgue suppose exist least point atomic give compact covariate continuous stochastic convergence uniformly bound compact covariate derivative every theorem use result put non requirement positivity leibl neighborhood existence test positivity construct create assumption pointwise determine empirical make model function matrix convenience sufficient algebraic actually subspace situation arise example covariate combination require know simply keep hyperplane b covariate theorem showing give b achieve term leave open bound setting rate achieve general extend accommodate reversible sampler model assume often violate shape locally outlier region globally poor prediction accommodate induce flexible introduce separate variance modify poisson misspecification propose determine metropolis proposal candidate block hyperplane update area unconstraine model candidate distribution proposal covariate partition subset n kk hyperparameter proposal distribution necessarily parameter high hyperplane covariate current removal component hyperplane hyperplane create region proposal distribution mixture along subset observation direction full mcmc sampler extremely fast unlike converge right component reach four construct strict block ensure autocorrelation drop three sampler lead situation restriction sampler noise admissible acceptance drop estimate unconstraine counterpart competitive art behavior objective suit
comment shall proof subject order apply regularity realistic case modification allow subproblem end two subproblem number generalize lagrangian henceforth assume sequence repeat assertion repeat replace precisely follow sequence keep mind q suppose continuous sequentially weakly compact weak k saddle lagrangian far turn subsequence weakly low semi estimate yield solve proceed saddle observe observe u p v prove part assertion see q subdifferential section thm thm lemma definition alg concern automatic adaptive function noise end mean type constraint alternate multiplier orthogonal projection intersection convex prove capability illustrate imaging relation observable unknown shall formalize identically distribute r encode functional inverse inverse apply estimator speak invert regression application primarily mind signal g characteristic structural assumption broad claim regularization recently different branch mathematic consider therein recently image neighborhood relation model framework straightforward reconstruction typically consideration imaging situation high dimensional respect minimize functional subject coefficient residual modification aim regularization variation cf norm hence convex smoothness texture relation account selector increase paragraph efficiently class estimator end incorporate possibly fit quantile encode distribution know parameter sound least constraint estimator rich signal visible suppose significantly differently statistic bound reveal application area estimation mind mainly normalized indicator residual resemble white set word feature illustrate example among residual resemble white noise prevent parsimonious reconstruction mr statistic various context incomplete overview classical mr consider indicator mr statistic call q reduce weakly establish image denoise employ diffusion equation spatially vary jump piecewise regression consistency general hilbert another attract function reduce particular multipli cf trivial choice consist segmentation translate testing nonparametric qualitative monotonicity similarly mr density multiscale sign test curve cover framework generalize selector interpret consist regularization functional lasso selector small consist scale sense constitute extreme practical cardinality set denoise million simple mutually perform solution iteratively poorly reconstruct mr approach allow put differently paradigm redundant mind convex convex cone euclidean realize employ appeal scale cone cone straightforward capable previous approach programming homotopy method author stress projection computationally application mind locally reconstruction use aforementioned graph large constraint bring cone exhibit face compare w fact turn cone projection onto simple linear algorithmic framework numerically many slack primal feasible slack appeal effect employ without consider alternate direction case occur form tackle orthogonal projection intersection merely increase considerably decompose contain never put position method locally visually algorithmic linearly problem lagrangian alternate assumption prove convergence give qualitative iterate theorem occur paragraph illustrate study problem nonparametric question linearly discuss notation subsection paragraph alternate multiplier admm alternate solve unconstraine penalize euclidean reveal modular replace projection paragraph stand grid moreover convex notation rewrite average arbitrary could useful different image transformation residual consideration separable product induce bound mr agree upon bound discuss follow saddle assumption refer instance dense need order requirement fulfil gradient base regularization slack rewrite equivalent characteristic region assumption set technique recall definition name stems augment saddle point saddle lagrangian already saddle versa existence usually hard assumption summarize set assume saddle due continuity minimization follow explicit third variable perform convergence subproblem sake simplicity ht size tolerance approximate iteration uv u v r kk generate weak cluster point statistic modular projection section algorithm penalize least square extensive overview tolerance terminate output inversion know introduce constitute efficient case euler functional fail serious try euler lagrange condition simulation regularize order hand standard smoothness signal end introduce regularization functional bregman satisfy semi example bregman incorporate word bregman differently symmetric bregman total sufficiently tv tv formula complicate mean symmetric divergence give approximate empirical mean trial regression estimator close estimator define bregman oracle practice reference secondly constitutes drive spatially adaptive regression become independently upper solid application mind arise spectra entity signal choose finally mr consist interval quantile mr statistic except special statistic usually hand henceforth argue smooth small exceed interpretation evident peak stay contrast level additionally maxima take indicate maxima imply c c l similarly reference concerned even superior indicate meet smoothness much result visual index increase side side condition interval intersection group simulation minute need step considerable parallelization section apply dimension typical noisy scale black h variation semi define range agree specify henceforth particular simulation estimator test favorable preserve estimator preserve essential detail piecewise reconstruction smoothly vary undesirable oracle visually perform bregman bregman smoothing image area pattern recover u h reference distance inconsistent human take contrast use author lie perfect match simulation list suppose superior mind evident rather suited distance remarkable equally bregman unknown far structural concerned superior natural proper weight function achieve substantial part g etc obvious standard image depict signal pattern yield power test order image mr average sake consideration precise center colored indicate location smoothed region incorporate local residual comment normally distribute distribution set scale certain dominate mr statistic multiscale compute normalize constant alternative approach would turn purpose eq group side intersection correspond hour parallelization deconvolution assume lattice zero circular deviation primal amount solve solution approach application technique model depict marker image area resolution pixel spread optical full half nm note fall range cover modify division understand pointwise involve firstly nonconvex apply secondly project onto intersection project modification iteration consequence modification consist square side length overall normally step iteration take minute need depict choose number appeal locally adaptive concern protein mark concentrate basically gap image emission constitute relatively capable image resolution reference depict comparison reasonable regularization relevant become one mark box visible mention aside transformation normality intensity
cyclic fig worker master choice either satisfy q straightforwardly mm long satisfie delay bound large compute sample say cyclic large delay penalty may dominate parallelization address drawback attention locally average architecture delay height span synchronization cyclic architecture worker communication master worker give section master simplex put theorems assumption gradient uncorrelated simply receive ii condition set jensen asymptotically number calculate delay graph give corollary irrespective adapt slightly give well rate expensive focus cyclic setting though principle average scheme denote master worker delay consider two regime rate attain roughly gradient dependence increase stepsize nd fast cyclic different
mechanism mab adversarial learning number include recently design offline sequential price approximation obtain multiplicative large dynamic pricing sequentially interact agent private value reveal agent price whether mechanism mechanism round output observe draw denote round item demand furthermore maximizer know sale define differentiable hazard price pricing price item stop case expectation pricing demand design price mechanism depend demand offline mechanism mechanism give seminal price let total price additive regret pricing demand price benchmark benchmark provide expect minus obtain multiplicative pricing regret offline regular focus benchmark address price benchmark fix price price price eq demand p price variable chernoff near price provide rest price p nr sp benchmark demand distribution price benchmark trivial bind price price mechanism upper free pricing strategy demand carefully optimize trade exploration exploitation
random eq subsequent bandit machine variable encounter action play sequence take round random random define two random q p hoeffding present regret bayesian play prior bound action stand leave absolute eq ct result present combination pac hoeffding definition
strictly map adapt model bias mean way impractical remark hierarchy box mathematically quantification expand important implement may implementation trivial program drastically careful tailor rna seq sequencing technology quantification consider read end uniquely begin read random length denote read note adjust generative choose read select among read read change reveal read note lemma supplementary condition equation transform unique map obtain read mapped formula measure exception equation derivation think relative abundance factor derivation relative abundance reveal evident abundance estimate absolute rather relative normalizing map read replace read sequence mistake apply improve abundance drastically address implement software package rna multiple ambiguity assignment necessary
explain example coincide introduce computational language positive production sequence l production clearly denote let order ordinal dimension ordinal recursive recursive ordinal conversely ordinal without notation k l l ij obviously preserve ordinal constructive recursively initial segment ordinal recursive range put di j finite inverse system quasi l x preserve order turn closed indeed quasi assertion l il x yx lx conversely actually l infinite anti viewpoint algebraic atom atomic
consider uncertain two cumulative f ref f ds response define ds structure box structure deal instead singleton cdf one need belief finite ds construct cdf transformation body law apply much early quantity interest measure cdf total amount integrate cdf decision denote deal interval drive
grid grid nod configuration require also scalability however well circumstance ex axiom describe first attribute graph exploit suffice sample together restrict technical scalability evaluation graph interest scalable also goodness generate compare statistic original graph hypothesis predict large graph estimate stochastic scalable graph previous graph scalable large model attractive fail capture characteristic order generalize attribute provably model power furthermore able real issue open currently aware scale bad observe edge determine exploit graph number know absence sampling adjacency trial generate real world graph million node paper restrict significant permutation sample graph target kronecker product np x kronecker take care discard later use th kronecker henceforth see rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle
may denote outcome load determine finite mesh inspire independent describe shape material modulus material stress linear euler mode independent objective shape maximum amplitude around surface half random field coefficient expansion field path squared exponential choice analytical detailed field random discretization involve cccc reliability table propose importance lead probability agreement strategy equal reveal configuration identify demand maximal whereas random configuration failure allow third provide prove problem reasonably rectangle rectangle usual reliability instead reduction consume expensive black
span capture orthogonal direction offer euclidean purpose pc decomposition disadvantage eigenvector nonzero sparse favorable interpret easy pc slightly requirement constraint maximization aware sparse pca high cost np literature rotation pc thresholde
clause clause gap reduction np hard hyperplane reader construction facilitate constitute hardness formula clause clause two whether variable follow clause four also importance become constitute correctness unit unit whose margin clause set coordinate integer true claim appear satisfy argument exist clause call absolute good one bad appear clause show many imply clause unit include rearrange w ic ic iy since
regularization increase face multi make le minima kkt minimum equality covariate discard instance goal exclude fold scheme fold use reduce nonzero adopt strategy lar framework adopt henceforth le choose le influential measure prediction minimize table simulation perform le increase std std mse outlier std mse detail replicate validation curve three replicate figure replicate le minimum vanishe regularization path mle consider binary class label employ notation construct linear net mixing derive entire regularization lead parameterization vary author website begin taylor follow minimize return problem chain rule relationship leave hand constant intercept regression coefficient
true interest maximize repeat reveal analogous contain simulated intercept ridge difference create thereby result poor parameter correction discretize surface discretization generate simple capture biological characteristic effect surface save true simulate mcmc use traditional confirm shift simplify appendix due inherent problem examine whether reproduce important characteristic fit data value paper use mode summary plot summary simulate actual term b compare summarize region suggest may fairly rich inform biological parameter shift vary show estimate contain truth vs traditional bayes develop characteristic process review gaussian web new several vector simulated case pose serious
smoothing penalty functional smoothed factor dense commonly store run gradient step projection show use penalty incorporate analog closely relate consideration address number factor extract amount scope several select factor extend imputation straightforward possible select amount two factor long variance explain first k proportion note algorithm component control sparsity smoothness seek way propose bic selection bic bic estimate freedom freedom iterative converge stable simulation achieve highly approach related achieve assess effectiveness simulate real example simulation generate k p snr center bic first spatio two consist region temporal activation pattern snr autoregressive autoregressive process var var demonstrate implement pca variation region activation pattern subsequent mixture never instead know equally spaced respectively possibility spatio temporal graph neighbor smooth near neighbor laplacian mesh space denote analogously explain spatial smooth recovery
width region array experiment shannon choose basic slide window start apply cluster depict count constraint capacity channel horizontal dot
ill detection pf ill deconvolution form function gamma essentially evaluate assumption paragraph statistic principal symbol order approximate theorem considerably impose asymptotic fourier fan et al recall ai exist positive previous variable negative principal symbol error asymptotic coincide usually multiscale r statistic almost ignore formulate differently asymptotically band moreover moment kernel negativity continuity non empty intersection solution argument I comment deconvolution scale simplicity band locally pf bands converge constant band situation confidence band bias error multiscale scale confidence display obviously say existence confidence denote confidence account construct symbol act derivative appear density study statement explicitly width influence pd reformulate easy additionally lagrange calculus polynomial restrict scale I q locally powerful summarize find kernel require define theorem prove weak
token later hold hypothesis interesting tend generate token theorem lemma prevent expect individual solve divide part contain internal separate box keep internal separate two define separate sub set inter separate binomial fix separate font variable token
operate try minimize regret adopt one subtle must handle adopt meta parameter duration policy slowly increase follow novel meta policy problem regret achieve secondary try channel availability channel evolve channel find receive instead channel get reward aim reward throughput slot designing difference reward know matrix express obtain policy ts show channel e positively
adaptation constraint method stability separation robustness neural use form one type feedforward neural neuron coefficient x sigmoid consider unless estimator bound ensure differentiable allow estimator bound design difficulty intuitively think interpolation sample suitably convolution designing boundedness thus serve modification wish emphasize subscript match convention statistic let finite estimator act ensures parametrize convex nx lx characterization kernel value hypothesis result nonzero characterization need robustness numerical calculus deterministic conjecture careful show stochastically true stochastically know vary pointwise minimizer limit
use systematic deriving net dimension obtain taylor expansion symbolic find see expansion derivative centre align r pp p condition conversely approximate truncation obviously analysis seek example derivative forward difference central mathematical software raise expect derivative behave surprisingly answer proceeding know derivative pose pose net h yu mu refer figure mh laplacian evaluate grid separation u value omit convolution operator net analyse character compare net noting degree freedom introduce oppose subsequent analysis fourier spectrum periodic behaviour behaviour width map also rewrite order elastic net part net carry dimensional case joint fitness character minima energy explicit continuous formulation du du norm square integrable operator order p du since term keep variation likely result belong incur incur polynomial fitness take generally fitness unique spline fitness mixture variational penalty transform continuous domain fourier act pass cutoff frequency increase fig likely fitness inverse write convolution make explicit discrete however continuous operator introduce extra consider finite approximate derivative mention delta cr cr cr cr r r associated area high low define toeplitz hold large computed directly involve modulus transpose hermitian transpose symbol implicitly produce align centre dft appendix toeplitz toeplitz represent rest successive rotation follow property matrix eigenvector plane easy basis also matrix mm dft since identity odd pair distinct conjugate eigenvector allow spectrum give call mn eigenvector hold eigenvector latter invariant permutation rotation origin eigenvalue odd span cosine n span span eigenvector generic eigenvalue nan incur compute superposition plane uniquely net eigenvector contribute eigenvector apply operator eq discussion fix eigenvalue quasi toeplitz polynomial later useful proposition show modulus tp modulus
noisy edge boost good correct edge describe entry think rank neighbor build row early neighbor degree way select frequent regular degree variance around top order look idea illustrated frequency decrease node drop experiment well type ii balanced error come error false pick threshold trade l type ii rely histogram neighborhood neighborhood degree one
satisfy another subset equally thus imply random conduct carlo simulation generate nk match start random record lp succeed repeat experiment
assumption write submatrix recall relationship p p expression string decompose image combine theorem eq apply would imply contradiction show determine therefore imply hence p p p v key p p therefore th analogous suffice generator induction yield iw iw argument denote closure algebraic variety sec def simple proof theorem first consequence closure closure agree topological closure agree follow equivalent n u j iv j consist p
label adjacency feature rkhs oppose regularization rather global interpret task estimate manifold helpful laplacian graph matrix diagonal differentiable laplacian think measure smoothness give eq small write formulation graph laplacian matrix function useful eigenvalue show reveal structure laplacian literature walk reach various property derivation eq average vector w st infinity inversion continuous I law let l laplacian generally matrix generalize positive machine th interpret th motivate context answer give graph proximity taking oppose interpret vertex connectivity entry jj ji metric path connectivity expansion equality side heavily heavily generalize
fisher jeffreys odd distribution lead e b lead update interpret failure reasonable default choice jeffreys possibility table prior match bivariate normal perhaps entail minimal odd lasso well contrast choice bridge extensive specification center multi hierarchy regime indicate subject assign treatment full exchangeable matrix prior treatment matrix respectively imply lead dependence notational ease gamma framework e l examine posterior ij p multivariate bivariate appeal may express combine kernel covariance hyperparameter conditional loop center conditional show inference datum exploit exact modern day machine
go indeed basic gamma payoff recently single stock insufficient explore payoff realize still trend development designing product acknowledge limit available information via payoff concept design simple asset financial product natural create exist argument mathematic research want last future unless appear unless belief market good probability payoff derivative similar third volatility product side payoff connect likelihood transform research payoff structure tool utility technical concept independent
action bold exist deterministic step say objective stage discount discount give expect select notice objective decompose vx successive state incorporate risk replace obtain objective objective analogously remark compare dynamic history allowed apply type risk due motivation markovian optimize risk history include behavioral economic measure originally value mapping monotonicity whenever translation moreover within economic cost reflect high axiom translation future viewed consider sure time axiom l homogeneou risk measure define finance neither especially mixed preference state win neither coherence without mcp first map without control chain investigate time
measurement make sequentially record record record value record break inverse scheme sequentially convenience scheme examine successive record q exponential study variable cdf characteristic percentile determine hour cycle pure determine
chernoff hoeffding light tail prove extend chernoff light tailed sequence sequence policy generality incur sequence easy exploitation incorrectly identify event easy rank correctly arrive exploration give horizon horizon technique partitioning epoch geometrically epoch logarithmic regret good need bind arm hoeffde logarithmic cardinality necessary identify reward available cardinality achieve logarithmic exploration see combine approach logarithmic large tailed heavy tailed chernoff true lemma chebyshev consider induction jensen performance heavy tailed exploration give consider see inspire
trace possible scenario current case framework specification undesirable behaviour framework logic social rule describe framework virtual automate reasoning consequence acceptable monitoring participant generate follow framework specification make crucial behaviour analogy engineering requirement describe see automate mechanism framework couple logic gap normally behaviour intend capture case case existence answer failure solve specification inductive suggestion improve specification secondly novel integrated logic semantic support purpose nature essential create domain demonstrate show iterative present introduce use
derive flexibility edge generalized operate joint among vector specify dependence attribute value network specify joint capture produce illustrate edge formulate represent feature distribution vector sum subgraph product
probability limit zero distribution statistic mapping illustrate behaviour limit amplitude limit behave one rule amplitude way zero quantile straight intercept true behaviour amplitude frequentist tend especially limit
take measurement sufficient lasso signal see show compute regardless scenario group configuration overlap yield length scenario apart last almost overlap apart element element common one list identical odd disjoint group overlap remain case argue become overlap ii overlap iv transform group account parent
case z pz z coincide multi recall x xy yy weight coincide acceptance multi obtain satisfie detailed guarantee target algorithm detailed I move simplicity generic satisfie detailed balance symmetric want generic inside integral describe section
two margin sequence respectively analogy moment central couple v df margin dependence copula use respect eq coefficient th moment copula moment bivariate bivariate page need reason likely use copula copula deal estimation bivariate parameter coefficient counterpart estimator three copula copula accordingly present estimation popular family copula copula copula correlation
dot summation cause storage avoid implementation locality many problem aim costly gradient address qp solve descent gradient obviously computational dominate suggest greatly benefit power demonstrate calculation iteration multiplication extract handle large often vary multiply parallelism adopt consideration demonstrate type multiplication prove solution multiplication adopt multiplication multiplication partitioned column multiplication partition partition summarize definition illustrate input partition group stage perform multiplication sum part summation result procedure often job responsible partitioning grouping intermediate detail ki step group individual sub partition schema often partition intermediate multiplication consider
hill covariate formally expression normality consequence propose adapt slope belong extend fulfil distribution address variance combine precisely follow result straightforward two define theorem entail permit asymptotic combine hill asymptotically estimator result direct convergence hold w tool minimize conditional hill provide asymptotically unbiased estimator continuous minimizing tw
balanced automatically create balanced fast function carlo integration arise carlo create variance unfortunately must good present create random mean random obtain output carefully analyze number b unbiased estimator multiply final error
national health research research centre health mh south foundation trust
rank behavior relax doubly long learn doubly doubly construct wise softmax projection know nonnegative row formally column hadamard division iterative normalization matrix convergence unique summary guarantee non negative doubly suffice reach matrix facilitate normalization incomplete term square whose constraint nonnegative normalize interestingly
fitting acknowledgement nsf dms give q l jj jj expression truncate two method sampling cumulative ex minus plus ex remark model wang south sc department flexible prior enjoy normal prior characterize sampler lead new approach basic autoregressive spatial structure appeal new improve predictive augmentation autoregressive estimation statistic large efficient advantage parsimonious often inherent high approach reference deviation flexible hierarchical prior conjugate prior typically metropolis hasting matrix identification zero zero inverse covariance undirecte attractive gain attention imply
simplify integrate part recursive obtain grateful suggestion work european european period series analyze anomalous model type stable similarity difference brownian model describe diffusion finance diffusion time period
corollary simulation wish empirical number entry per denote result section turning case stationary vector var process zero mean driving corollary miss var drive suppose vector row drive parameter let theorem corollary corruption deviation extension identical omit problem relate via lasso recently bound fully I row var devise corrupt remove exist vector I estimate additive observe additive noise pair j covariate covariate response pair obtain j j jj matrix cc project descent nonconvex case versus rescale missing covariate autoregressive represent trial column finite uniformly scale bound corollary technique hold
adaptation stick breaking process asymptotic parametric receive lot attention recent year recognize bayesian collection range seminal rescaled parametric discuss estimation mixture prior number normal remain univariate case surprising study dp high
example size existence path strong connect entropy spin birth size common underlie principle path connect entropy discard create contribution entropy interaction stop build recursion sum respectively ise grid bm huge mean message
dimension infinite principal approximate multiplication sum hence tail carry matrix augment thompson demonstrate technique adapt bernstein yield version tail prove construct simplify probabilistic sum prevent infinite arise
name normalization quality assessment assessment effectively preserve embedding second global overall assessment therefore natural task learning validate effectiveness manifold advance store set issue science understanding cause curse dimensionality complexity pca multidimensional lie linear embed ambient meaningful embedding manifold family method diffusion map dm tangent alignment variance unfold riemannian manifold great interest nature intuition successful detection recognition hyperspectral crucial natural learn embedding label method fully unsupervise freedom underlie dimensional unknown quality learn inspection intuitive qualitative quantitative dimension address assessment cast sample pairwise preserve neighborhood neighborhood
arm must idea availability reward bound primarily nonetheless ucb match binary among bandit family interest consider policy draw arm ucb arm ucb call optimistic act instant reward high possible value compatible past ucb call ucb latter ucb online horizon prove constant ucb variant needs tune tight regret bind arbitrary expense front proposition correctly ucb numerical ucb ucb tune
direction current stopping change nothing else analog variable properly direction duration device analog store basically consist set usually thin change device circuit typical circuit depict explicitly suitable locate decrease first vertical connect negative dropping across cause consequently passive element decrease similar increase state application time adjust suitable analog store goal propose fuzzy binary gate primary artificial purpose logical gate depict binary receive datum come via strength correspond efficiency neuron neuron thresholding write total set logic become logic two input e create neuron weight logic layer neuron create logic
marginal likelihood small divergence sense might indeed finite maximizer eventually theorem remark order originally motivate match rate however improve difference come ultimately subproblem version third regard complexity considerably space correctly specify converge identification would arguably close prior hierarchy propose prior incorporate effectively reflect unknown likely reasonable recommend consist binomial conditionally prior denote prior include able component say information first problem poisson quick comparison context example gaussian
functional category gene near original contain miss capability profile experiment replicate mixture component five five
value variable variable threshold cite quite small set major drawback deal index hypothesis free tuning model length whose set let set index distinguish framework order variable asymptotic hypothesis method order procedure procedure prove powerful procedure let subspace denote organize simulation suppose suppose focus hypothesis test subspace nan alternative collection subspace collection level level reject procedure collection subspace successively soon kx k
graph output dag condition l occurrence path extra illustrate equality establish present dag jx circle dag p dag variable function size restriction throughout correlation version section contain consistency weak correlation partial correlation nan ai denote version simply adapt decision conditionally role impose maximum skeleton satisfy correlation allow grow polynomial sample represent set pose maximum size oracle upper correlation tend avoid identifiability bind assumption outside range assumption condition similarity evident difference dag version unknown datum maximum denote step strong skeleton algorithm skeleton contain sep definition sequences sep grow linearly low study consider require additional tuning directly tune adaptively modification apply simulation estimation version significantly moderate feasible graph procedure dag size generate
involve criterion sample yield sample figure versus versus give simply increment increment run sample show detector present training create dissimilarity require number pareto dominate sort non dominate sort sort construct pareto comparison involve create need calculate involve find dominate test front binary front training front comparison bad anomaly threshold determine phase sort criterion experience requirement obstacle sort order pareto searching list dominate front criterion pareto front hence average occur pareto front attention canonical anti dominate sorting criterion pareto sort nonempty add front
predict target apply logit maxout perceptron maxout applicable binary baseline problem result task report give adaptive hyper parameter tune loss kernel kernel kernel kernel average kernel maximal experiment perform fold fold model optimize report fold svms train descent use rbms hybrid c average hybrid maxout mlp maxout logit maxout svm public drug annotation task time repeat fold
adjust expect rough estimate distribution beneficial adjustment computation adjust variance conservative credible rough importance g quickly determine reader comparative construct toy gold comparison understand property method mixture component dimensional evaluate b marginal two mixture combination component generation proceed independently probability specify support prior computation simulation close rejection follow adjustment marginal adjustment inference package abc mixture relationship observe summary panel b c rejection follow adjustment indicate dash quality kullback grey rejection adjustment panel illustrate bivariate contour indicate true rejection rejection follow regression adjustment panel addition
limit requirement execution important aspect implementation ideally format interpretation hasting rest discard enable post degree retain freedom considerable storage capacity capacity disk throughput limit sample write dominate execution second pattern average moment histogram accumulate simulation record storage
atomic hazard binary process underlie de ibp extend introduce negative ibp line choose select ibp customer control negative binomial general use may control count allow representation particularly simply contribute variable poisson enhance flexibility place gamma construct count make connection time appear contribution mark process count topic diverse corpora poisson commonly model place produce binomial gamma large thus
show alternate admm note new subject semi definite vast norm subgradient minimize substitute back condition lasso pool unchanged give pool average find optimality satisfied solution n path recall full solution far assume assignment henceforth backward approach
context know concern wiener set error procedure derive limit adaptive suggest average criterion restrictive available old mu achieve mat
mostly combination stage random population age capture use record thus implement method general census record head part methodology survey carry current detail briefly day seed seed datum collect report extensive degree sequence degree experiment obtain group panel panel population tree combination addition participant say preserve clarity combination panel panel eight panel nine evaluate circle
processor gb tb disk execute map make run serial worker ensure contain block mb vs default mb block size accuracy allow expense worker variant use bag instead distribute illustrate block block different line size bag instead block train run different measure accuracy large accuracy started yield result ensemble member core size size ccc size straightforward replacement number ensemble member evaluate evaluation may ensemble evaluation ensemble start thus per first subsampling random block datum accuracy use full code train size member training train block ensemble total time local member size ensemble vary point comparison good serial model total
large choose first section term finite except due transition initially assume know transition lie cl function regret know incorrect calculation become independent player increase regret transition except slowly linear horizon term slowly version prove regret horizon fp fp take threshold mix exploitation fp player pick pick arm arm end play optimally else arm play optimally play large function due suboptimal bound theorem give fp fp threshold tc horizon fp upper exploitation step select possibility know optimal imply either enough vanish gap belief fp belief correspond get bind clearly diameter regret number partition proportional tradeoff theorem balancing set set suboptimal balance tradeoff sublinear proportional player future research section inefficient approximately quantization relative iteration modify exploitation solve observation transition increase percent example solve transition exceed policy
education etc model equilibrium describe observation calibration epidemic importantly collect population equilibrium result follow likelihood present transform trajectory practice point spaced model age proportion percentage individual give number population person model year incidence collect new south validate apply date collection result health cross case gender age normalise roughly capture health person health incidence rate obtain gp observation spaced assume full activity available though parameter observation vector vector ode system st gender assume observe assume beta furthermore gender bold corresponding incidence category state vector incidence incidence gender code code decompose incidence apart would system model ensure point state least finer solver group gender model specify beta parameter matching obtain comprised model age explored perform bayesian estimator correspond mean mmse posterior mode explore resort draw paper procedure non ode adaptive monte history consider present present monte carlo combine posterior give particular introduce background essence construct trajectory ode epidemic
limit expert expert switch expert expert time course expert length define sequence summation abuse ti virtue direct forecaster expert track record loss issue return conclusion set dependence condition expert restriction sequence keep share turn eqs initially static always tw f tw share assign exponentially forecaster apply base expert sequence ti eqs comment
payoff value player opponent aim game interesting existence consistent calibrate name abstract basic structure central manuscript payoff subsequent establish geometrically characterize player usual payoff convex compact equilibrium scalar determine dependence simply organization summarize intuitively standard theorem fx scalar interpreted state player affect carry cf later consider thus minimax cf player may minimax remove game natural target project normal always appear choose orthogonal attempt force manuscript role play turn manuscript organize present background
run examine segmentation minimize chain method collection sequence mcmc compute empirical mean test sequence expect hamming empirical rely across compute distance sequence optimal sequence high aim provide address issue display skeleton depict learn contiguous segment box group behavior label behavior two category show common ar identify group dark various behavior appear movie four split green motion category raise attribute raise one exercise slightly counter motion subject three run motion category split also correspond body run subject perform subject splitting phenomenon gmm gmm initialize component pca primarily bp ar hmm hmms frame six movie hamming frame hmm ten mcmc demonstrate ar hmm difference display map gmm bp hmm component hmm state behavior namely hmm assume see strong band imply bp well specific portion bp note bp merge behavior hmm discover dynamical formulation prior subset binary time novel ibp utility bp focus switching approach equally
vx vx v x vx g vx vx thus first follow completeness conclusion final conditional projection completeness vx second hypothesis measurable positive definite g e mapping obviously cauchy inequality hc te hc w h also note cauchy inequality give h b g h w expectation g g let tf dc tf c tf tf cf hilbert schmidt mean square square g subtract hand multiply subtract additive separability c eq eq q part condition follow implie know I completeness part hypothese r hc iterate expectation hc g follow divide zero eq distribute imply chen global identification assume completeness rely
recursion demonstrate depend curvature determine bind follow constant stepsize sgd serial case fairly pay stepsize proportional counter result slow demonstrating eliminate complicated protocol decrease number run stepsize follow exponential standard precise real function constant iteration appendix discuss gradient less sufficient tell iteration give eliminate stepsize phase exponential converge shrink stepsize iterate ball suppose choice stepsize suffer much stepsize algorithm sufficient assume stepsize factor epoch epoch require previous epoch final q select run combine guarantee rearrange step protocol stepsize serial incremental eq put recursion
yield order plug find outperform counterpart first scenario isolate cluster isolate isolate heterogeneity sc covariate percentage pt p pt pt sc sc sc sc sc sc sc sc sc sc sc sc sc close percentage truncate spatial simulation page cb counterpart spatially structured finding rank cb triple exhibit performance closely cb plug mle bad classifier sc sc outperform mle classifier sc spatial albeit variability estimator sc discrepancy mle may sc sc plug find mle chapter sc weight scenario well htbp sd five plug present different spatial isolated isolated area highly spatial heterogeneity sc spatial structure sc well estimator p pt pt sc sc sc sc l sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc close percentage truncate close percentage htbp five estimator different level variability spatial isolate sc isolate isolate area sc structure spatial heterogeneity sc spatial structure generate hidden sf expect count percentage loss p sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc truncate second entry percentage close percentage heterogeneity systematically note variability ensemble estimation plug cb gr plug heterogeneity substantially increase dispersion effect increase heterogeneity simulation car car scenario indicate table page different plug expect benefit level function show level plug systematic trend nonetheless notable whereby high similarly cb plug although triple goal term percentage expect consider column latter scenario heterogeneity play substantial whether sf observe sf plug affected level expect count appear percentage estimator sf systematic mle notably count albeit heterogeneity chapter also next ptc pt p pt pt threshold number percentage truncate close small uk year surveillance publicly available classification health service medical practitioner probability contract evaluation use full retrieve uk health set cover four period thesis longitudinal classify accord year department health period day member plan department health rate day seven discard national disease compose statistical illustrate inherent change level mainly concern risk oppose trust year threshold expect optimal estimator scale factor p pt loss threshold c truncate close second digit percentage close small percentage observe case trust per day ny rate uk day trust thousand use assume dispersion cause disease variance gamma normal close datum fit cb triple vector specify discuss chapter plug choice third threshold scale classify identify substantially level choice high national also substantially low positive become solely classification affect posterior percentage sd sd threshold dash indicate red sd sd threshold dash indicate red particular poor sd sd threshold dash red htbp sd panel dash indicate red plug threshold remarkably yield great classified exception trend conservative ensemble indicate conservative classification particular gr find cb estimate show cb gr plug behave however small misclassifie area gr plug yield set plug estimator page aforementione across cb plug exhibit classifier behave plug mle classifier increase order posterior regret page show ensemble estimate consider trust panel figure page classification ensemble compare figure page plot distribution see general classify rr posterior mean visible since produce identical classification class
pose check radius language regular least break language string middle since piece say string may show arc rectangular regular suppose language language contradict rectangular trajectory b language contradict deal language rectangular free formally construction rectangular trajectory length sufficient represent language trajectory rectangular trajectory free string contain conclude free apply deal rectangular trajectory conclude modeling issue state reach state self self string production rule never terminate instability desirable derivation terminate finite criterion define matrix number production number constraint terminate derive mode syntactic classify target arc generate iteratively process parse parse top sec give syntactic discuss mode sec syntactic estimator version operation infer production string call stochastic parse context syntactic track sequence state state semantic tracking algorithm recursively sec string store define marker end partially probability incomplete illustration syntactic trajectory estimate string target syntactic produce trajectory syntactic build hypothesis detail provide specifically syntactic multiple mode
standard image optimization numerous optima evolutionary population iteratively select combine produce increasingly skeleton acyclic
norm formulation norm motivate problem section measure data image operator lead write denote wavelet regularization tradeoff minimizing serve lasso penalty wavelet coefficient reality pattern plausible commonly small wavelet scale
sound conceptual justification neither frequentist problem justification experimental evidence idea probabilistic role play inductive inference variety laplace borel kolmogorov build apart selection mathematical implementation example abstract relational might bayes updating etc borel kolmogorov notion probabilistic model build theoretic equip reason frequentist claim probability frequency update probability update fact first lead abstract boolean allow semi measure ever theorem algebra
satisfy moreover chain explore surface also surface automatically incorporate toy experiment dark light dash influence evident density prior wrong location indicate dash degenerate plot peak near wrong seem lie peak point lie hyper small value hyper surface though hyper surface figure happen multiplication likelihood sharp contrast observe count strength whereas constrain multiple degeneracy expect c integrate observe noted purpose however happen wrong show relatively constraint merely able albeit integrate wrong method
abuse terminology dependence clear goal sharp bound discuss relatively quantity picture special operator functional study q square quantity additional factor involve control map correspond sample sample theory relate consequence example useful general recover noisy observation square unit hilbert convex fairly derive translate upper norm example subset f j l mm useful pack remainder begin background set
integrate context correspond make hyperparameter hyperparameter iterate keep respect keep fix step become course maximize find em well provide expression challenge need take explicit system numerically operation triangular case association cluster hand trajectory task location distribution analytically express gaussian approximate posterior mix probability hyperparameter typically input correspondence equation belong current sometimes
variation hard specify change display location surface prior height height height height
hold consider cover ball clear radius volume easy ball center cover manifold ambient ap ball every point bind invoke lemma project onto tangent action clear manifold onto ball invertible distance point derive point use lower get c consequence bind decrease noiseless proceed dimensional constant rapidly rapidly grow invoke hold manifold desire large invoke homology start result tell kernel positive volume appropriately clear lemma conclude around use clean point denote remain estimator analyze eliminate eliminate since within contradiction keep nb
krige poor choice piecewise approximation true mean bad another choose directly high well variant right hand three stationarity evident advantage fields define expand weight change interestingly replace incorporate dependence general drift concept behind markovian transfer care need take advantage stationarity underlie well ts describe direction random diffusion stationarity define locally flat field manifold still
asymptotic normality semi parametric rao equation form presentation x j simplify explicit q j dx dx f moreover note equation case asymptotic efficiency new type estimator functional choose taylor expansion dy behavior
particle least parameter propose particle ratio jump number go adaptive approximate abc line tolerance tolerance generate compute weight sample decrease reach feature see generate particle twice particle combine constitute ns algorithm stop result retain particle step heavily equal quantile simplicity case avoid particle suppose negligible compare algorithm particle x kernel yield
survey interaction depend fit marginal adaptive penalty equality penalization vector penalty form parameter estimate zero parameter zero need separately concave differentiable coordinate wise coordinate ascent cycle take minimize approach sub section local perform computationally
article seek iteratively approximate strongly optimization smooth domain accurately point unbiased independent estimator subgradient independent identically scale measurable scale dimensionality oracle
record sufficiently dim distant enough neuron center prototype total neuron neuron neuron densely rf size neuron densely extraction neuron densely present distant stimulus neuron map collect stimulus visual stimulus finding match vertical horizontal map contain correspond match response neuron contain fig whose visual fast neuron response neuron exclude neuron spike stimulus spike extract eliminate feature neural activation map feature match actual stimulus stimulus fig white color indicate reach spike reach spike
remark spectral sum independent matrix surely equal ni ni mt q plugging get like bind random regard prediction result excess expectation rademacher theorems part rather lipschitz function sample derive derive depend argument definition remark institute consider approximately reconstruct partially approximately receive much attention surrogate reconstruction trace max present reconstruction rademacher unit publish specialize approximately reconstruct unknown rank interest reconstruct partially observe noisy quality observation search error np work focus trace nuclear norm surrogate rank trace sum norm value singular quality reconstruction largely drive compressed sensing condition empirical
multi server column suppose help one histogram evident join histogram experiment prediction within prediction histogram real figure deviation paragraph histogram fine grain big wrong histogram join histogram predict hundred tuple output tuple observe deviation prediction method small join operation often contain tuple tuple prediction accurate I contain tuple query database soon behave percentage prediction correct make join add column thm evident guarantee spirit conclusion al online query properly database demonstrate use significant improve histogram join advantage within predefine range collect pre capture collect histogram join multidimensional multidimensional histogram exponential representation join operation join database exponential interesting highly
without give leading ij kernel f evaluate chart spectrum lead eigenvector kernel simply kernel snr feature zero center commonly kernel polynomial kernel radial basis heavy rbf rbf paper obtain eigenvector take basis match hyperspectral background interference spectrum modify match sense interference linear subspace lie hypothesis
belong depend evidence probability estimate accurately bm illustrate subset effectiveness cut false alarm include retrieve high collection belong subset indistinguishable fact optimally rank way index share value decision subset retrieve disjoint subset subset occur document term orthonormal either represent respectively term clarity weight depict vector relevance relevance non relevance relevance vector relevance occurrence belong common basis space vector random collection vector vector correspondence
show satisfied solution bind eigenvalue condition nn concentration get nc min immediate consequence construct dual pair r pc projection notice ratio triangle distribution probability kn result zero variable concentration bs state notice ds assumption sign j go provide assumption lemma eq q union bind theorem say magnitude large magnitude magnitude theorem hold trivially corollary constant specify equal concentration optimality get random concentration vanish exponentially fast success recall dual variable satisfie satisfied hence rest direct consequence prove lemma exist contradiction variable equality orthogonality projection inequality
condition compressive pursuit bp mod briefly early recovery support denote sec mod try outside among solution enough may recursive recursive compressive cs compression cs instant bp practical reconstruct measurement bp actually linear q rip geometry recovery bp work sparse reconstruction partly knowledge call compressive sensing obtain condition mod show weak mod cs try e solve refer mod mod cs introduce retain also parallel support solve threshold
condition variance q ordinary least error triangle differ design apply ordinary require dimension covariate slightly direct argument simple characterize excess ordinary hold show fixed bind bind lemma appear similarly ny come therefore analysis estimator expectation bias triangle paper fix pick condition q various aspect simplify ignore treat overall relatively mild essentially effect isolate simplify view nn second noise unique covariate suppose
intuitively infinite see note prove recovery adapt crucial ingredient nuclear mb easy suppose choose necessarily satisfy hold easy obey well mr selector lasso choice q slight modification bind thm proposition reconstruct unknown application quantum analogue sense recover sparse fouri almost isometry rip recover norm nearly optimal class orthonormal obtain entropy low reconstruct machine filter netflix remarkably turn choice matrix uniquely reconstruct nuclear element case measurement suffice provide incoherence
profile usually describe game pool multiple user utilize resource propose convergence fair outcome common relevant resource allocation control wireless communication player specialized game involve profile simpler applicable almost yet specify closely present different require game specify profile organize define presents game namely present multiple outcome also demonstrate formation game establishe establish finally conclude remark finite action profile product player preference induce utility complementary pure nash equilibria nash nash always refer pure nash equilibrium first action profile value game satisfied payoff dominate j n form interest large accord one game satisfy payoff difference player profile difference establish
extract perform contrast panel superior second method unlabele classify al unlabeled randomly unlabele example supervise set average repetition test rate prediction result suggest improve prediction functional supervise functional basis
algebraic statistic design collect appendix mention consequence presentation ideal polynomial finite symbolic without medium sized table strictly vanish binomial vanish log odd versa odd algebra output computation emphasize usefulness definition eq therefore obtain normalization define integer call markov goodness geometry basis goodness independence represent q quasi independence encode except represent force diagonal probability typically use notice issue outlier model zero view detect mention
speedup htp figure effect start segment red line line generate uniformly thin plot leave right htp origin classifier training vector class cast q l detail reader reference survey advance htp svm logistic setup continuous remark l svm htp svm jx tx jx able gets update ii derivative perform dataset instance selection list testing ta htp ta dependence ta iteration find datum half second let remark scale htp use train approximately second l loss lemma remark paper method minimize nonsmooth prove linearly nesterov efficiency smooth composite remove importantly contrast author achieve
regularity positive detail autocorrelation slowly alternatively assign experience function degree choose mle feature suggest regard e short xt carry interpretation notion loss obtain different example let p newton method many goodness context even theoretical clearly distance difference shift sample efficacy parameter obviously number assume ahead model separately pt misspecification integrate shift stationary memory example aic figure observation close strong close well matching interesting well comparison matching size dot solid attract considerable admissible distribute stationary identically aic still lead line solid dash kalman pt kalman utilize likelihood method em dot figure kalman filter unstable sample aic kalman produce outside range model truncate lie replicate simulation first estimation true realization panel correspond result panel panel observe pt pt demonstrate substantial figure pa time perfectly specify parameter
regularization tuning view use parameter selection degree freedom allow criterion evaluate regularization variety penalty carlo simulation situation datum degree freedom fundamentally important dimensional traditional procedure information criterion unstable inherent drawback impose simultaneous model considerable lasso penalization regression elastic net penalty minimax elastic net solution close present efficient entire solution modeling identify assign include criterion cross validation freedom g directly sparse analytical stein specific show degree freedom degree fuse base differential parametrization enable freedom minimax procedure via bootstrap approach unstable estimate present iteratively degree extend wide net furthermore obtain degree algorithm perform remainder briefly describe regression iteratively compute degree generalize path section artificial dataset conclude
play role life study ise weighted link information log graphical field distribution lagrange multiplier ratio interpret log odd ratio conditional correlation despite similarity exponential relate moment difficult compute marginal family repeat marginalization conditional let triangular matrix conditional third family regression family marginalization assess logistic original may da dd
say subset coupling fail happen fail coupling note variation uniform uniformly variation fx always couple w imply give rough description strategy describe coupling evolve couple markovian couple sup record specify coupling show keep occur consist difficulty partition showing remain worth point dependence coupling get correct analogous markovian walk couple essentially modification gibbs
inductive favor multiplicative research question transformation ability suggest robustness pre energy would transform dimensional fundamental many vision task include role multiplicative play infer detect rotation share multiplicative code method light review operation perform cell implement multiplicative interaction suggest utility arguably primitive tracking view scene invariant description action recognition relationship single carry contour consider correspondence namely area pixel go depict almost understand movie understanding decoding frame correspondence task keep mind build vision system lot progress building task reason help eliminate recognition consistent feature plausible feature overcome engineering domain degree place construct module raise whether amenable may open road perhaps beyond vision paper recent task introduce mapping mapping illustration unit interact
boolean learnable positive unary free positive example learn represent query xx xx xx xx let l l r subtree root return filter construct serve filter subtree root node select part select invariant one choice library n library book attempt bottom two common root return path title move author title expression presence reasonable schema algorithm schema line least desire completeness completeness choose free unary learnable positive e set symbol positive learning setting exist consistent sound need query follow decision positive consistency query consistency np outline hardness consistency use minimal example indicate symbol yshift cm xshift yshift xshift c yshift cm xshift yshift cm edge edge yshift xshift cm edge yshift xshift cm yshift cm xshift f edge yshift cm yshift xshift yshift xshift xshift edge xshift yshift cm xshift yshift xshift c yshift xshift edge yshift xshift f cm xshift edge edge x xshift edge edge yshift edge xshift f xshift cm c yshift yshift
e word ultimately scalar structure retrieval implementation answer implicit good ask another question lemma example value relevance optimize estimate key subset recall subset recall result prove retrieval ir system decide relevance relevance subject measure end define research ir describe axiom perhaps crucial ir task ir reach good thank classical unit top probability operator classical hypothesis express quantum investigation phenomenon
I bernstein inequality state sequence let martingale difference sequence I entirely together
finding treatment portion apply practically area roc curve assess intermediate auc roc good easier determine range several among thorough roc find extended status disease death represent diagnostic marker study disease classify subject generalize time wang nonparametric estimators develop inference procedure process correspond facilitate dependent summary organize follow section
diagonal correspond hermitian trace pure course example occurrence eigenvalue assign certain event latter event space algebraic represent matrix event rule trace q compute allocate may odd diagonal sum assign state prove every space dimension great space probability vector basically probability calculate trace call use superposition suppose event occurrence two probability multiply occur multiplied occur law interference range usefulness mutually exclusive describe interference violate kolmogorov axiom admit space general store acquire eliminate precise exhaustive nevertheless former amount set hypothesis decide topic must relate
acoustic source problem minimize correspond observation create measurement receiver location solve equation differential regularization typically achieve nonconvex figure measure especially dimension accomplish strongly increase pass hybrid outperform deterministic good nearly hybrid perform although show pass practice involve prohibitive make quick progress solve good focus inexact gradient method optimization objective inexact nesterov case convexity assumption optimal inexact bound author notably error encourage achievable might analogous proximal convex non composite noise subsequently convergence rate bound certain individual concentrate us concentration hold high select element bind individual far refine
non show event topological respect topology infinite equivalence topology topology except hausdorff unless anti separate every also vice anti topology reduce discrete close open open topology class every every topological union class closure give union intersect aa desire establish field subset monotonicity aa aa constructive natural obviously every observe additional completeness order every supremum full full case prove make completion element satisfie show similar result follow natural additive component recall possibly family number outcome subset super additive bb bc write component component taking hold arrive conjugacy note box alone additive full box check extension low envelope mass moreover already event explain proposition
h h maximize assume v value namely form namely eq life span maximize gain remain step time agent fix life span gain maximize recursive clear reinforcement horizon type worth point behave differently reinforcement signal assume signal together meaningful plain reward sum theoretic meaningful namely gain reward planning ahead immediate expectation grow
symmetric possibly capture penalty transformation requirement shift transform use characterize denote set positive piecewise example huber notation definition inside inside maximize huber obtain take huber representation take need ensure call turn ensure penalty def subject denote integrable measure
similarly constant less exponential go histogram convergence early density estimator estimator attain argue metric study far aware target analogous rate asymptotic target order vector value stationary prove sequence asymptotic histogram yu doubly asymptotic histogram iid
plot slightly redundancy total information decrease increase relate decomposition note redundancy give position fig near find group redundant simultaneously interaction obviously set vary bin use illustration discuss herein address system change simple interaction focus interaction boolean gate beyond variable individually relationship decomposition show magnitude indicate presence node indicate redundancy develop interaction contrast interaction show sum word correlation interaction variable feature apparent total varied approximately linearly regard whereas measure variable neural mutual information development correlation total varied approximately connection dual highlight gate correlation variable gate evaluate interaction know variable compare act correlation
flow far boost efficiency present lattice continuous transition calculate autocorrelation several autocorrelation variance estimator consider autocorrelation bin size fig comparison conventional autocorrelation become metropolis heat
explain yield latent pose ground truth frame yield square iterative standard general small aspect iterative self regularity impose proportion satisfactory describe residual datum give covariance novel difference dimensional variant cca include characterize rank powerful
preserve release highlight technique thresholded predicates polynomial threshold learn base utilize view privacy analysis sensitive extract individual individual record offline interactive setting query mapping answer release extract approximate answer query medical analyze release grow explore show permit surprisingly least dimensionality central private develop hardness release efficient tool datum task predicate derive release answer challenge access access get oracle query answer theory give access query release face apparent access impose release release ensure privacy many efficient run another often sub size oracle query description work explore learn private release release relate task obtain differentially release algorithm briefly way differentially decade aim back work similarity work insight future view connection technique release set explicit modular reduction release conceptually arise release include dependence database release aim conjunction contingency query study differential literature take universe attribute count item database attribute answer past tailor
concentration discount time illustrate parameter long vs iteration exhibit autocorrelation figure three parameter computational reason accordance methodology concern nonetheless seem explore discount behavior power expect expect concentration instead number model roughly achieve seem concentration three top nine feature way three look learn collect last iteration express iteration nine express distinguish digits version competitive reach burn iteration run per iteration average iteration stick break stick break essentially complete stick break chinese restaurant motivate three stick break type law discover process power law useful discussion helpful suggestion experimental national fellowship knowledge science foundation award stochastic carlo many conditional form require notable step discount integration discount
instead return efficiently build require subspace improper checking elimination probability improper previous chernoff bind overall collect conjecture inverse time example bound demonstrate runtime pair preferable hypothesis write therefore perfectly erm time agnostic minimize mistake obtain learn preference idea hypothesis class addition complexity solve erm overall erm erm predictor predictor map
take decay adjoint map embed hilbert rkhs eigen decay relate interact give rise functional number versa costly function various author lie ball span sample together theoretical pair sharp rate track asymptotic accounting introduce smoothness use among principal recent devote noisy consider via approximation derive rate estimation certain paper trade lda approach inspire special observe oppose aware pca working framework per optimal eigenfunction analyzed sample al derive rank couple scale invariance combine extra weighting also estimator eigenfunction lie rkh different interesting question whether work emphasis sample
bind albeit word despite error end inference suppose expectation moment matching error break carlo leverage efficient although already suggest effectiveness bayesian reduce solve system q perturbation diagonal conjugate matrix product avoid costly cholesky factorization simulation length vector store g matlab also note conjugate gradient employ estimation use marginally chi freedom imply unlike drop reach estimate sufficiently accurate even expectation propagation work
question group panel participant participant group panel consideration call effectively cope thin sample estimation adapt possible function estimator attain infimum choose effect certain risk hellinger kullback risk naturally hilbert turn derivation refer work risk density dominate divergence dominate risk translate square total variation kullback leibler application binary naturally slow prohibitive covariate yet decay slow orthogonal series achieve near decay estimation computationally instance fast hadamard estimate basis operation appendix therein orthogonal recursive particular entail estimate basis coefficient compute stage decide remain likely significant allocate high significant spirit originally
strategy equilibrium anonymous sized mixed strategy nash anonymous strategy player base insight anonymous game sum bernoulli first second vanish exponentially player iterate involve dynamic implement nash equilibria sum ess weak slow analysis approximate equilibria existence efficiently computable cover variation player play player player player set pure player payoff player payoff player literature equilibrium normalize mixed pure strategy choose player iff among vector nash x strong notion support nash simply nash anonymous sense dual player play strategy play utility player player way play player play
likely regardless network direct tie offset odd capture intuition limit degree poisson tend limit additional offset asymptotically analogue n appendix theorem effective interval argument develop primarily establish insight question effective focus asymptotically distribution derive little foundation formal illustration interval interval estimator v examine desire study e v word mean expect number actor level substantial natural model desire value simulate evaluating individually coverage sample interior convex realization mle mle mle frequentist report
propose greedy construct nash grouping since action nash equilibrium another nash customer action customer customer choose action choose group customer greedy table customer customer since customer customer choose accord next prove contradiction inequality equilibrium grouping suppose equilibrium equilibrium grouping contradict equilibrium grouping affect customer request find customer nash chinese pre customer sequential chinese restaurant queue outside restaurant generality rest customer know customer customer know decision customer current customer table restaurant notice grouping customer customer eventually end number customer predict decision equilibria chinese restaurant game equilibrium chinese game begin formal chinese begin customer group customer nash equilibrium perfect nash nash equilibrium refine nash equilibria perfect equilibrium chinese restaurant game construct equilibrium grouping function customer equilibrium grouping notice equal equilibrium grouping generate remove expected customer grouping procedure implement j choice customer
prove semidefinite idea ex mm theorem lemma definition base active unbiased estimator hypothesis give weighted forecaster exploit recent spectra random hypothesis active learner active empirical access sampling oracle provide predict unseen cost obtain quite task speech speech need fairly extraction label datum also good unseen label oracle unlabele active algorithms oracle learn provably broadly active membership query algorithm stream pool query query label might necessarily
converse consider sense give increase probability converse fig scaling possible recall random model fix measurement tuple satisfie noise sufficient condition reduce match converse bound technique admit critical minimum numerically see fig critical small converse converse achievable increase probability converse scaling respectively increase number phenomenon noise per hence cf degradation depict fig setting converse analog however hamming choose say program decoder need decoder free constant choose decoder decode converse disadvantage converse corollary previous section draw sense analogous product sparse set restrict channel model noiseless sense precisely pmf unity derive reliable challenge long informative rest dependence explicit ease exposition decoder reliable weak fix measurement appendix utilize split albeit control allow success seem plausible min decoder see surprisingly require sense draw match theoretic analyze set work study distance proposition firstly decoder event tight case speed theory large deviation large deviation speed n bind speed open secondly small devoted code theoretic interpretation understand geometry optimization correspondence minimization rank decode form code element whose rank exist whose identical vector assignment interpret rank
improve particle mass curve proportion close optimisation bring histogram final weight concentrate around unity mean large size proposal adjustment particle practically adjustment make improvement compare evolution several magnitude step smoothly impact look particle weight prior kernel challenge sequence value ensure prior set state equation provide concern invariant rotation shape kernel figure likelihood em em adapt initial iteration algorithm constant visually particle updating filter nk nn draw particle distribution spread particle negligible particle weight proportion leave carry mass adaptation highly adjustment weight particle specific adaptation use display whose choose constant result
decade address directly application obtain ranking overview well aggregation ranking gene statistic position position ranking aggregation extract list chain assessment stability ranking compare g order list gene brief list suitably triple multivariate x call permutation permutation mean false exchangeability definition exchangeability much exchangeability weakly exchangeable permutation e absolutely continuous exchangeable weakly implication exchangeability give exchangeability variation iff turn space algebra consist weakly weak define hausdorff q variable give clearly ed ed max x x mainly exchangeability exchangeability random visually plot plot weakly exchangeable overlap exchangeability exchangeable pair panel provide terminology exchangeability define exchangeability universal set gene consist patient rank test ranking position gene exchangeable weakly without subsample ranking
compact infimum attain k u I I I correspondence c since show I correspondence apply continuity pz singleton unique uniquely unique neighborhood correspondence notation er h uniqueness continuity u denote column row g g result j j g g g show inclusion unique case statement unique imply neighborhood h lemma product l h cf show hypothesis suggest support recovery consider strictly w w w kkt contain decomposition support sequence converge contradiction assume extract subsequence subsequence assume n contradiction support however overlap dual take form q l point assume reduce simplify w substitution eq role form w yield unit ball face unique write q one reduce situation active analysis obtain group admit optimal use duality complementary equation share yield add side expression derivation uniqueness incidence q invertible span positive unique pseudo latter necessarily remove consider situation decomposition necessary sufficient support lemma incidence support row index whose index element uniqueness uniqueness notice contribute objective convex problem admit kernel consequence solution kkt condition sufficient ensure uniqueness objective strictly row conclude sup paris france berkeley university california berkeley usa centre
accuracy additive experiment standard toolbox author website construct gp th interpretability flexibility full experiment intermediate contribute relate particular degree quantify et previously show learn call learn set use sum number x pre weighting search also force hull neither difficulty hard contrast optimize
extend multivariate classical univariate indeed replace u cumulative reduce multivariate without normalize fix limiting assume vector function infinity chi degree segment start simple maximize boundary
label sample label sample belong separate row constraint minimization global subproblem obtain hard svd thresholde global solution first singular zero solve subproblem subproblem variable prove keep minimization subproblem decomposition subproblem round round optimality global yield therefore decomposition error keep decrease round
successive path index modulus modulus path eq q cascade modulus convolution pixel yield coefficient convolution modulus intermediate computation computational complex wavelet energy decay numerically depth obtain iterate wavelet satisfy signal translate modulus become translation
implementation discrepancy fit simple kernel estimation choose extreme quantile discrepancy principle quantile combine kolmogorov distance mode suggest distribution lie since almost surely prove bind fast behave although become estimation sufficiently kolmogorov detect kolmogorov leading principle goodness form result large threshold law iterate logarithm analogue slight essentially pp detail sn lower go fast criterion choose
sde appendix predictor loss adaptation essential cost section key lemma base trait evolve dynamic describe initial trait value root equal adaptation optimum pair trait transpose eqs bivariate employ argument q trait reader treatment consider involve eqs standard regression theorem present establishe
truncate select rank maximize capacity thereby reliably subset like geometry difficult challenge comparative order establish sparse acknowledgment partially center fp project htb several mutual simple create quantization error
distribution spatial result average respectively minimal study introduce heterogeneity keep interestingly network size spatial network conjecture ref network process exponent far due heterogeneity aspect network enhance reveal master analysis quantify laplacian strong link
domain link important conjecture discover attribute relational static largely ignore utility incorporate relational relational attribute value might g research author secondly might activate throughout person email additionally change time predict anomaly prediction attribute set relational change link depend predict decay link window temporal representation temporal classifier temporal dynamic link present datum social biological work select relational increase temporal relational temporal ensemble transform leverage constraint finally explore discover temporal pattern scalability temporal representation ensemble mine temporal static temporal assume strict temporal ignore relational propose flexible capture link moreover evaluate static work discover pattern attribute centrality network temporal link weight past temporal window link node form
player address spc game static behavior spc software access strategy gain know article present incomplete software game spc connections necessary equilibrium keyword nash issue mobile wireless solution power business motivate localize extend network area macro network weak cell access datum part member member share start member share member share share member member make associate lead exchange technology moment share mobile sharing mobile mobile access spc sharing problem
hope line hard method begin discussion mixture gaussian dp mixture arise coefficient covariance bayesian set observation initialize alternate step value follow ic j ic assign k function datum datum attempt minimize popular simply cluster minimum index centroid updating quite precise connection equal straightforward cluster equivalence explore differently ultimately obtain apply nonparametric briefly equivalently choose generate observation cluster cluster discrete arise place dirichlet covariance dp vector mixture utilize state
require shape density level indicator clear provable require compute principle proper prediction agree well first sample prediction distribution shape crucially pre rich intensive organize construction combine kernel density discuss practical choosing bandwidth present possible technical give construct method measure agreement multivariate depth see test construct sample distribution argument random exchangeable let yy exchangeability closely relate level prove augment could sample compute inverting
formally evolve walk graph input seed either stay node evolve hold seed random dynamic thus interested graph seed random set brief discussion global spectral arise generalized determinant matrix norm conversely solution sdp run procedure notably sdp nontrivial heat walk see regularize eigenvector intuitively act penalty ridge regression intuition precise mind context predictor pair ridge minimize
differentially output abstraction show automatically efficient release mechanism interactive mechanism maintain give sense datum sequence sense distinguish may set formally capture database database database tuple property construction iterative database every tt imply database construction database exist sequence differentially private release query exist query release interactive time adaptively efficient private interactive first multiplicative structure sparse multiplicative compose write sx sx si hx hx counter add element add element cause structure present sparse work universe run multiplicative update variable delay universe necessary simply multiplicative cardinality universe depend carry
detail h analysis remarkably expect involve probability need double summation form routine numerous database hash estimating similarity equivalently practice store small bit hash care g involve pairwise care
permutation define obtain accord cell cn n sn ct sn common increase sn si n sn si sn si si jj minimum induce table integer demand integer counterpart real bound calculate equal procedure approximation give bound seem perform table generate count count precede fix define sample define bound induce define determine employ iteration importance sis determination sequential adjustment cell appear early table et currently ss li otherwise n u n sl n u I sn l sn sf return combination integer properly recognize chen call interval gap good knowledge tool tool integer discrete
removal along simply projective characterize implication exchangeable action permutation exchangeability consistency example infinitely exchangeable exchangeable uniquely characterize dimensional projection follow cd induce cd simplify infinite exchangeability measure poisson variable ignore realization random x cover x straightforward
integer nx h h sd nx self operator sd nx h nx h h h h arbitrarily section start denote weak brownian operator motion converge weakly consequence show converge limit natural sake write condition satisfied prove algorithm preserve sd nx sd triangle prove weakly motion show complete mala algorithm hilbert suitably scale markov infinite dimension follow show mala approximation mala probability reach target target motivated differ study diffusion rwm work hybrid hmc explore invariant case extend challenging start stationarity ode combine hierarchical herein challenge direct applicability measure exist rwm mala acceptance degenerate explicitly rwm study ask metropolis
build first order numerical method match mm corollary section supervise optimization fit penalty term regularizer encourage r matrix vector machine processing range composition prescribe regularizer induce mixed identity matrix choice give rise kind prove composition e fuse well norm multi many cost per iteration descent acceleration number reach value essential method certain
constitute aim concern whether get weight system partially observable thus effort achieve htbp meta initial set reference control action function update newly map q parameterize dimensional discrete step iteration illustrative property visualization purpose limit cycle htp controller know simplify problem significantly relationship controller action take function constitute reference signal start action state time point trajectory logistic map depict figure act limit explore map standard value htp htp htp analysis behave
I mat since lattice site intensity neighborhood structure assign site index neighbor rectangular grid give neighborhood rectangular mat generate rectangular grid consist column properly mat neighbor list corner cn cn neighbor ci ci cn interior neighbor j ci neighborhood corner c neighbor cn cn n ci neighbor ci I neighbor cn neighbor cn ci neighborhood triangular mat list corner neighbor cn cn
close hope htbp diagram bound red green curve envelope source form iid vertical red line would hope fit model close except end interval smooth concave another concave function pattern might mark line symbol iid symbol algorithm repeat graph particular represent algorithm partition concatenation transition symbol fail essentially uniformity skewness graph htbp mark source generate draw markov chain iid show stationary estimator detect figure
ten well heuristic performance increase node almost achieve number mm mm joint scheduling power problem wireless multiple single present scheme interference achieve wireless essential computation currently wireless world hoc wireless wireless employ wireless area wireless pose recent convert careful communication scheduling
datum use miss build routine frequency initially complete part separate four mis x likewise guess miss imputation sort amount variable random forest response criterion meet algorithm representation vector sort index store previously mis mis stopping meet newly datum present categorical assess normalise root use notation proportion around observe part variable get meet imputation
restriction prediction value analytically integrate volume see value goodness could cause lead inaccurate use technique fold cross validation exclusive slightly set contain sample create set hold sample hold time function cross choose adopt stacking hold eq use hold variance value finally one greatly calculate prediction hold alone low quality evaluate goodness represent monte give fold
cost cost reveal regret total add loss incur offline cost portfolio management regret incur cost incur solution use differentially private want provide sub convert online private sub regret two popular namely ascent fairly strongly framework differentiable regret differentially private online privacy preserving algorithm private bound offline differentially large learn world critical issue several ad hoc notion stand break netflix challenge publicly two instrumental discard hoc relatively sophisticated notion diversity attack pursuit theoretically sound notion inspire definition notion accept standard privacy notion privacy year community several interesting problem concern among interest offline program interestingly handle handle well error mention
evolve order column field formulation evolve discrete dynamic temporal process walk paper nonparametric multivariate covariance covariance readily specification dependent loading sparse gaussian function quadratic dictionary element methodology advantage employ collection cope rely another fact prior model handle presence limit observation cope computation update finally prior covariance addition theoretical google present value discrete tractable categorical letting goal good setting accurate prior collect subject analysis section space represent aim building nonparametric covariance dependent limited loading build loading index factor predictor despite latent loading challenge large far element coefficient comprise dimensional th column characterize predictor location combination tractable formulation factor induce decomposition function increase latent predictor factor ensure constrain loading triangular however task decomposition unique interest characterize decompose dimensional finite
jensen divergence hellinger distance also yield show enyi entropies shannon cross entropy formula whenever enyi entropy multivariate calculus neighbor although applicable kind intensive enyi close exponential available close
collect sensor subject design learn adversarial fashion provably optimize objective exhibit generalization publicly corruption encouraging indicate efficacy superior baseline method miss imputation fill well imputation work address issue local feature batch setting theoretical online useful corruption problem corruption well component wise product give give corrupted subsection batch type hypothesis arbitrarily reveal learner ask simply distinction fix weight making assume predict hinge see address change corruption give corrupted corruption predictor make provide implicit depend consist hypothesis function
bound incorporate fr e rewrite correspond ij ji symmetric symmetric establish ranking strongly differ establish relation exhibit relation set adopt joint feature mapping explicitly outperform main structured object string pairwise kernel paper pairwise compatibility bioinformatics various kronecker kernel discuss article utilize within naturally set run regularize square kernel apart mention solution predict dyadic setting movie argue specific well exploit relationship space object simply train dataset multivariate structure pairwise kernel basis dyadic pattern match relation intuitive near learn protein like r contrary consider learn nonetheless course prominent place special kernel relation enforce end least
irrelevant lead instead row fact optimize relate variant function discuss connection material algorithm objective symmetric two auxiliary upper introduce ard nmf half prior regularizer behind regularization term fact easily auxiliary objective proceed auxiliary j close first monotonically lemma rule ard regularizer take regularizer iff auxiliary recalling update term denominator tolerance vector fix value calculate kn case regularization auxiliary mean q replace table upper new function summarize algorithm update implement regularizer loose convergence result show bound original function g choose solve cubic equation rational derivation ard counterpart regularizer reader sum ard ard ard w input data hyperparameter nonnegative relevance nonnegative tolerance calculate hyperparameter
noisy weight play role network adaptive quantization error reveal distinct square adapt diffusion adaptation exchange collaborative strategy enable real incremental strategy cyclic cover node generally enforce hamiltonian cycle cyclic robust failure incremental network adaptive diffusion originally extend encounter also online consensus agent main emphasis role adaptation network original quantization study degradation particular perturbation incremental strategy extend square analysis stand alone counterpart explain useful along link exchange estimate work account algorithmic
build assign tn j ii unnormalized normalize instance commonly asymmetric walk normalize laplacian govern normalize unnormalized walk normalize correspond diagonal graph normalization closely connect limit graph primarily limit typically fine local proper without build provide theoretical harmonic semi label point aspect study weighted laplacian paper exception manifold boundary space specifically series involve normal reformulate implication laplacian believe popularity machine boundary necessary main laplacian explicit also interior manifold example
novel size discussion corrupt algorithm subspace solver augment lagrangian gradient subspace state main adaptive tracking solver least derivation admm extract residual dual rotation track vary behavior subspace change true new random figure successfully identify change track h ratio corrupt inexact multipli write want comparison rank previous entry matrix select value maximum oppose bernoulli challenge outlier outlier outlier noise perturbation perturbation admm little reasonable computational outlier fraction second second subsampling second sec e e sec e sec sec e sec sec sec sec
hold refer perturbation method histogram concern subset make hence concrete motivate relaxed privacy differentially private term statistic risk eq private randomize risk function achievable hardness describe differentially private mechanism use derive minimax hypercube neighbor hypercube namely differ eq q neighbor upper e obey kullback bind affinity take expression differentially
enable posterior various far datum analyse radial make survey simple model numerical integration epoch integral integral function uncertainty deviation consider reliable see aic biased marginal however biased converge estimate extremely logarithmic grey converge density aic uncertainty estimate aic however estimate reliable make posterior much investigate accuracy combine epoch one clearly close omit fig receive combined estimate grey blue
maximization expect value log maximize aspect highly user lead monotonically increase disadvantage disadvantage avoid quasi acceleration yield second constant expectation complete therefore hierarchical conditional f characterize require likelihood function moment numerically care naturally sparsity find handle start remove get prior mean work practice disadvantage enter perturbation assess whether enter optimization
discard large residual unnecessary keep residual lipschitz continuous discard residual time median correspond criterion ol objective aim indicate median absolute parameter control removal residual linear interpolation coincide criterion everywhere huber affect discuss consequence compare equally indicate landscape ols criterion complex objective minima backpropagation minima ols backpropagation work study optimisation start free bundle start free base dynamical quadratic objective piecewise proper optimisation minima speed become study good objective try free simplex competitive
assertion fix robustness theorem special borel follow robustness convenience value yield complete algebra metric b totally see see respect weak combination
require enable knowledge information snr contain hyperspectral draw convenience respect denote relate remain towards general theorem due matrix p nm complete indirect reference parameter matrix show unitary explain successively generate unit explain present situation need bring modification handle statistically rewrite p kk give k set core see vector von dimensional sphere sampling strategy rank derivation
incorrectly entire work glasso update monotone glasso minimize glasso coordinate w equivalent minimizer simply need achieve covariance matrix update identity know simple update operation primal lasso glasso cycle around column repeatedly perform implicitly compute solve warm start column update work glasso update every actually procee establish glasso solve coordinate reach conclusion elementary argument align intend ascent dual objective plot though coordinate update solve instead present framework consider primal stationarity condition sdp rewrite q wise eq symmetric denote one operator observe box transformation solution notice optimization suggest glasso hold optimal exclude diagonal hold fix equation express optimization p glasso solve dual
widely game environment grid node combine heuristic first find later necessary keep track list heuristic queue node effectively couple path heuristic admissible never remain heuristic among node turn flexible go back stage artificial intelligence game playing reinforcement observe state generate word transition bring reward short later contrary given receive immediate make enter reinforcement environment agent receive current
infinity way subsequent motivate correlation response adaptive unit ability lose extend multiple correlation output make well imagine process able measurement task method make enhance enhance prediction change similarly extend variance dependent multiple output extension include distribution rejection process combine flexibility process adaptive signal correlation input length heavy tail distribution expensive numerically unstable computation bayes carefully multiple gp model gene model financial review notation
introduction use justification coefficient build present ensemble classification learner bag forest boost choice bagging grow combine boost sophisticated try build manner equal weight ensemble rule rule base learner eq q indicator evaluate attribute constraint assign attribute vector meet constraint single attribute constraint indicator rule emphasize parameter rule set terminal root sort decision tree build cart regression algorithm contain terminal node make expensive control diversity observation subset decrease potentially less extracted tree clearly tree terminal node size prevent capture subtle distribution size determine grow branch termination meet terminal node select cutoff attribute choose split child split diversity time huge avoid capability boost diverse
depict circle branch decision place tree decision appear hand branch decision small decision tree subtree subtree immediate problem root decision combine node combine chance subtree partition write instance fig utility recursive definition past configuration arcs event therefore associate tree event represent arc possible decision tree easily extend tree convenience tree event arc precede event consistent tree really restriction inconsistent brief overview standard probability calculate expected chance utility well subtree expect procedure take arc expect either would em parent west east em child circle child height child circle em child node parent child node draw circle child node parent edge parent solid edge child draw distance node
performance aspect model notation prediction denote selection fold leave validation indeed however say impact final care particular concern provide enough parameter inform issue may analysis calibration size validation measurement partition outcome partition unclear relate admissible partition provide insight influential behavior know priori frequency capture behavior consider
compound process eq f respectively poisson process rate result fourth second asymptotically respectively match proceed prove convergence use device omit proposition stochastically bound unconditional h brevity omit technical bootstrap unconditional weak consequence unconditional asymptotic sum distribution follow zero surely immediately ng total characteristic one x side immediate third number sure side conditionally bound upper x conditional nz nz b index therefore measurable hence n integrable page equation replace nr far previous vanish show page proof proof statement compact point whenever imply characteristic z n iv dominate almost zero strong n z automatically w condition vi event imply condition vi condition imply vi result follow proposition validity regular vi observation follow equation true compact interval achieved able vi probability increase sequence whose subsequence subsequence borel vi almost therefore
party get verification american rate vote score conservative vice compare rating give former range relationship score interpretation correspond qualitatively infer agree remarkable base vote co course interested compare infer trajectory score score compare measurement rate year compare show trend
latent factor contribute nlp benchmark method develop binary predictor nlp bit optimisation base approach spike small bar real attribute count patch generalise bayesian spike overcomplete basis low case model overcomplete reconstruct hold specie property result dimensions nlp rmse data nlp nlp popular consist spike apart spike deal provide effective reconstruction good predictive behaviour model comparative spike much well
consecutive level subtract quadrature entry dimension quadrature rule adaptation user correspond sparse quadrature product quadrature original divide old set new candidate candidate backward validity expansion formula difference index intuitively contribute integral error candidate multi iterate active member tolerance propose find drawback parallelization parallelization also evaluation nest quadrature proceed advantage quadrature yield integral simplify notation inefficient quadrature surely overlap integrate total local indicator indicator lastly summation manner original experimental exploring characterize posterior ideally narrow range pointwise constant grid number increase arbitrary monte instead must resort monte sampling pointwise construct reasonably approximate posterior perhaps burn mmse simple powerful metropolis mh metropolis generalize improvement mh delayed combine mcmc exist active involve derivative posterior balance even proposal million estimate acceptable accuracy mcmc thus computational surrogate describe first illustrate use algebraic nonlinear use estimate designing experiment leave surrogate consider nonlinear scalar parameter denote additive measurement infer uncertain utility equation appropriate figure
exact paper completeness condition condition solution dual distinguished ambiguity rank guarantee clear exist matrix projection mn n contradiction innovation way construct minimum equality lemma let establish well establish infinite subsection also exist e sum converge n w consider candidate satisfied condition show subsection equality projection orthogonal complement next q eq last sparse incoherence conclude two inequality assumption power simulation agree low success minimum generate adversarial matrix method relative
k rw trivially inclusion node include proportional thompson analysis rw require fraction equivalently feed result obtain correspond repeat close degree implementation estimator publicly propose characterize far technique cover node degree average real c rw sample rw function graph tail correct behave analogous average node square b degree average node configuration pairwise edge bring well theoretical finding life graph formulae analytical expectation thick plain background average result line lie perfect estimating degree traversal follow coincide rw monotonically decrease exhibit bias estimator bias course life substantially different depend node connect similar indeed social tend degree degree quantify coefficient computed graph value indicate purely affect traversal
evidence literature aspect wikipedia well wikipedia article high article grow collective wikipedia influence opinion analogy google huge portion page site dominate political subsequent google major stream relate wikipedia automatic detection characterize wikipedia characterize page page article measure page level c measure user serve evidence usefulness previously automatic topic engineering perspective
optimize criterion test evaluate model spherical fitting evaluated measure table test trace spherical input convergence discover split include east variability ht correspond third variability exploratory functional cluster term prototype representative cluster spatial cluster process association among analyze spatial exploring similarity curve datum alternative spatial
square signal low signal paragraph illustrative concentrate application vision see compute positive whole mark simulated maximum type c c omit c error prefer contain square result signal give square however directly table good power already follow probability tend zero exponentially improve rapidly simulate ii simulation center lattice simulate principle shape concern triangular lattice site cc threshold signal maximal certainly effective pixel please
system find integrate neuron fire tend integer period find dynamic er structure favor uniformity reflect remarkable include marked cluster signal uniformity er explain along us spike element instant th spike element express size window respective eigenvector pca integrate sis dynamic observe result curve different parametric dynamic
bivariate scale define argument specificity application compact input six multidimensional polynomial interested characterize variable orthogonality one ref joint integral quadrature bernstein maximum thank bernstein polynomial output transform basis expand polynomial eq identify multi index
rectangular kb ab strictly diagonal invertible lemma analytically invertible hessian inverse positive dirac hard good important previously investigate differ considerably cite evidence neural amount generate x try infer gain dirichlet belief opposite belief result approximate compare monte carlo elliptical slice return sampler laplace estimate deviation approximate gaussian wishart bridge two apparent bad
though offset dimension slope stability algorithm recently investigate depth sampler normal behaviour rare event early set truncated type via rejection truncation recent guess available truncate smc hand sequential step monte truncate distribution initial sequential monte easily define estimate perform provide newly previous particle truncate move approximate truncation covariance particle lead shift imply region prevent version backward art rescale multiply long scale positive transformation scale mean truncation smc covariance multiple depend two single step start observation initial distribution possibly require smc therefore provide depend simplification derive argument trace provide though component order start set rewrite function depend replace next matrix way iteration yet however next complete iterate numerically n even decrease approximation conditional complete iterate equivalently multivariate normal turn single mention demanding case algorithm remain space probit comprise rescale check likelihood unchanged root
polynomial raise argument eq c x bx assume proof lemma write raise similarly term outer hand g additionally summation distinct correspondingly term algebraic plug sum expansion subtract remainder eq ne apply three find remark moment p hence lemma permutation size apply conjunction lemma k theorem eq expression panel plug ht mark large right marked square plot hyperparameter convergence mark plot show use large square bootstrap use adaptive output mark plot proposition conjecture remark electrical engineering science california berkeley electrical computer science california berkeley department department electrical computer california berkeley simple assessing involve increasingly demand computationally subsample bootstrap robust specification often rate bag incorporate subsample computationally assess modern parallel architecture furthermore applicability favorable statistical bootstrap bootstrap subsampling demonstrate superiority massive empirical extension series inference exploit basic capability
lead topology restrict subset illustration topology variant arrange line element intermediate latent choose embed illustrate sample exist circle example element position neighbor complexity take take I intermediate embed latent
likelihood eq independent flat conjugate prior gamma form incorporate probe set probe variance probe signal allow probe maximize limit sample ordinary parameter limit microarray robust overfitte optima compare moreover take explicit manner principle prior probe microarray probe hyperparameter external microarray collection previously side improve gene publication provide treatment publication validate compare preprocesse preprocesse probe probe level contamination comparison publication preprocesse probe effect probe contrast probe probe standard preprocessing provide array algorithm local background correction utilize non array level array summarize probe expression robust statistic probe preprocessing method experiment target spike receiver characteristic roc publication publication spike preprocesse estimating outperformed provide explicit genome array probe know probe error error genome alignment probe presence snps sequence good assess probe reliability detect result relative different source probe contamination detect source seem fraction highly contaminate detect remove probe level external information genomic fail portion perform publication rigorous algorithmic tool investigate understand probe probe contribute reduce probe noise array present chapter strategy differential introduce utilize probe analysis microarray database probe lead verification alternative microarray preprocesse analysis interpretation microarray tool preprocesse method publication probe strategy array experimental probe preprocessing array convenient access algorithmic microarray preprocessing probe view wide availability investigate genome control normal disease source valuable unique expression need traditional large scale collection chapter thesis exploratory response genome scale across wide collection summarize observation activity reflect exploratory research hypothesis material detailed widely cluster dimensionality visualization collection open possibility investigate discover previously connection gene expression traditionally relatively set particular disease cell detect gene differentially express predict disease outcome potentially increase collection cover thousand drive analysis formulation question traditional insufficient approach propose study gene group predict gene network base increase integrate genomic detect process disease mechanism implication perturb cancer cell line enhance detection drug functional genome wide lead breast remainder section particularly closely contribution thesis detail biological activation pattern diverse database contain concern interaction instance functional molecular classification human category micro chemical perturbation joint related increase statistical build activity typically ignore interaction gene pathway database molecular cell relational gene guide interaction improve activity predefine utilize paradigm genomic pathway
see super helpful factor endow sphere action sphere non overlap number rapidly large sphere packing action place element metric packing contrast sphere cover sphere third os enyi form every action independence clique translate algorithm performed simple os enyi observe neighboring graph randomly except high armed bandit observation run number axis run variance algorithm side utilize observation improve standard multi armed bandit gap clique tend almost action study problem broad regime bandit provide upper dependence
derivation drop form entirely analogous dimension stem n k ks ik move chance poisson expect eq define vector scalar consist uncertain change truth leave wiener measure function change rate consist deterministic deterministic mean sample process gps beliefs obeys operator go bellman bellman expansion eq integration remove loss correlate
plot depict contour kde mixture c point region much affected htb decrease lipschitz hold huber express explain translate robust necessary let hilbert necessary characterize principle satisfied differential condition name represent combination kernel suppose furthermore q give interpretation I robust sense additional also satisfied convex since assume strictly satisfy I strictly n imply convex convex fortunately iteratively square classical weight least generate iterate intuitively
suffer disease constant patient incidence overall call inverse duration eq correspond often statement odd incidence disease duration example disease read equal product incidence duration incidence article overview new ode express transition model
corollary nearest nn widely mining application aim understand reveal spurious variability contribution statistical certain subgraph graph point finite tradeoff conservative pruning able guarantee removal cluster level salient nearest nn link mining interestingly particular like understand graph reveal underlie spurious variability spurious structure contribution graph sample level satisfie boundary unclear might mild recovered subgraph
sample account fact center around clean manifold explore devise analysis assume confirm origin practice tool cloud et condition noise arbitrary model contrast explore accord corrupt crucial result careful analysis nonetheless also analysis present population eigenvector despite difference technique bind recover free multiscale pca study parallel develop address examine recent wu tangent basis neighborhood absence hybrid zhang assume collection recover flat see idea denoise smooth surface al graphic problem well tangent work approximation top value neighbor work often refer size answer consider perturbation tangent trade seek small avoid curvature tangent plane eigenvector tangent space curvature trade subject decade dynamical main true tangent define orthogonal distance angle compute perturbation theory ease presentation order measure tangent ex aid quantity denote due subspace general useful subspace encode subspace curvature recovery curvature trade readily apparent small value neighborhood small ill intuition dominate approximate neighborhood radius denominator bind control numerator intuition serve enough curvature contribution conditioning denominator force curvature enforce bind principle curvature approximate must curvature become large ensure probability curvature uncertainty principle intuitive notion combine curvature perturbation accurate tangent uncertainty principle computation homology manifold explain detail section principle geometric perturbation review accuracy estimation present modification account noisy introduce practice apply theoretical algorithmic consideration technical riemannian locally reference tangent manifold origin rotation align axis direction principal coordinate axis plane manifold
pt report predictor median preferred incorrectly variable ideally discard glasso setup rsc achieve recovery definition rsc inferior three result confirm glasso rsc number unlike large explain slight method however tune infeasible practice serious finding selection inferior experiment conclusion find winner term speed appeal reduction provide evidence cognitive carry world publication paper control paper variable take predictor yield intercept make response result neither penalty impose website structure dimension reduction rsc describe find assess performance leave square newly
year overcome several pattern aim galaxy neighbor neural support map universe galaxy add near optical kb deep survey like systematic specific region emission often method use statistic evolutionary observational shift filter specific dramatically value previously mention probabilistic increase computational overall hereafter design accurate relatively scalable produce survey method type tune automated ease exploitation survey outlier limitation mention region happen section worth never account possible structured feature principle discuss implementation use provide description selection find determination final galaxy together thorough performance method reconstruction determination error outlier describe find dm aim case local reconstruction otherwise measure order dm machine hereafter ml method problem extraction point ml derive case source knowledge kb useful unknown treat analytically dataset ml three extract base kb kb degree run implement avoid like extraction knowledge without target say extraction practice approach drive validate subsequent aim object subset consist belong say unsupervised determine cluster represent dataset tend clustering figure efficient clustering define supervised mapping regressor thus target explicit provide training regressor map response variable
horizontal dotted capacity width compute channel channel interference section wish compute case channel factorization follow many analytically available case analytically available see bc bc extension x q accord cycle
desire requirement tp l f n tp p condition g p sufficiently l la k independent hoeffding absolute inequality probability nm f mainly scheme bind term necessary incur sign capital letter symbol font matrix zero respectively large singular nuclear value phrase numerical use proof fix w ij ss obviously whole setting
without reversible spectral self adjoint finite one argument obtain claim prove integral observe gx gx k write integral spectrum spectral geometrically ergodic
ga denote point clique scalar equation translation without loss x l k easy q eqs firstly j kt secondly r sequence surely obtain surely therefore part equality without chain vertex chain bin bin chain equally likely node bin number average chain average chain procedure demonstrate special discuss ij j h let hypercube direction almost align small direction xu exist bound follow eq constant task node vertex hypercube define unit constant b eigenvalue notice due thesis number node coordinate vector node sdp k r ng nr xy dy tx x ip coefficient arbitrary conjecture claim cloud euclidean distance various area localization closely dimensionality
moderately alternative satisfy two main alternative satisfy restrictive critical value statistic critical value available assumption nh consider nan alternative sufficiently moreover lack drive specific alternative choice propose statistic correct estimation elegant optimality necessary suggest hence possible restrictive allow absence theorem establish optimality alternative p n sequence theorem detect alternative detect popular process variance constant alternative polynomial short term statistically negligible multipli order covariance alternative j coefficient process tool may contexts alternative case test q statistic study wu er von statistic introduce observe asymptotically critical test alternative nan white gaussian hold consistently asymptotic ii imply standardize establish involve due statistic equal simulation aim propose valid penalty test preliminary practically white process investigate latter order give box label second since kernel k
clearly restriction different test availability widely software flip flip replace observation cauchy datum replace randomly select label keep scale two class circle htp eps eps scale eps scale eps eps plot contamination class represent circle simulate contamination mis capture aspect mis nature tail additionally attempt simulate extremely error scale center cauchy multiply typically datum contamination occur uniformly I calculate cauchy gamma interval empirically effect contamination nest boundary
graph smoothness work addition branch devote analyze spectrum good introduction particularly emphasize promise constrain suggest way cluster task work graph view intuitively efficiently graph analyse effort idea essentially spectral walk mixed random walk graph author permit efficient clustering result post work close sense use unified find representation multiple graph directly develop co regularization conceptually similar adopt related paper first despite nature exist effort combine layer mostly way art graph equally layer role addition respective layer theoretic beneficial processing multiple graph network believe effort mobile phone dataset field represent technique
mark mark black nine method vote member empirical random member theoretic complete plot dataset perform approximate gaussian exponential enable estimate gain extensive gold evaluate loose monte carlo objective preference select continuous eight version decision expect three challenging decision boundary second uninformative uninformative capability multiple disjoint use depict uci multiclass dataset distinguish letter vs process preference aggregation fig significant random preference domain
point indicator additionally square shall frobenius tv distance real mle initial hide process order keep instance various mle existence invertible normal interior yx yx remain open ball center mapping space state tail theorem good qualitative guide mle even hmms sense asymptotic mle quantitative abc eq consider q analyse abc define mle measure corresponding take one immediately law variable crucially thus statistically identical q denote perturb estimator reveal mathematical tractable similar spirit author view hmms try section likelihood perturb abc mis raise still general answer let directly law asymptotically consistent though value maximize estimator long asymptotically consistent follow abc mle accumulation belong subset accumulation lie neighbourhood neighbourhood go finally one mle spirit
experiment adapt critical temperature lattice wang design efficiently surprising must temperature many peak large peak comparable wang might seem sampler achieve mechanism learn choose length sampler sampler thus acceptance peak learn length problematic adaptation length manually wang figure principled addition temperature change investigate bias ise auto trace considerably wang effective clustering show exploration ise cube ij sample low speak space visit simulation densely wang substantially reasonably cube spin hard worst would hope arise run train likelihood rbms bipartite undirected variable side visible unit unit
high frequency network combine handle direction determine structure recent study c gene investigate profile bn relationship program reflect expression rna isolate month old real rt implicitly inherent uncertainty justify author parametric resample level bn structure edge several reveal relationship normalise algorithm et selection edge line represent threshold limitation computational fact permutation computational complexity second large extremely value positive gene study choice
duality employ average principle rule formulate aid calculus role learn principle free reduction also principle formalize principle brain optimize representation identical optimize complexity free
simplicity assume mixture support contain handle get state next many exist modulus vanish pointwise summarize surely range finite root I abuse notation I vanish nearly theorem middle difficult mapping drive rate must finite mixture optimal rate allow rate restrict subset ultimately illustrate
introduce amenable determine minimize et match limit case norm intermediate specific minimize interval soft restrict show figure value approximation relationship quadratic activation solution activation valid people undesirable bias many signal processing statistic attempt statistical norm smoothly different capture desirable many norm basic architecture modify continuity near demonstrate norm constant avoid bias one compete goal smoothly deviation scad penalty region cost function define scad individually region give activation activation argument thresholde restriction condition scad attempt something close coefficient calculate cost activation invert cubic root calculate generate thresholde positive correspond outside range analytic function obviously fitting circuit aim modify robustness function quadratic smooth transition
obtain reservoir international competition future evolution reservoir extensively speech recognition outperform isolated recognition reservoir complex recognition comparable good method couple road building basic reservoir suggest new computation system reservoir loose reservoir dimensional turn turn idea reservoir computer fact important good reservoir rather good eq change one neuron experimentally dynamical reservoir design remark computer build dynamic couple dynamical external much come dynamic experimentally adjust instability adjust external read construct weight reservoir subset dynamical satisfied experimental realization reservoir standardized test evaluate linear capacity often benchmark reservoir computer memory input input memory reservoir unable task task task reservoir auto move average system reservoir drive two widely see instance predict trajectory benchmark publish recognition digit recognition exhaustive benchmark road master isolate digit moderately hard imagine recognition nonlinear memory give independent reservoir road cat couple semi optical recently delay feedback loop demonstrate analog reservoir task isolate recognition question road map leave open reservoir noise exhaustive remark distinguish reservoir noise reservoir computer continue
bernstein approximation approach knowledge chance constrain knowledge moment study constraint high hybrid surrogate let take convex surrogate copy objective chance constrain optimization infinity obtain essentially theorem structural assumption inherent address indicator investigation beyond scope classifiers eq eq yield yield find sufficient low begin extensively use sequel argument classifier simplex theory marginal rl lipschitz em convex successively contraction
large state eliminate bad datum bad view share mathematical compressive nonlinear goal sparse recovery bad network consider analyze measurement mix norm bad observation linear mesh improve almost mathematical early error generally generalize iterative convex programming noise measurement iterative measurement program iterative compare propose minimization additive consider estimation data attack formulate attack rely signal compressive sensing extend restrict isometry sense overcomplete error correction rely condition rest bad detection
integer result one remarkable feature mh ratio example mh greedy trade decide add subsequent employ speed observe safe exhibit aim subsection estimation replace employ result screen nevertheless qualitative coordinate show pt know sparse theoretically entry depend choose numerical norm value run replication readily lasso regularization fold fold ten calculate cross validate package prediction b case right sparsity pattern recover report deviation particular illustrate well right illustrate evolution hypercube correspond sparsity
reformulate absence consider statistic
gain every past determine choose decision feature value variable denote transaction clearly gain become classification transaction consider f transaction dominate past transaction branch past transaction gain tree construct decision construct transaction describe transaction simply tree transaction gain branch add node past transaction successful terminate test perfectly transaction classify transaction compare root tree discuss decision avoid overfitte etc transaction trust able return recommendation transaction confidence recommendation input e transaction perform algorithm knowledge quite recommendation confident recommendation equally random recommendation completely confident denote successful transaction transaction input recommendation discriminant analysis refer transaction input transaction space recommendation greatly power separate successful clearly minimize transaction otherwise transaction may overlap potential transaction confident word determine lda centroid transform transaction measure distance transaction centroid potential transaction successful transaction group lda perform discriminant analysis distance clear eq classified transaction safe trust recommendation construct accord information gain accord gain
level bandwidth choice observation sampling pt median lin lin band build describe sample covariance draw realization variance explain equation band highlight smoothing validation validation lead empirical get always confidence band estimator smoothing coverage smoothed interpolation high pt area lin bandwidth band band interpolation coverage strategy weight empirical expect one population transmission cost network noisy thompson population sample curve consistently although present estimator band asymptotically correct band conditional difficult survey good rigorous theoretical reveal far superior performance interpolation even low smoothing beneficial building confidence band
easy depend autoregressive gaussian definite definite wishart multivariate gamma chi square integer square univariate chi square distribute likewise outer product wishart wishart distribution definite chi freedom wishart mean wishart wishart assume dependent though effort value everything write replace suppose kt nk ij kt u wishart constraint function univariate wishart generalised wishart time variable major axis eigenvalue like draw collection index time collection definite process matrix index draw
extent decision detect come alone insufficient essential aspect historical go probably accuracy importantly one place epoch indeed digital concept article build song audio content community algorithm achieve comparable even provide acquire community valuable improve raw query cover song identification outcome consider cover song community song within show measure centrality discriminate original community light stand promise cover especially acknowledgment author thank review project reference complex cluster music retrieval cover song explore cover song automatic underlie piece pose song match similarity song community answer start art system embed similarity content recognize cover
include source transfer different enough sophisticated transfer introduce source encourage measure supervise effectiveness measure performance interesting transfer benefit target existence limit source whenever transfer whether adapt transfer perspective work summarize transfer also method whose solve tradeoff identify good source formalize set finite highlight task advantage identify combination task task confirm adaptive transfer tradeoff organize notation transfer problem report challenging report conclude appendix consider discount process close bellman v action span basis orthogonal projection transfer source build
w g notion key generalize chen identifiability compact strongly identifiable fix fix assertion exactly distinct point e proof subsequence distinct strong identifiability specify practically condition eqn next unbounded support convolution function thus fx p smoothness tail behavior I ordinary density density normal full symmetric dd md g measure discrete subset remove density p g laplace density unknown framework endow measure measurable eq study neighborhood van analyze behavior divergence hellinger density mixture view notable concentration term oppose divergence mixture separate family simple hellinger metric ease consequence
q writing c I ib ic q gradient decompose update conclude proof life predict outcome paper mathematical artificial market supervise probability feature fusion contract price outcome market inspire real prediction market reasonable efficient linear aggregation well logistic aggregation specialized instance experimental market outperform synthetic extensive evaluation detection ct improve adaboost positive volume online market information market yield interest department health care make informed price market probability outcome market capable introduce mathematical simulating purpose mathematical solve contract show market participant forest turn linear aggregation market regardless market contract furthermore section market regression classifier respectively certain accuracy predict outcome region classifier obtain whole become sort ad hoc trend
range genomic finance distinguish produce interaction party component infer exploit concept statistical causality maximum principle major theoretical difficulty reliable analysis interaction comparable present establish quantitative correspondence boltzmann machines physics graphical analysis bm mathematically computationally challenging infer correlation scope letter symbol correlation configuration bm field ise equilibrium many design tackle advanced message pass expansion graphical lasso running procedure quality popular principle one rescale fluctuation separately projection uncorrelated contribute say notably large pca determination retain mp correlation sampling configuration generally spectra coincide retain explain achieve time pca may illustration pca look principal vanish conceptual bm appropriate detail necessary mechanic model originally rely informally speak attractive direction tendency align norm characterize attractive pattern auto bm direction orthogonal generalize attractive bm interaction equal know priori generic bm allow infer parameter component small limit show ml attractive correspondence pattern correspond normally discard remove address many inference pattern due analyze pattern send possible supervise rigorously study see review validate ise
present regression effect provide rigorous building expand lattice random impractical little advantageous remain one fourier log spectrum arrive example date example page make mention context separability thresholding model obtain field formula random proceed describe recursive estimate maximum result asymptotic estimator section illustrate study conclude extension imputation supplementary begin basic concept field focus field hold index lattice field may regression correct upon weakly stationary effect lag center variable second order define depend let via spectrum inversion notation integral frequency expression compactly
corresponding equivalence write transform st transform st equivalence equivalence matrix ht st equivalent define class finding small however end vertex connect column switch st graph contain connect switch st gram symmetric sign without form block three compose ht solution hill local optimisation follow decomposition find conjecture
member sequence linearization characterize arbitrary viewpoint quasi write false sequence mx false conversely false condition infinite sequence condition condition assertion theorem hence condition theorem desire conclusion enjoy construction conjecture
filter band insufficient critical filter threshold iterative correlation signal recover knowledge structure analogous cf signal encode underlie structure matrix importance pixel order critical generic diagonal interpret accurate scheme filter interest unfortunately complex operation inversion often matrix box routine read calculate seem e r obvious multiplication vector invoke example instead canonical
strictly sense verify explicit wise row transpose section th singular belong class invariant unitary look reproducing endow respectively satisfy proceed q reproducing derive explicit complicated regularize dual norm choose respect homogeneous pair straightforward eq equivalent satisfy compact subset recall strictly positive constant exist exist satisfy norm constant conclusion positive shall contradict proceed convexity contradict banach
rank frequency plot level monte distinct must also include bin zero list please display significance reference goodness fit testing hold remark know root powerful perfectly bin great number draw law corpus randomly bin indeed consider model integer nonnegative number sort fit bin bin aside draw bin fall three frequently occur law level via simulation great summarize classic whether first bin mean total little bin demonstrate experimental bin statistic significance monte simulation bin set four goodness fit distribution statistic mean relatively insensitive truncation particle observe bin dot square display count plot along significance root significance monte poisson simulate report reasonably ccc square square dot poisson line suitably proportion nine population proportion genome formulation suitably entail draw model goodness confident suitably figure c calculate
cut mp combinatorial structure need cut term cut solve graph cut pass elsewhere cut moderately special case analysis technique novel rather rely map combinatorial case mp execution map prove strong statement fix energy construct magnitude belief correspond apply broadly could region strict mp mp connection cut cut idea path strategy scheduling capacity scale graph cut broad cut high high separate dual regard delay message safe believe concept explore non throughout project thank anonymous valuable lead paper message preserve ij simply plug update ij ij b final lemma tt message x ta except position message I b subtracting belief final prove message homogeneous binary factor tm income form true message opposite plug plug message jx matter homogeneity homogeneous pass mp belief pass belief incoming message pairwise factor minimum sum pass belief previously still case analogous lemma homogeneous equivalent max flow min find phase belief variable message incoming message would opposite thus nonzero edge iff merge encounter cut assume eq cut bottleneck capacity easy position capacity
learn discriminate action due lack detail model four dataset dataset process manner stop character baseline weight representation dataset corpora corpus corpus compose corpus large category corpus page ng compose l corpus document nb category nb sentence document compare measure
direct employ operator add hill scoring annealing begin take step decrease undirected learning structure possible strategy candidate keep step hand short try clause move candidate clause algorithm pseudo pseudo atom prevent predicate dominate graph selection markov network regularizer norm impose force extended technique first learner clause perform one learning contain thus focus address challenge group exploit sophisticated whereas search performing design pre address potential sophisticated avoid maxima discriminative structure learn alternate type move locally current order optima alternative increase structure al domain attribute employ constrain search table attribute single entity table describe relationship proceed stage dependency local second dependency join require dependency stage dependency join attribute space constrain orthogonal characteristic use bayesian network convert graph structure structure modify score pre roughly find promising space approach constrain potential parent par rv proceed stage form potential parent par chain potential similar candidate add beyond capture currently parent parent par rv parent relation lead high network group learn structure network identically process induce template clique markov network template dependencie ordinary clique par par rv consistent template far search relational promise community search
likelihood greedy neighborhood conditional log square loss neighborhood recover elementary entire recovered contribution rigorous algorithm condition also local greedy require significant improvement global weak marginally greedy mle impose explicitly impose condition condition entry theoretically
conditional variation tractable family representation high variety vision financial social network graphical contextual occurrence opinion formation technology social involve estimation draw np high observation typically design structure seminal structured span tree various broadly classify two category combinatorial typically solve penalize ise algorithm exist estimation high tend graph consistently complexity demonstrate fundamental local wide enyi r enyi graph almost indeed whenever random law cycle small applicable elaborate relevance aforementione extensively topology model topology various phenomenon social formation technology concrete use ise voting reveal political decision scenario e network popularity new regard distribute model finding imply
predictor construction recently binary predictor may take trick note feature sort calculate motivate consider wise piece wise function see piece piece represent piece domain multiclass shall predictor apply variant piece linear piece practice example train multiclass predictor pair piece advantageous generalization provide defer number error iteration analyze mention solve eqn np follow tell accurate illustrative find fail example exist entry gap equals denote eqn solve q pc matter regularization regularize rather produce use show
isolate angle big advantage generate graph examine affected rotation beyond goal article example power sect promise rotation drastically ratio isolate especially rotation angle check effect sect deduce isolate limit isolated node become dominant proceeding note deal size case iteration lower bind bind size cover measure describe sect width column adjacent slope united segment fig application measuring carry low segment yield much small interestingly frame curve consist two subsequent segment setting
simplify include physics index asset price economic citation serve completely current dynamic principal generalization contain temporal configuration ignore future variable markovian conversely describe construct reaction hereafter preserve upon protein reaction energy dynamic optimal reaction system give reaction cut energy half transition reaction time complementary superior histogram invariant reaction coordinate rescale insensitive detect reaction
statistic
function graph maximally clustering naturally multi criterion information independence need clustering impose style graph want cluster something know criterion information independence second far independence clustering variation metric section aggregate ex world country model force service etc structured format language unweighte people trait american weight skew correlate trade relationship clustering substantially optimize linear combination edge allow free package balance modularity et clustering group independence modularity vi distance china cluster france clustering modularity statistically thus aspect like less aspect share cluster trace highlight spread contain root root group explain alone combination potential drawback aggregate clustering necessarily aggregation explain adjacency weight axis eigenvector length matrix significant flat without cluster hand close
sensitivity residual explain account outli nonzero estimate minimizer nc ml nc n denote membership satisfy cost rewrite nonzero yield outli reduction np hard mean hardness latter line iteration guarantee due practically feasible remain eq robust mean convex jointly solve resemble lasso recover interesting robust cluster couple mahalanobis covariance euclidean mahalanobis distance c outlier identify outlier replace entry whole iterative solver develop present limitation constraint membership assignment assign cluster vector fractional membership soft differ hard binary alphabet box th replace leverage sparsity set present section still soft assign cluster soft interpret unobserved centroid subsequently outlier probabilistic draw membership
limit state proposition recovery alternative algorithm minimization implementation sparse section satisfying pursuit kk ax dd assume situation integer nj entry block representation loop approximate convergence analysis upon situation true approximation quantitie bx bx bx bx applicable exponentially along since follow bx cover block theoretical sensing b block sparse absence observation small possible pair answer aside condition validity optimize allow yield recovery problem may think norm recovery validate provide sparse course noiseless discussion besides assume mainly sake notational
allow formulae finite give integral finite substitute bs ds ab sample elementary yield finite density calculation adaptive analogous analogous reference analogous proof whereas analogous analogous recall part density obtain n n ns dx u sn ns ns ns n ns use trivial eq every b nc nn since arbitrary show k already converge zero closeness argument q b n dx x soft du use substitution elementary verify nc nn complete z c observe lipschitz n n I clearly assume bind consequently nc atomic hard unknown density weakly atomic par I atomic converge absolutely next absolutely total mass absolutely limit mass density se se eventually get dominate ie absolutely preserve complete closeness soft converge weakly view move fact I atomic mass atomic turn part total total absolutely absolutely part sn sn converge convergence absolutely soft since total preserve complete immediately closeness convergence cdf converge large cdf prove nx x x prove immediately proposition n l normally
technique issue worth investigate constitute seed author thank warm author paris author partially grant grateful whole stage helpful recall assume infinity go constant define upper bind write let impose finite disjoint n ij n I kk inequality follow go go use exactly combine arbitrarily choose enough equation together go n n j equation corollary statistics abc nonetheless algorithm statistic bayes asymptotically true quite appropriate author bayesian pick necessary
optimal bound policy optimal mdp tight mdps much mdp reinforcement learn policy finite alg approximation alg look perform time step operation step utility loop operation inner loop perform time mdp p ni expect calculate stochastically simply sufficient mdps accord loss mdps subscript
occur alternative merge cost lagrange multipli true rank set goal underlie dimensionality perturbation appeal clear typical cross final error rate term way stage chain well mean intrinsic desire provide actual theoretic
draw object factorize new compatibility fix functional thus distribution involve apply general marginal field paper mean marginal summation computation graph propagation belief propagation product algorithm pass exact tree cycle bp update pass apply general approximate source order message update require notation local function detail clear constant choose ensure provide flow cc message iterative equation local update global update pass meaning structure unique mild potential least long type contraction uniqueness compute marginal potential normalization see structured graph define graph description multiplication original iteration inspection incur per prohibitive although certain exploit special structure purpose bp communication cost update number along limitation sensor cost stochastic propagation adaptively randomized form usual bp update motivate observation namely pass along formulate incoming
line citation acceptable long consistent reduce font size point year use cite pp mit book realistic new recall journal error compare area introduce survival experiment cox hazard cox series
large insensitive say large instead predict occur mix data assumption homogeneous markov mix transition markovian ar error coefficient transition address parametric elsewhere assumption constrain ols common assumption enough moreover predictive bind analysis traditionally model minimization residual see aic really prediction
radius region factor present straight bound appendix brief binary sample unbounded discrepancy believe discrepancy sure super polynomial purely jensen inequality th well hard constant kernel describe page kernel principal shift invariant include kernel nice sometimes negative kernel super set unbounded infinitely min run hence attain run discussion lead I length david
subspace distance reformulate figure seem obviously proportion e computational view important provide stability algorithm paragraph discriminative aim good discriminative axis discriminate group criterion situation interest unsupervise furthermore model discriminative must approach axis keep orthonormal conditionally soft partition discriminant extend sequentially select subject first matrix soft conditionally q ik dataset preferable ny maximization problem remain fisher aim optimization procedure construct first complementary subspace discriminative axis solve problem iterative eigenvector associate assume orthonormal discriminative span discriminative axis lie gram schmidt allow basis linearly independent discriminative large soft matrix subspace stop axis model likelihood conditional provide conditional expression vector counterpart discriminative axis step remain result vector ic em fisher em efficient selection work popular keep high partition also generate covariance multivariate normal parametrize empirical directly fisher adapt strategy determine
toward preliminary average express cone describe see rewrite w balanced conclude w worth use make distinction main work relate allocation core game result core highlight solution principle principle solve infer possible select say derive augmentation fluctuation value formalize fluctuation fluctuation convenient fix u generic inverse q square augment independent run dynamic satisfy equation let formally q obtain mind ready controller drive zero allocation allocation converge direction previous controller
reasonable considerable average uncertainty variance bridge logic concavity even information typically predict square optimal solution realistic example bayesian outperform counterpart estimation conclusion reach example arise merely bridge marginal case risk conclusion penalize particularly regard bayes classic diabetes lar conclusion wrong specific merely practitioner broad dimensional case mixture many share favorable induce tail include relevance machine normal exponential gamma inverse pareto mix convenient lead poor miss paper discussion include supplement extensively poor behavior sampler difficultie something bridge excellent mix favorable mixing
small cause discuss sec rw recommend set category case set c equivalent volume proportional sec set one size sec resolution discuss increase decrease small category weight rw unbiased note collect weight attain weight increase empirical facebook suboptimal sample cluster show two base c j latter rw show rw optimal plain gain compare rw come irrelevant demonstrate facebook among category observable facebook well evaluate allocation gain control demonstrate insight nod densely h partition category nod dark light extreme scenario partition scenario purely cluster arguably remain ec ec resolution category respectively normalize square obtain short pilot estimator show plain curve advantage category neighbor naive dash moreover advantage
accept reject reversible preserve simulate hamiltonian dynamic preserve volume theoretically algorithm name carlo updating via hamiltonian updating depend energy hamiltonian always energy discrete step l discretization hmc proposal though distant previous hmc simulation inaccurate acceptance take successive undesirable slow mix hmc generate trajectory loop back doubly close side markov ergodic result chain slow hmc powerful usefulness limited tune parameter preliminary problematic simple short rely autocorrelation preliminary run turn extension averaging devise take us step long initial use convenient dot initial position system quantity proportional progress algorithm run quantity proposal move unfortunately guarantee converge issue slice black solid solid exclude possible sample position correspond exclude satisfy balance subtree momentum vector final trajectory process stop since blue x ignore begin
sort rewrite pca basis fundamentally different proceed approximation signal follow observe q keep monte simulation decay plot ideal energy linear approximation patch size example decrease e decay approximation small comparable ccc gaussian approximation comparable approximation numerically evaluate mse minimal proceed independent realization apply mse ideal good approximation decay fast linearly plot figure eigenvalue typical almost htbp ccc indicate number show good term parameter typical ratio mse approximation mathematical analysis decode impractical assuming describe single analysis conventional failure order cs decoder error term certain respect optimality say mse equivalent instance optimality index constant otherwise side one draw similarly
fisher maximizer maximizer right hand write fix evaluate however invariant respect th th column transform satisfie maximizer correct value determinant limit covariance column determinant indeed opposite formula deduce evaluation expansion initial describe refer algebraic ode ordinary satisfied gr rank ode call ode reciprocal reciprocal numerical procedure system toward along direction differential first study group manifold type differential operator delta analog
unit applie ratio bound dimension hold rescale rank satisfy convex program regularization deviation slightly mean computational achieve much worth observe interpretation rank consequently numerator noise second arise identifiability avoid restriction turn previously involve plus sample equation tail norm specify choice regularization regularization eq great pca see bind rank dimension roughly degree expect see order state regularization covariance concentration establish population version differ focus decomposition noiseless focus identity exact bound nuclear program enforce difference et three see incoherence consequently provide noiseless condition guarantee interest observation distinct leverage notion restrict sparsity rate plus minimax et sub involve bound limit decomposition nearly fraction zero analysis allow high simultaneously inspection corollary go set regression know goal share perturbation share sparse matrix maximum rescaled design uniformly mean note design program regularization q great leverage deviation bound base perform might relaxation provide norm
match point vector curve point construction surface difference tangent yield let set three origin fix thus kernel inner theory explanation simple get euclidean definite decompose bilinear define original explore form consist eigenvalue generally I theory carry
avoid stationarity figure non stationary harmonic signal signal model classify fit parameter bad true parameter maximum wrong improve separability harmonic series space black dot show avoid bias simply model describe white mle solution exploratory ar example bias good would identify correct tune however example large overlap remain red triangle green panel fig parametric subsequent comparable axis series right color shape represent section applicability growth neuron spike less stationary transform
consider instead compute norm minimax modify enjoy follow exercise pure ensure player resort modify derive result payoff round simultaneously accord mixed monitoring observe subset payoff ambiguity uncertainty follow convex dx first behavior also rewrite compact linearity extension case mean tend surely strategy round player respective mix full take end get account yx ba empirical joint action round interested rewrite linearity strategy player player proceed provide reformulate indicate repeatedly probability entail claim cauchy schwarz value continuous yx yx mm lemma entail let prove continuity read stand value norm pure action robust satisfy necessity hold exist set define distance give individual compact indicate corollary attain minimum whose action denote show pure action concentration argument hoeffding
whose absolutely lebesgue definition side complete condition fractional condition affine fractional analogue reduce take eq consequently interesting corollary optimal clear clear proportional previous one implication drift fractional brownian establish pg brownian motion case corollary
accelerate stochastic increment change reach implementation penalty normalize form follow collection behind ratio update average fact control control generally way prove frequency unfortunately transition find drift joint general able irreducible section prove directly case assumption restrictive imply implication frequency bin suppose schedule go meet notation use prove go imply follow proportion reach precision satisfy except proportion
world assume initial gmm effectively cluster localize turn decrease network gmm flexibility specify tailor evolutionary intuitive third final experiment gmm ba ba heavy act massive vast majority connection many feature skewed exhibit web email country country tend fast rich get rich node inherent attribute make likely web massive primarily web site connect google dynamic piece ba graph always begins iteratively enter tie already form tie ba produce tail degree si ba graph simulation leave panel red parameter cumulative gmm specification curve gmm ba si ba gmm counter increase figure scale likelihood simulation ba parameterization gmm recover generate classic ba gmm connect base structure base ba gmm hold constant classic ba gmm conduct ba base increase gmm function base pseudo code implementation include ability like fit method fit slope law ba piece ba gmm scale estimate figure
set input program input output implement program correct correct correct pair complete intended specification output concerned conversely correct output possibility belong care correct satisfying definition space find fig vector imply implement conversely fact capture definition role quantum setup studying introduce map error formally replace space consider refined comment capture anomaly detection perspective anomaly detection batch anomalous pair definition weak efficiently train quantum speedup map detector aggregate average turn quantum formally weak classification output weakly classify classified advantage classifier minimize heuristic input n use construct representation weak cluster computational set input training training easy idea perform weak clear sign hypercube label indicate assume assumption advance weak classifier weak training set process create accurate classifier know classifier open purpose include identification overfitte space quantum approach boost term accuracy speed select shall generalize value introduce weight dependent classifier difference strong express definition correctness pair average opinion know whether compare strong pair versa correct formally high set challenge construct beyond training solve find weight assign penalty
posterior root suppose start simulation well via vertical shift put recover calculation also proposition finally constant paper improve early geometrically chain applicable illustrate improvement analyze normal toy put drift normal alpha elementary compare chebyshev minimal minimal ep base magnitude remain less reality proof auxiliary note assumption special assumption relate calculation drift x v x geometric drift lem well bind entail qx let fix apply write sm
recover behaves miss cp less however sharp error factor recovery recovery increase tensor datum reconstruct tensor factorization literature alternate entity square fitting commonly speed suffer fail accurately component correctly ii missing imputation suffer poor therefore alternate least simultaneously contribution develop opt couple opt incomplete miss opt numerical notation discuss analysis opt algorithm miss entry compare factor couple euler letter denote capital matrix denote
fourier fit along various degree training bin split dc lead solver time spline high degree regularization cubic bin unnecessary embedding compute generalize fourier normalize train relatively expensive store cc cc cc fourier degree accuracy accuracy accuracy encode fouri embedding split dc training
estimate idea paper notice correction take weight construction tensor point becomes weight likely field effect I tend seek pattern include locate direct hierarchical draw distribution markov carlo interest estimate rough estimate point knowledge within density orientation orientation field evaluate orientation choose proportion mix property monte indicate knowledge alternatively little known nature extend start continuous birth death control enable range birth death brevity key detail fix birth death rate maintain detailed birth death indicator indicator signal point parameterization describe event occurrence birth reference length distribution respectively integrate field length field integrate assign choose birth density include update calculate death rate ensure hold death variable allocate
make clear bind orthonormal family tf take mass fx theorem times recall main difference argument show probability low application argument minimax batch infimum forecaster unless omit tf apply constant online forecaster dependencie take achieve instead standard gaussian jensen get proof present page f tf define lemma prove definition last meaningful elementary true equivalent c imply c dc tc c c choose eq remark fact multiply side conclude concentration argument stage sequence lie outside conclude dedicated term leave output note j
variance strategy analog var thus basically estimator implicitly existence strategy b estimator strategy type restrict study little suppose ask prove strong impossible plan overcome occur support version chernoff involve set incoherent analysis prove know ahead condition estimator probability prove automatically support highly quasi isometry property obtain sign jointly already deal unit norm sparse uniform among subset sign last concern nonzero define introduce x range range strategy recover pattern sparsity e hold support great reader relevant us coefficient strategy x strategy exactly sign normalize standard gaussian coherence proof
scenario middle side wrong edge length bad number discuss section bad affect merge serve cycle lead guarantee short path affect performance correctly recognize either short combination shortest detect middle cycle long shortest short use locally graph establish every diameter strong distance guarantee analyze count hide least recover event occur diameter decrease increase event cycle increase state choose c fraction node imply nr function enyi recover representation v accord homogeneous edge length approximately length distance length select effectively dominant presence subsequent shortest occur meaning enough second event occur middle bad analyze cycle short strong compare recovery minimal possible node component parameter equivalently consistent topology recovery length participant recover participant suffice recover availability short distance make discovery proof theorem test middle short
characterize indicate clique node program ax basis consider differ selector reason lie fact enable computation network try represent identity people frequency interest clique every view clique receive measurement order interaction represent reconstruct provide figure observe clique htp illustrative example scenario signal correspond assume denote interaction frequency row entry whether set column equal normalize summarize canonical transform omit mean matrix transform call transform row exploit transpose matrix transform construct observation basis way partially rank usually interpretable information approach sequel compressive compressive correspond rip I every subset column constant column index sparse unfortunately basis small failure let inequality every depend problematic constant clique rip try arbitrary
orthonormal pt denote bb coordinate q bs output suggest available could incorporate subspace specification tend theorem terminate give one stop iterate iterate sufficiently typically zero intuitively simulation stop always interesting question algorithm eigenvector analyze property assumption establish yet principal rate suitable normality assume though assumption could orthogonal factor dimension spike initial apply algorithm constant study iteration theorem probability state spike eigenvector hx x least satisfie decompose component coordinate average coordinate term factor arise separate first eigenvector rest regardless establishe understand upper compare eigenvector cp note bound
show ranking demonstrate select slightly adaptively comparison instead comparison propose address rank pairwise ranking mean rank uniquely pairwise pair primary objective need correctly structural constraint suppose may embed space wish discover knowledge practical restrict pairwise comparison view rank ranking specify ranking bit many sort human pairwise obtain pairwise comparison impractical situation thereby induce object object rank audio difficult section object specify rank select comparison select need pick select impact form response denote indicator tie allow quantify query reference also reference point object satisfy relate embed limit rank learn
development coin infinite formally bernoulli process know problem replacement pool chance unknown coin distribute equally likely look
argue end great transaction transaction contain end hence th replace transaction long transaction case transaction invariant th iteration stay contain begin algorithm transaction begin argue transaction begin enough transaction transaction begin iteration argue th complete invariant thesis perform online greedy grow assume add transaction build conjunction class constant vc always build transaction item interested use vc much dataset equivalent vc also tight bind empirical vc monotone vc construct algorithm guarantee collection construct obtain algorithm exact appropriately formalize transaction build ground index thm know mean particular satisfie f sx guarantee def def stress absolute interesting know advance minimum build use sample dataset transaction depend transaction less different relative constant pf sx x let sx
factor cost mini optimization minibatch minibatch pass data minibatch pass suggest imply communication small minibatch large cost mini cost per yield overall still empirically note minibatch another algorithm use algorithm single family delay minibatch minibatch backpropagation differ substantially communication cost format run complexity since compatible yield combination training programming rapid online high batch percent performance reveal drive benefit gain bfgs effectiveness figure discuss loading scheduling affect approach oppose nonetheless scheduling overall paper choice linear system upon training predictor ik ik ik present learn predictor dataset million hour careful synthesis conduct describe show choice see grow
involve area book function involve histogram sequence topology notice main together rate estimator assumption hereafter differentiable nonnegative tend argument uniformly f xu vx number cm iii iv usual infinite space dimension ball al ib hereafter illustrate condition induce principle abstract metric element together
show finite lagrangian eq derivative respect belief lagrange relate lagrangian solution beliefs lagrangian combine approach belief versa constant ensure appendix continuous derivation present lebesgue whenever correspond random continuous pass value suggestion factor pmf restrict real find code however base always always sequel solution approximation want minimize constraint setting suppose ix ib aa constraint fulfil stationary exclude lagrangian belief exclude vice versa hold include special appendix contradiction marginalization indeed x marginalization constraint together imply cycle message pass iterate belief bp bp cycle point initialize
ideal differentiable everywhere desirable purpose estimator vanish encode digital error simply uniformly digital practice atom example piecewise image atom atom efficiently encode atom mb residual mm triangular causality aspect consequence depict along atom patch causal bi atom linear prediction residual assume describe dictionary update compute iterate later know image patch dictionary patch arrange bi predictor template outside patch atom k example quantization dictionary learn alternate estimator mapping optimal ml parameter part significant fix late atom prediction see use atom term share decomposition assumption encode sequentially encode distribution encode suppose encoding coding encoding pre encode th laplacian form universal counterpart original convex support able minimize zero example already encode encode
resp resp mean description additive criterion kt ji histogram histogram worth standard wasserstein distance accord accordingly scheme criterion weight criterion satisfy relate dispersion high respective weight criterion restriction cluster restriction close respective weight perform accord use lagrange multiplier method sake without prototype accord rule md property convergence stationary convergence partition criterion decrease converge inequality inequality hold finally hold finally decrease bound converge stationary equality know minimize equality operation computation wasserstein distance operation weight allocation operation order algorithm histogram htbp eqn accord prototype eqs win cluster k
generate sampling ip mean bivariate logistic column fourth one show fix row sample marginal sis rejection polynomial time term arbitrary implement method interesting table condition thank section definition question sequential bind notice importance sis constraint equal distribution sis table uniform contingency
paper restriction adversary end necessary separable tight state topology topology allow require restriction player minimax restriction necessary q distribution tx x tx joint distribution restriction adversary round come apply nevertheless restriction strategy satisfy bound answer question even though supremum immediate question regret play minimax adversary latter case yet holding achieve importantly infimum move minimax move learn restriction online external obtain restriction sequential arise affect roughly speak introduce sequence tangent selector tx sign arise eq mapping conditioning make sign difficulty sequence bad notational seem employ version depth select sequence notation write tt understand restriction satisfie restriction pair recursive eq gain understand tree root next say child via p reveal greatly simplify nevertheless describe allow unified notion rademacher
away hz negative triangle inequality object general find know however manifold obvious point always calculation disk choice benefit least utilize online identify apply tx come semi matrix identify mining mahalanobi tw yet need remain tractable spread consist work approximation factor greatly typical operation complexity development mahalanobis distance geometric rotation identify r r induce tangent identify horizontal tangent tangent vector horizontal satisfies riemannian element class elementary computation hz sequence normally invariant dominant complete fact subspace stable equilibrium average invariant indicate unstable equilibria simulation loss coincide horizontal apply course hadamard way nevertheless violate moment begin preliminary matrix ij I tend parameter average origin ii ii ii assumption curvature positive
code adapt perform per empirical weighting noise standard sx repetition range weight experiment guarantee analyze rademacher complexity marginal degenerate netflix dataset empirically fill zero rip rip rip ts recall maximal q proof case row turn possibly marginal uniform modify proceeding j fp mp f mx fp r assume lipschitz bound sample term expectation ij sn identical argument proof
conjecture measure conjecture normal next approximate even proximity simultaneous describe extent measurement abundance protein true difference difference unknown protein protein compare bayes abundance measure laboratory institute breast er pr breast cancer transform positivity abundance abundance protein iid health condition cancer condition cancer respective distribution likewise standardize
research increase cost crowdsource lead worker worker less isolated effect keep still need extension view worker another example worker leave choose want stop batch middle systematically worker come repeat task amazon difficult task start practice way condition payment complete batch assign take batch complete task want worker task assign might worker finish restrict model crowdsource time tail law finish optimize pricing continuously stay available fast complete worker response time strategy crowdsource assignment crowdsource systematic crowdsource paper well crowdsource together importantly establish optimality voting majority voting majority worker crowd error since regardless voting provably upon reliable worker response learn worker worker idea expectation maximization consider label response give algorithmic base commonly crowdsource classification step probability maximize answer estimate probability recently number algorithms variety crowdsource em apply classification find get considerable advantage despite popularity give guarantee initialization make difficult task understand matter question task allocation inference together devise algorithmic total cost want strong performance fundamental contribution design reliable crowdsource allocation allocation optimal low propagation bad case distribution oracle estimator target crowd collective less task inference runtime factor conversely worker bad constant budget main show non bad use strategy attribute fact amazon worker exploit worker message worker truly benefit
unary form discuss histogram define ease exposition domain overlap frequency rectangle kind pl lipschitz lipschitz f convex r r application n minimize provably achieve goal set r incur minimize erm solve pt prove bound relate function fu l fu solution least serve proxy original decreased haar wavelet wavelet serve tool regular signal purpose haar wavelet piece wise vector parsimonious representation haar orthogonal transformation piecewise dimension wavelet transform several exist life mostly nearly learn theoretic tuning architecture tune histogram operator maintain feedback record abuse expression although available period assume independent initial histogram theoretic tuning histogram width histogram present learn setting handle reduction recovery static histogram multidimensional histogram formulation query fix goal incur unseen distribution query n ns rf cardinality query l problem solution
live negligible run could high evidence properly familiar uncertainty live sampling procedure converge measured uncertainty increase live explain independent live merge live simply merge sort step reach uncertainty necessary validate analytic verify indeed analytically scale cube box I fractional analytic examine evidence volume volume analytic uncertainty respectively agree gaussian nearly examine
determine content local object propose close roughly current compute move say phase greedy keeps find object close write comparison phase note phase terminate set probability happen sum use terminate word selection object define geometric fact number h x law learn call take place return zero occur source follow relationship proof show search cost upper us part stochastically comparison assume strong object content lift object embed target leave include upper np characterize consider approximation algorithm space exploit heterogeneity
choose might sensitivity select supervised learner sensitivity marginal risk ef rf x f notational f using let hx fx n constant h denote define everything adaptation h logarithmic powerful certain boundary support number supervise
section follow horizon integer consequence conclude use deferred expansion probability quantity analyse rigorously quantify next quantity appendix probability integer explicit result quantification mean drift next approximate u z z w thus conclude use computation give proof estimate express equation suffice bound corollary p p right hand equal zero consequently schwarz schwarz end covariance sd hold martingale rescale finite time generality integer satisfy k convenience suffice establish k iterate computable constant increment bind integer hold prove suppose state drift bound k schwarz inequality suppose binomial equation conclude noise converge weakly motion weakly prove sd sc priori lemma term proof conclusion condition cauchy readily consequently conclusion
potential estimate weight significant however pixel nature part finally combination potential main determine configuration former future contribution herein could determination configuration alone significant problem option handle configuration pose minimization general hard approximate decade graph differ substantially class complex pixel take move part effect appearance shape drastically one consider label label say scope reach addition centroid ideal relative nonparametric convenient ideal location tie directly pixel adopt hasting mh mh eventually detailed clique gibbs exist walk probability configuration propose markov act move say proposal dirac normalize part
say margin marginal margin information marginal function conversely value close essentially reflect subsequent intuitively margin condition opposite indeed ensure whereas quantify extent condition latter nontrivial policy straightforward nontrivial show condition bandit solve covariate point wide range local surely piecewise subset covariate partition theoretic sense collection hereafter call bin analyze convenient bin th time bin arm expectation successive arm sequence therefore observation static lead policy precisely integer fix static bandit arm successive yield mean parameter policy initialize initial arm outline
motivated arcs line indicate conditioning figure illustrate associated diagram chain implie unless state independence model g graph independence change independence pair directly figure path ci em trivial extension primitive induce note primitive induce primitive induce lemma property maximal graph side pointing towards imply primitive hold ij contradiction connect path must ji j l identical edge contradict short li primitive induce connect necessary analogous contain primitive induce adjacent node primitive induce connect give internal internal may pick
scheme result even section one present introduce crucial polytope eq unit take since always program introduce express convex body definition give hull word perspective minimization face pass ray point face reveal index nonzero figure concept figure see hold close face geometric consider construct hull face figure red plane origin grey space intuitively subspace outside face far lie outside face lie see close lie approximate restrict finally sufficient subspace direction depict dot line blue point coherence direction onto polytope help understand blue draw polytope blue almost volume red total volume propose understanding limitation ssc analysis restrict understand spectral gap ssc ssc reader indicate ssc corrupt trajectory research subspace ambient beyond spectral metric subspace detection hold intersect ssc affinity subspace
significant find separable plausible impose hope study nonnegative great machine currently heuristic approach popular nmf separability hope concept thank david discussion lemma proposition fact restriction constraint remark remark theorem support nsf grant nsf grant support part grant dms nsf innovation fellowship nonnegative factorization problem nonnegative make sense minimize norm rich mechanic etc past decade popular variety heuristic without restriction optimally study problem give constant nmf small complement substantial give identify algorithm provably believe nonnegative entry matrix refer factorization factorization nonnegative rank goal nonnegative rank note make ask norm inner dimension minimize nonnegative problem exactly decomposition fundamental context heuristic find extract relationship segmentation history name self resolution biology g research natural factorization nonnegative e chemical law mass optimization call slack boolean polynomially deterministic complexity equivalent real rank matrix application statistic quantum mechanic run factorization small nmf problem large
modify full edge code adopt storage graphical edge update integration available platform package frame edge code v v around set end break gs gray fill gray corner font node execute execute execute cell anchor south node anchor west west south anchor south anchor
regret sense assumption extensively sparse advance dimensional j oracle roughly thus sublinear sublinear thus seem ideal support paper regret achievable logarithmic indeed derive call bound belong learn discuss see deterministic online counterpart set latter express number forecaster observe pair fix capital paragraph integrable conditionally forecaster regression depend risk fx p tx notation typical probability thus trade coordinate df scenario oracle derive via selection argument fix design body dedicate computationally nearly prove reference work account regularization focus weight sharp design reach guarantee regularization efficiency algorithm sparsity oracle inequality almost approximated numerically reasonable online inspire langevin properties technical focus sparsity bound comment rewrite form
differentiable close differentiable lagrange term throughout index tuple negative integer pp univariate taylor theorem generalize eq polynomial define multiply side q like shift dimensional express product similarly follow eq multivariate representation property recurrence one taylor description algorithm notation point query tree r point query give notation relevant query g rr obey additive disjoint specify type notation represent change change use four different I moment centroid field form representative point inside accumulation inside like expansion way light choose format inform science perspective structure discrete aspect one expansion kernel sum express taylor infinite term tolerance series representation kernel region expansion geometric u expansion reference query expand thus impose along expansion center far expand small truncation order incur discuss evaluate
grid fail reconstruct graph toy root ss belong short path connect distance write equation row equal ss semidefinite hard trivial periodic connect four close neighbor ss p expansion taylor expansion power label node neighboring denote neighbor node denote periodic symmetry know write expansion write calculation expansion fashion term appear thus irrelevant computation configuration zero contribute two basic configuration must negative front picture configuration count track type produce counting express c ex ex basic counting configuration figure ex thus negative spin neighborhood second picture take count state type record ht negative boundary number ex thus write basic state one build positive negative table ex expansion e e write follow expansion put everything together obtain expansion put everything expansion put everything involve
bar question tuple system game consider chain piece chain look table possess tuple entry entry tuple simply form tuple output network sum approximate function I probability train td tuple generate playing since game van need td game seem organization behaviour present
could topic appear release propose two change whereas frequency area probably stage people individually another alarm dynamically value increase false meanwhile early topic value job advance practice far concerned text image video etc topic stream focus reflect behaviour model per detection
assessment may attempt truly use package test speed repeat work processor small critical conventional challenge produce stream release library generator library work graphic work generator small size short order exceed limited individual statistical quality adequate case statistical application default package feature perhaps adapt period optimisation quality gpu brief
draw represent sample suggest advanced algorithm region guarantee definite regardless collect quickly model online complicated anonymous website course user visit website ever website result objective ad site sale version week ad effect ad build ad decay week week consist five ad stock ad stock depend ad along nonlinear representation give structure high attempt estimation prove work conditionally data likelihood algorithmic software posterior mode hessian proposal mode covariance time inverse assume log generate efficiently dramatically collect draw
solve effort problem algorithm distribute exist multiple parallel build directly scale guarantee algorithmic challenge computationally guarantee subtle spread may retain divide divide expect preserved face divide problem whether design relate statistical property freedom overall machine algorithms factorization wide practical filtering recommender reference therein link web video surveillance graphical modeling focus rank noisy factorization recover matrix corruption outlier arbitrary significant propose factorization problem completion vast majority tackle formulation solution formulation rely nuclear admit variety matrix completion inherently repeat truncate decomposition limit scalability previous attempt burden approximation justification factorization propose divide randomly factorization subproblem subproblem parallel regularize formulation combine subproblem provide statistical underlying variant set new novel characterization illustrate experimental collaborative video finally sec matrix th
accomplish structure minimize svd individual first remove individual iterative supplementary reducing via red value monotone decrease thus improve joint individual structure algorithm pattern depict object common independent add represent size row joint visually phenomenon contribute structure correspond measure present example effect microarray show estimate true correspond negative variability scale permutation material rank joint permutation identify variation explain figure amount share variation type show responsible variation gene considerable amount variation influence residual order reveal share pattern joint
version deterministic require movement optimal reinforcement learn agent stochastic path thing description bit dot position ms pac position whether blue second kind feature necessary rl action north south west however typical consist hundred still need mid knowledge module hand combine module combination pac attractive illustration environment life characteristic task environment task separable mutual rl utilize technique extraction architecture shall consider kind advantageous break tie e decide like go dot direction near go towards go opposite example construction publication dot distance blue extend rule rule easily decision high condition fulfil execute make action module intuitively activate important rule module execute rule create assign action rule
potential analysis signal current search important propose anomaly show physics search feasible manner grateful access financial institute physics box classification use high energy fall machine event systematically inaccurate mc dependent search semi additional gaussian recognition anomalous excess misspecification supervise fail correctly anomaly suffer
chemical biological medical goal software derive reconstruction topological descriptor topological descriptor show improve pls input guide selection approach much sample correlate least pls nonlinear limit linear combination largely outperform pls remainder follow introduce pls discuss
simply condition satisfied argue p random measure also absolutely respect lebesgue rotation result hold absolutely apply desire matching scalar precision discuss amount recommend side free evidence nan various evidence obtain plot particularly summary bayes course reject achieve search favorable prior know alternative explain bayes factor harmonic overall bayes specification efficient imputation inner approximate different adaptation liu outer show stick breaking eq label track tie three set n
machine approximation basis pursuit forward forward greedy algorithm perform forward weak eigenvalue due circular requirement require control taylor behave extension occur address first part analyze backward statistical general reader case backward log come strong guarantee objective outside model well counterpart theoretically experimentally condition impose strong graph recovery node
scalability large comparison fix case large accelerate complete ratio os imply reveal use rigorous deal question broad trust minimization scheme factorization gauss superior rank viewpoint trust implementation guess stepsize plot trust region scheme trust scheme stop x trust competitive although superior per iteration suffer scale iteration scalability issue bottleneck rank iterate low issue accelerate additional heuristic lead thresholding project fix technique stop either absolute trust region addition stop max op create recursive relax criterion trust stop increment duality gap suggest competitive rank considerable rank rank increase perform surprising rank effort full moderate however heuristic truncation truncation iterate move compute
goodness compete conditional evidence maximize exponential family optimum posterior theoretic goodness fit frequentist since bootstrappe estimation need thereby evaluate related model nest apply give goodness bic nothing theory incorrect book
work branching represent uncertainty planning horizon limit planning especially exploit water reservoir option price resource also complexity problem schedule paper branching generating scenario need ask scenario tree tree draw scenario branch detection change scenario generation scheme tree inside procedure paper structure procedure good specification amenable fast quickly generate decision large conduct various guarantee potential decision stage impose successful remainder follow explain procedure implementation numerically describe propose report conclude explain say scalar constant distribution problem assume scenario represent determine transition arcs determine path particular obtain multiply arc
way implementation question high efficient sampler improve approximation serve abc computational abc compete computing day despite drawback abc sufficient method principle beyond triplet tuple fall uncertainty helpful feedback calculate bayesian likelihood rely estimate high application risk unlikely free max modeling motivate predict environmental heat impact event natural spatial spatial statistic model design suit focus increase probability high event modeling take maxima temporal block year maxima block environmental maxima location develop several location theory rapidly spatial infinite maxima estimate promising limiting field
I occur soon occur avoid every link rule side repeatedly side detection sense attain quantify delay change apply repeatedly choice mix open plan nan hypothesis composite rule arise multi decide population hypothesis advantage counterpart minimize order suboptimal usually continuous rule main find discrete sequential maximal kullback leibler negligible sequential hypothesis nearly detection proof lemma h well definition time therefore without restrict ct tc c suffice acknowledgement us paper
shall call course toy polynomial feature complicated g mode would relevant statistical property expressive power bayesian integration overfitte conjunction rich perhaps allow parametric restriction difficulty lebesgue derivation respect measure rigorous may issue feature jacobian correction fully determinant product exponential nonparametric shift ignore n important exponential laplacian fall issue multiplicative derivation implicitly dimensionality involve improper however argue space improper
regressor rbf result outline suggest task optimize nonlinearity matrix non significant regressor analysis improvement entire discuss differential equation time series mild behaviour follow stepsize generate training depict last sequence used unit modeling comprise sigmoid matrix randomly assign respectively unit feed constant respective linear feedback uniform include state equation correlate regressor space weight first discard design train ordinary find show compound regressor variance training training fed reservoir cause generate regressor low mse design analyze easy variance drop regressor autoregressive generate training observation probably generate rbf flexible likely autoregressive may performance ht regressor free examine regressor table locally regularize smoothed compare sequence order locally rbf
hide several perhaps common however convenient construction space consist bi sequence course employ time define history convenient adopt finite set word symbol exclude word space bi x primarily interested shift sure convergence probability w follow fact concern word probability throughout mention uv uv primary type state simple focus restrict hidden markov output although generalization set alphabet symbol matrix represent state state normally I xt overall overall symbol label edge depict length reach two symbol label transition matrix state alphabet single parameter operation think random walk associate direct current determine probability hmm output symbol labeling generate matrix interested symbol generate observer hmm direct access symbol state symbol
adaptively learn explore posterior proposal distribution give covariance theoretical motivation choice scale dimension present recursively online recursion mechanism path metropolis hasting framework posterior design algorithm smc develop allow achieve first rao via kalman factor present idea discuss secondly know sir latent latent perform weight adaptive rao particle appendix detail rao sir construct discussion reduce variance importance sampling conditional gaussian kalman filter increase advantage formulation importance likelihood beneficial perform give convolution integral algorithm rao present path acceptance involve development kalman scheme conjunction sir particle decomposition filter filtering seek exploit utilize sir particle resample kalman filter easily sir particle context sir
interact collect consideration precisely ignore fact communication network differently depend amount artificial ability predict remain behavioral phone record link decay advantage fact link dyadic communication suggest bias self foundation fact research social phone share strong connection without necessarily phone mention tie behavioral observational really produce observational behavioral interaction produce dynamical pattern formation decay social structural property already validity cell phone interaction accurately predict measure traditional close one cell phone communication study rely yet usage email cell phone display basic small community connectivity paper insight bring mostly due fact binary weighted supervise science social discover scope million million communication traditional regression technique tackle dyadic structural link without incorporate assumption form homogeneity size remainder organize briefly social connect identify factor decay largely literature computer tool hand describe basic distributional feature choose task decay section analyze
iteration least solution ti ta let furthermore last eq next subset equation rip furthermore q fact j md optima although fast optima pursuit pursuit corollary paper compress goal recover propose hard thresholding lead iterative extreme like htp novel pursuit like add exactly residual unlike omp remove simple prove term restrict isometry property omp rip locality hashing provably proof technique novel algorithm pursuit provide experimental provide vector demonstrate learn greedy call pursuit add large gradient method guarantee omp isometry popular locality provably connection thresholding recently popular extend
residual prediction filter steady application impulse system divide sparse whose impulse sparse system impulse response include acoustic wireless channel compressive sensing attract sparse recognize signal communication propose estimation work group homotopy lasso ii recursive manner
ever quantile rescale accepted category latter acceptance abc color contour ep posterior across diameter run ep abc exhibit extremely poor triangle posterior computed stress later scale end value several million auto case show fig contour ep ep abc hope fairly since acceptance enough adequate obtain varie range truncate dimension locate variance third ep arguably rather poorly without help produce gaussian already may propagate wishart matrix consider accommodate treat constraint ep abc simulate ep simulated model maximum alpha distribute innovation incorporate certain easy discusse illustrate composite technique intractable review sake
denote b solution depend solution must small local discard two eq condition solution unique see decrease point variational eqs fix require efficient new lagrange multiplier iw iw I remain denote similarity recently pair pmf variational pmf variational consider govern relate present input layer factorize specify relate make thus model paper pmf distribution place contrary factorize approach restriction posterior delta function optimization find search solution forward hyperparameter validation initialize warm dm b I decrease sign linear ignore eps
exist neighborhood divergence follow unique upon define divergence maximum belong obtain mind get formula instrumental divergence type element dual divergence divergence normal location p n x
friend friend comment link produce united site american english produce united site english analysis uk additionally user web interface facebook mobile character comment q character country mode index user share share comment character uk american english include examine uk post comment country regard estimate combination mode write comment user long comment mobile interface user difference search cause datum analyst likely want hypothesis comment length mode software base compute four mean level seed generator compute comment could factor different four comparison analysis difference factor inspection confirm attribute chance account random factor confidence quantile variance
conduct line combine nevertheless preprocesse issue preprocesse incur cost process many cross validation set truly large dataset cost preprocesse magnitude loading gpu preprocessing easily load hashing replace need mapping hash practice bit hashing inherently discuss training unique interestingly difficult researcher obtain truly dataset experiment hash gb may overcome gb format still
stimulus stimulus numerically onto grey stimulus density stimulus front consecutive stimulus strict sense process probability stationarity ergodicity realization characteristic ergodic show forget learn multiply integrate consider average prove ergodic ergodic replace brevity integral convolution dirac minus
variability base software thank provide thank stein anonymous helpful early article acknowledge cancer uk supplement explanation study apply fellowship grant grant identify individually affect involve binary property suit differ place implement search define group predictor jointly influence testing obeys violate offer introduction study drive genome ever detail aim detect genomic variant link detect understanding variant effect pick fairly readily subtle association obvious view regression predictor either association predictor absence mutation variant neutral association abundance examine currently available characterized influence fit possibility affect specify examine
player strategy advantage good exploit whereas large play play player select response action rule strategy use belief strategy treat environment game stationary implicitly observation poor play treat weight play opponent probability play formula section automatically adapt opponent player belief stream available class fitting stream fix multinomial assume frequency datum describe real stream adapt player strategy simultaneously widely impact old window jump segment window form approach window complicated window method another introduce geometric discount old recent distribution evolve advance result drift
use directly bethe usually analytically contrast use move location bethe energy pseudo moment overcomplete representation convenient express affine rectangular gradient bethe eq zero linear free place marginal realize converse whose exist poor sometimes surprising claim contrary matching reach fix point target belief set belief parameter belief propagation marginal sufficient bethe local hessian bethe hessian subspace consistency hessian
violate hyperplane separate via check constraint automatically separate check algorithm origin run positive aim hyperplane put policy candidate hyperplane wrong side hyperplane find optimization negative direction hyperplane policy perceptron number collect policy perceptron rewrite xx u constraint ellipsoid program separation oracle separate hyperplane get complete carefully iterative call infeasible policy reward delay choose present reward action delay modify incorporate delay tx w ta tx tr delay unchanged thus replace acknowledgement several reference value variable
challenge give rise class precisely interested risk infimum hausdorff equation find hausdorff use metric assess beyond sometimes differ phenomenon deal hausdorff distance sup eliminate additive manifold relate deconvolution problem support noise deal purpose field geometry paper draw way quite exception estimator define ball choose share certain topological closeness
amongst actually include follow verify order l l c separation able thus considerably mean great meaning present individual aggregate population determine nature heterogeneity base point uniformity range entire conclude interpretation population heterogeneity support diverse temporal due aggregation population bar time hour great mean aggregated metric uniformity reasonably represent confirm concluding homogeneous present population population l l quantitie source datum eight heterogeneity see magnitude argue interpretation homogeneity study well pdfs respective could careful population constant fig kde confirm lack var observe quantifie support diversity variation justify patient bin measure homogeneity support correct diverse less pdfs point homogeneous totally figure confirm visually minima integrated essentially see pdfs see relatively small homogeneity independent detail diverse information make overall finally homogeneous small table really interpret uniformity support range give imply population diverse move include appear great apparent contradiction imply somewhat homogeneous calculated diversity diversity section confirm patient patient ten bin conclude population support temporal aggregation exist individual population diversity reflect represent represent finally patient patient estimate scale represent population filter relatively essentially essentially individual proportional contribution aggregate heterogeneity homogeneity calculation difference particular substantially information include patient heterogeneity population estimate member overall sample mass pdfs bin filter
select message length notion empirical bernoulli plus message mass computable bernoulli variable description kolmogorov normalize finite distance two new notion explain former individual kolmogorov object former finite receiver review kolmogorov string finite string alphabet string denote string letter string identify emphasis convenience alphabet encode theory neutral alphabet cardinality encode bit length
complex signal corrupt independent complex mr imaging visualization sample magnitude remain contrary noise mean variance magnitude consequently design magnitude mr gain attention past strategy distinguished treat square mr chi freedom proportional noise magnitude appeal aspect noise rather enable follow maximum likelihood locally field make standard effective follow filter adaptation filter develop effective approach distance magnitude mr relate transform significant reduce sharp mr denoise account subsequently unbiased coefficient square coefficient exhibit signal easily thresholde transform
large I decrease tw get bind notice w w h reduce applicable use simultaneously actually probability fix small refined deal integer assume cover w dimensional w pass point whether grid upper combining lipschitz scale get fourth unlike w lemma combining lipschitz bound simultaneously tell cover class cover
satisfy unknown modal hypothesis algorithm confidence tool suffice learn fundamental fact kolmogorov distance asymptotically support kp vc independent drawn follow mi mp vc also follow subset distance say close increase resp ready state agnostic learner hypothesis aforementione domain interval hypothesis obtain tv stress monotone perfectly agnostic easily result agnostic non final distribution one routine place know whether hypothesis another hypothesis testing accuracy hypothesis generate hypothesis routine sort draw hypothesis hypothesis sake algorithm modal distribution result property subroutine ignore mass use learn matter identify bad mass interval ignore use take kind deduce bin get probably ok give sample essentially logarithmic factor output theorem confidence partitioning consecutive disjoint empirical roughly care need support mass may error roughly global error across
principal try analyst quantify promise central definition quantification scalar seem fairly quantification first exploratory correctly central answer either justify interaction several issue subsequently appear literature arguably dominant thought projection pursuit school certain structure interesting design preferred direction maximize away central school argue search expect view literature development methodology making point unimodal modal fact development pursuit non index note index similar normality development serve demonstrate realize nothing normality exploratory pursuit success really multivariate normality simply
transform reward game uniformly locally lipschitz choose carry reward map continuous bound bind take bound map local reward smoothness total maker strategy much bad choice function measure implement exp behaviour maker performance sequentially interact budget belief period range include range market maker market market start asset behaviour belief note knowledge price interact market share oppose infinite case market bandit maker exp
disjoint also generate overlap index among use vertex represent block connect block dependent vertex correspond instance additive various linearity consequently include graph special graphical correlation establish graphical matrix construct specific link large cluster explanatory form structure corresponding focus assume grouping time invariant study belong unknown function unknown spatial another modern robust noise structure meet sparse eigenvalue modern however hard different grouping proper grouping medical care service item item less medical price mat material implicit price consumption service fourth group grouping close actual put service service grouping drive lie situation operator sparsity quantify rate novel context
kernel propose svms rkhs parameter quantity describe paper interesting area volatility finance consider concern range chemical logarithm ratio receive source single input reflect detail show leave svms look black middle value
achieve optimal estimator allow increase generate consideration hold impose remove optimal kl true estimate go denote likelihood allow assumption increase glm whose true derive I combine many grow input target smoother generally encourage expert ii study increase derivative polynomial pay bad optimal evidence upper derive polynomial fix rough statement true minimize ii choice vary rate constant proposition achieve parameter smoothness smooth target
matter happen term favorable accuracy investigate artificial numerically estimator tend perform sample compare theory variance pe parametric parametric relative exist let come standard book express upper let denote asymptotic normality book depend ratio affect therefore occur pe note superior pe divergence imply overfitte occur relative ratio pe eq holds bound monotonically approach pe advantageous plain pe divergence mm define express ratio thus include density minimized deviation model significantly overfitte long include produce experimentally evaluate homogeneity task propose estimator two homogeneity test sample p homogeneity different homogeneity two test
toward divide interval average policy g see latter action play rarely update rarely might convergence possible normalize obtain equation describe express strategy toward rescale set term increase tendency profile exist show emphasize system energy temperature maximize maximize entropy relative recently free principle suggest review learn receive joint generalization idea two reward select second action agent yield term bi equation examine proceeding elaborate rest theoretic equilibrium ne ne increase equilibrium ne
xshift xshift right pre edge pre xshift bend angle auto edge cm edge yshift pre edge edge leave si leave pre xshift node distance cm bend angle auto node pre node right pre node yshift var node var u cm leave right u distance si pre xshift pre grey red link maximum capacity link infinite capacity j group structured hierarchy flow literature solve projection ball community sum sum problem depth overlapping show efficient exist generic exploit cost significantly well practice similarity network simplify equivalence priori proximal operator regularization explain section compute flow cm step jj uv tv e j u solve weak optimal cost flow capacity constraint arc flow exist algorithm return cost reach flow bind achievable cut part exist detail optimal cost arc remove optimization call correctness far appendix min max flow experimentally essentially max despite case guarantee weak adapt graph complexity
practice although effort improve model gaussian study wang li regularizer large microarray study parallel sphere et attempt limitation evy describe utilize optimizer train constrain effectively broad parallel performance term speed execution open optimization rely model sample must know flexible yield reliable enough curse amount spatial unless simple obviously satisfied practice grow fast performance try population large population selection pressure maintain course size level univariate solve estimate good necessarily grow validate dimension complex considerable since gaussian curse dimensionality usually model freedom histogram freedom however notice previous unimodal multimodal limitation curse specifically suppose population move search guarantee variance although develop likelihood improve effectiveness high search still validation curse restrict application dimensional exclude fitness building cost however multivariate rapidly increase step whose fitness evaluation consume multivariate runtime concentrate analytical data complexity last generation individual complexity please l model share differ parameter share computational via matrix factorization factorization likelihood estimation step computational complexity cubic whereas graphical factorization relevant algorithm state idea variance cost complexity say idea different manner analytical calculation difficult offer analytical computational consider fact representative sufficient solving grow univariate overall grow multivariate fast normal idea practice overall multivariate thus experimental comparison cpu nature search computational population univariate find complexity speak gaussian effectively criterion advantage
shrinkage allow conjugate lead gain onto model flexible prior shrinkage obtain scale variable representation tail shift strong shrinkage density neighborhood tail equivalence three hierarchical mention early ab equivalence work mixture lead flexibility shrinkage make worth prior take mix identical ng first fixing mix global act hierarchy choose mix
efficiently range encode arithmetic decoder character arithmetic tell character character arithmetic know character point arithmetic decoder need typically character know reach technique store additional along file length although arithmetic achieve compression factor prevent file disk overhead storing limitation prevent overhead arithmetic string ccccc ccccc cccc cp ab ac da algorithm consistently compression create character alphabet character character string represent character character character likely occur character assign predict character entire one history input occur character character immediately match history character character short one match less occur chance create entirely context match various context partially responsible implementation technique model count context character low model character bayesian nonparametric model share similarity implementation compression good way compression performance file however metric use
zhang project matrix apply mathematics pp zhang l z low analysis zhang z nuclear problem mc rank great science especially publication al nuclear low suitable strongly convex rank recovery sufficient lead recovery rapidly recovery rank approximately occur science range machine vision attention publication recovery nuclear matrix low fundamental impact engineering recently network keyword low capture recovery strongly convex exact guide choose namely completion
curve correspond transformation support visualization possibility extend optimal design another design constraint design design convex origin ray candidate difficulty approach difficulty constrain problem acknowledgement author
handle instance hz rational cover regret rate mirror method majority analyze purely stability underlie batch pick introduce notion online stability relate derive setting number open question sufficient rate guarantee show counter otherwise example learnable exist loo verify online potentially loo exponential learnable randomized loo stable open covering existence loo stable online section institute university stability quantify output small characterize convergence necessary sufficient necessary stability online hypothesis good stability stability uniform set online popular learner category mirror majority guarantee property example suggest might set sufficient class learnable learn sub exponential online stability exist observe sequence pick hypothesis incur loss next functional mh goal
compete conclude bold letter number upper letter stand write stack successively triangular express must coherent upper vector frobenius dot b tb fact histogram practical application particular case scalar histogram represent column vector canonical row column research contingency two margin histogram define real matrix program positive homogeneous matrix point convex highlight parameterize notation mmd mr two argument distance whenever distance name variation wasserstein recently extend un histogram histogram carry whole onto extension later
take decision statistic say say separation blind decision kind exist indeed critical test sensitive grow critical quantitative regular uniform rotation theory chapter kind invariance therein invariant rotation detect deviation process preferable completely ignore mask must localize wavelet alternative spherical spirit wavelet angular spectrum noise although purpose ask kind regularity procedure regularity sphere coefficient supplement precise construct approximation meaning compare function q modification characterization wavelet xu second distance sake see ray ray nearest rely norm require smoothness framework type efficiently handle test adapt usual wavelet never recently tight frame orthonormal produce enjoy fouri obviously space sphere basis basis spherical lead invariant wavelet define analogy sphere extremely diverse inverse especially full context region happen sphere supplement empirical coefficient jk f simulation dyadic lead concentrated wavelet choose profile represent available supplement
appropriate dataset split amongst appropriate input dataset overfitte version suited learning problem seem fact lar learn category example feature classify classify classify exactly un qualitative left histogram decision illustrate ability detect svm lar baseline mainly show adapt behaviour difficulty classify confirm histogram decision input always classify input seem identify good behaviour direction concern acquisition feature main filter embed involve classifier classifier
consider response error denote distributional cumulative two regularity I vector vector contribute significantly predict response situation arise study cancer consider main seminal regard nuisance situation inference shrink coefficient full towards restrict utilize nuisance covariate plausible generality restriction constant follow shrinkage life present positive shrinkage validation describe section square define eq degree stein sign could unity view
prune gram level frame compatible sense invariant permutation correspond column take choose decomposition variant attempt find second variant decomposition hadamard output remove transform attempt tree example complementary properly proof rational quadratic form theorem long history design projective candidate gram matrix block rational rational signature agree criterion rational signature divide signature follow fail gram sometimes backtrack existence pair back tracking explore decomposition fast search
mechanism split propose scalability parallelization diversity use interact year create proposal induce range sampler development parallelization constrain chain generate distribution iteration move chain emphasize chain entirely proportion update proportion therefore indicator mcmc law similar employ approximation chain costly particularly graphic iteration single chain distribute iteration tradeoff chain multiple processing consequently collect update processor graphic wang available additionally multiple processor benefit hence flat enough chain explore mode method reaction chain provide reaction move potentially reaction mechanism space proxy movement acceptance acceptance make move rate move adaptively encourage recommend although find example test stochastic standard rate accept move particle walk chain history covariance structure target proposal help space mode yet chance eventually act standard deviation large chain computation storage history chain recurrence formulae compute covariance justify certain
log respect define let large sample exist eq hellinger study asymptotic smooth I large dedicated position conditional extreme observation slice c conditional slice variance remain stable extreme gaussian behavior statistic slice eq explicitly sake situation distribution drive opposite
object distance correspondence space quantitative rank score read agglomerative hierarchical greedy structure merge view output construct fine trivial object binary child node terminal node commonly refer dendrogram ultrametric topology ultrametric define strong topology compare geometry ultrametric symmetry though note triangular inequality ultrametric triangular ultrametric discussion linkage hierarchical terminology subgraph threshold linkage single linkage arise dissimilarity member hierarchical characterize linkage dissimilarity hold cluster employ contain dendrogram inclusion relationship partition early work field first major context dendrogram sufficiently common interpretation hierarchical type interpretation interest level approach semantic attribute approach significant hierarchy apply dendrogram wavelet ultrametric sense
signal denoise choose eq necessarily derivative nm n q def problem lagrange read lagrange multipli optimality reach two substitute dd get sufficient condition number optimizer penalize square also condition gram matrix connect general positive
optimal next empty target suffice size notice achievable q ensure step new optimal finite length end polynomially norm compete finite different rate optimum loss suppose fail constant adaboost respect loss convergence rate first convergence achieve round show polynomial minimum solution upper rate adaboost near kind derive decomposition intrinsic value feature rate doubly example rather training table c reference bound om norm reference parameter study national foundation grant helpful discussion appendix optimum table adaboost step bound round get loss round add begin edge round generality pick c pattern claim round fact strong claim recurrence prove claim verify round edge correctly incorrectly round update add
justify estimation concentrate observation previously paper aware formal proof lemma positive denote prove step back last equality isometry equality last fact hold actually unique next mean mean rather g identifiable unless additional constraint go denote lebesgue first compute distance minimize lemma asymptotically signal align population I cg c ccccc estimate illustration I generate sequence panel align function fourth finally examine error compute estimate panel converge size grow domain specifically focus alignment dataset berkeley berkeley growth subject case use growth leave top leave
microsoft microsoft com kernel integrate adopt transform efficiently scale massive extremely dimensional especially compare hash cm sketch projection empirical usually bit storage bit hash far especially internet inherently store context translation corpus practice long capacity example potentially unique architecture consequence emphasis learn technique rely solely parallelism expensive requirement often induce additional approach challenge pose leverage area increase make storage prohibitive representation compact
rank modify subtle leave implementation element eq suffice qr norm run comment regard first role randomize algorithmic leverage score column clearly unless diagonal span relative leverage cross qualitatively approximation construct randomize euclidean sketch approximations cross see theorem provide parameter statistical leverage section return least q relative assume treat running time score coherence rank ji ij c ij score satisfy within treat overall running application scale generally either preprocesse develop problem roughly hand score discussion decomposition compute importance computational probability bottleneck leverage depend fast sampling approximation problem numerical randomized algorithm traditional implementation generally quantification case algebra implication
bind statement observe assumption imply measurable apply jensen assumption see conclusion conditional summation entire define noting otherwise expectation martingale sum proof note complete common inequality see ct see equal hand berkeley support aa support microsoft fellowship google fellowship support laboratory office contract nf descent situation optimize long suitably quickly distribution enjoy strong implication optimization dependent stochastic paper analyze solve statement collection close contain statistical convex wide explore literature procedure possible receive instead index relaxation circumstance know receive may sample efficiently combinatorial design converge computational access source randomness study effect assume distribution point incremental apply problem objective access certainly control stochastic statistic study book nonetheless
evolutionary time objective view set attract significant recent extension mean expand propose cluster quality preserve cluster framework static spectral association normalize cut framework smoothness cluster similarity cluster choice temporal membership simply optimization example association q adjacency cluster membership find discuss approach share similarity pair association history work cluster attempt modify cost snapshot none aforementione address determine ad hoc manner accord preference temporal beneficial allow choose adaptively check test satisfy evolutionary preference temporal undesirable exist evolutionary agglomerative impulse sum filter calculate filter output affect adaptively dissimilarity cluster centroid time accommodate cluster deal adapt finally evolutionary addition aforementione involve mixture method semi evolution treat evolutionary problem follow already convert proximity proximity matrix denote realization non stationary discrete assume evolutionary algorithm identity row add describe propose
setup hessian available limited easy imagine comprehensive study nucleotide covariate might additional insight nearly ideal burden interestingly generic start variety I optimizer fail converge similar curvature likelihood em advantage analytic derivative surrogate numerical death flexible derive discrete hope sophisticated realistic death apply researcher estimate parameter structure currently available maintain publication grateful comment grant gm gm nsf dms california usa edu usa edu california usa edu birth death process continuous track particle biology particle death limited particle death times augmentation find algorithm close form remarkably conditional express computable probability arbitrary representation transform probability efficient
cl cl thus much day minute opposite represent sample rate day cumulative minute intermediate consumption distinguish occur cluster represent sample cumulative minute proof light everywhere four adapt gx ng converge surely check condition fulfil absolutely surely consequently gb ga get kn r n deal one get eq h instead anonymous valuable suggestion company allowing illustrate
potential theorem follow strict intersection quadratic ignore l qp decompose k kp qp z kp h tp kt kp kt qp recall decrease first invariant small see game possible assume positive say inequality follow receive major university sc degree mathematic ph economics department currently dr research interest lie system game university universit paris paris france post flexible sup university universit de france sciences fellowship candidate science receive degree ph matter physics physical division wireless technology laboratory currently physics associate system game theory statistical theory communication physics sc
usually one security credible indeed set credible credible resource contingency security frame address rapidly could analyze computational resource output main point significant turn distribution generate draw strongly sometimes engineer possible identify engineering distribution however try draw contingency instant base security correspond describe believe web attention expert random sequential expert pick
capacity tv denoise propose generalize tv multidimensional sum euclidean increment see reduce tv intuitively increment variation case imply increment profile piecewise profile share point natural classical investigate affect penalization differently position uniform unweighted suffer position choice group sharing profile additive profile length though leave weight signal change capture formulation profile lead error amplitude apparent profile dimensional tv denoise reformulate change reformulate convenient variable ii rewrite matrix attain jump center specific design group solve purpose solver want reach make sparse memory number select statistical criterion however none rely affine change section respectively suggest group efficiently place perform operation repeatedly implementation follow group optimize turn group converge report amount n
appropriate operator choice example reduce classic convex detail proximity call unconstraine major penalize constrain analyze convergence generalize differentiable begin match behave main must term bound increment compute helpful ease notation extend last z z r j j last tx
eq degree freedom property limitation deal sparse situation often small setting expression mutual test condition statistic nan compute detailed compute denote time usually observation present conditioning table configuration test permutation advantage permutation large assumption condition parametric
define infimum keep mind assume without generality make use consistency confidence define fact vector recall w quantile theorem along consider ensures far prove type set percentile summarize condition bootstrap routine bootstrap problem entire confidence detail well coverage limit leave open keep good empirical criterion nh independently divergence agreement show divergence regular exponential satisfied without parameter crucial presence criterion generate
many site site site depend note positivity concept neighbor suppose pi study minimal first case site
z z p take concatenation classifier base derivation omp capable recover even sequentially convergence almost termination stop criterion compute p form namely four proceed need replace traditional hinge hinge solution obviously algorithm quadratic impact determine weight influence cost impact previous classifier determine relationship hinge classifier high yield positively classifier imply correct misclassifie make representation classify
exp sparsity ability gaussian use high testing log difference use easy log likelihood validation graphical outperform generalization model tend moderate explain tend predict explain currently generalization latent variable expression encourage know gene interaction high wide observable marginal concentration decompose concentration matrix reveal model
zero assume estimator available ic time need statement kkt adapt refer kind regularization oppose realization exist state ic introduce q invertible inside provide bound eq inequality paper select subset sign consistency oracle asymptotic ols straightforward carry traditional actually grow polynomially sample selection consistency tell relaxed tail hold stationary tend oracle weight
continuous measure posterior consideration use work benchmark represent specify discussion moderate simple cg wide setting cg uniqueness quadratic restrict space restrict space restriction moderate principle derive frequentist statistical inference organization example bayesian confidence posterior represent frequentist represent pre modify procedure encode confidence moderate incorporate frequentist see provide undesirable inference undesirable little plausible posterior framework remark acknowledgment innovation innovation frequentist david institute
two mixture component normal mix leave histogram right histogram datum exercise break specification informative si mix mix correspond setup amount information exchange marginal posterior empirical evidence si exchange si mix mix mix concentrate si present previous raw chain burn period discard solid distribution gibbs set ergodic setting mix mix second mix h production business cycle great advance allow period regime seminal dynamic observation expansion phase business successfully estimation regime cycle verify contraction use high number business cite selection regime joint number regime multiple dirichlet vector autoregressive international economic united european
nonlinear high space carry truncation reduction suggest combine representation capture characteristic pair simulated comparison show propose nonlinear machine learn offer dynamical many estimate reduction unstable preserve stochastically perturb find remark sketch remark l department mathematics department college usa ac uk control draw progress statistical reduction method behave infinite dimensional balanced truncation carry implicitly nonlinear belong reproduce hilbert capture output approach control dynamical long benefit controller simulate cost simple important might light underlie linear system detail model proceed dimension system preserve essential directly balance system considerably
second one global decrease root stationary first generality transformation problem transformation invertible give vice versa therefore consider rational disjoint subset equal number instance x j j equality always equality apply equal
still effect identifiable condition disjoint relation x back graphical check know conceptually ordinary author like corollary model signal acyclic dag one occur probability characterization inference focus formalism particular notion intervention kernel direct identifiability passive back thought causality recently attract attention signal process researcher state imply relationship theoretic conditional
geodesic pair classify social interaction formation node leave component synthetic graph observe discover center generate extend treat set dp parameter set control concentration package figure depict summary box display degeneracy relate empty complete graph sign degeneracy show sign degeneracy vector lie relative hull generation realistic family extend right initial htb cccc observe line box plot include variance network degeneracy htb cccc observe indicate box
arm player player lead upper exploitation spend exploitation epoch fact exploitation epoch combine regret get reward cause mistake lead upper b theorem imply part regret cause epoch cause mistake exploitation epoch lower arm regret cause play arm epoch cause exploitation spend bound dt lt I thus
bayesian paths lda illustrate constrain dictionary learn topic coefficient manually expect conference section induce efficient consider function many sum additional interaction order make express potentially subset main many difficulty vector reproduce hilbert space complexity address impose structure among see reproduce space arguably since learn belong necessarily predictor reproduce hilbert space predictor parameterize consider state function admit f expression eq whose element k long assume space look j j j norm turn solution detail mkl define set pm desire functional limitation complex expansion functional expansion different subset theory function sparse predictor feasible theoretically
setup detail associate two come mind fit experiment experiment share wish briefly outline algorithm iteratively vector km classical iteratively centroid replace centroid computation thresholding formulation rank regression correspond constraint adapt multiple response regression modify speak neighborhood worth collaborative scalable motivated local identify estimate regression report well complexity mean ce method near exploit estimate coincide size method individual collective
digit share digit common ultrametric also ht digit first digit digit ultrametric produce classification see measurement say speak level level node course potential empty observation number precision point big storage give fall advantage viewpoint store fast easy
learn removed turn disease rank remove list disease purpose disease I gene curve rank mkl hide positive confirm level mkl variant illustrate learn formulation variant roughly one association mkl set assigning source try mkl mkl consuming mean decide restrict experiment refer kernel mkl heterogeneous information information sharing disease run experiment assess allow disease test disease weight across disease disease variant additionally control disease description disease differ limit disease association dataset association gene variant share disease propagation cdf compare average disease four fact outperform confirm knowledge disease phenotype weight sharing across disease fact outperform surprising illustrate fully description beginning method low rank generic practice available confirm approach comparison fair improvement run picture
lemma useful regularize satisfy exponential bridge kernel kernel verify lebesgue nonzero almost everywhere satisfy condition compact belong lebesgue argument similar manner brownian satisfie clearly brownian
implication dimensionality give algorithm construct kk opt cluster indicator correspond row opt matrix eq orthogonality opt opt drop eq combine conclude opt f opt k kk norm kk slightly give nm jj yy independent property strong version work introduce turn prove svd value clear matrix k svd certain algorithms frobenius norm svd approximation find target denote compute almost rank k give take input whose denote inverse follow projection random use event hold respective probability hold union event let r sampling critical replacement rt ip tt tt randomize sampling take sampling three lemma
show reproduce competition phase pre competition phase forecasting available different forecasting scheme training validation set partition three exclusive figure day testing figure use tune select neighbor perform performance suggest origin denote ahead forecast origin forecast day day day forecast three point forecasting ahead criterion I learn test competition generate forecast competition advantage design make pre competition model final competition b tune present discuss strategy forecasting criterion section provide average rank nan state equivalent hoc present strategy forecast configuration selection amount computation time hoc error htp post availability make pre competition sake consider move exponential smoothing nn avg exp sm configuration present hoc summarize forecasting configuration round rank htp
cover x class p md sufficient show md ip slope bx bx bx f show adapt collection eq functions bound slope bx bx bx e bx b bx b bx x bx bx produce cost bb mn mn b b cover function b bf mn x b b bb argument volume hyperplane origin complement ball ball hyperplane spherical hyperplane volume volume ball let half parameterized let spherical denote measure complement incomplete reference relationship pack every use lemma cover number b b x bx b argument part volume complement spherical minimal boundary maximal packing b follow h yield
perfect arbitrarily attempt able like obstacle project version switch optimality f show optimality decrease approach restrict true online effectively difficult online scope projection convergence guarantee obtain schedule subgradient project direction issue boost gradient single hypothesis small step learner error convergence functional restriction iteration rely describe restriction ff f gradient add dependent projection weak threshold train learner schedule
cost thresholded order magnitude furthermore graph amenable parallel separate optimization problem sub number may impossible operate distribute nested suffice operate independently path distribute architecture operate graphical small connect large split exploit induce thresholded covariance graph induce thresholded permutation appear decomposition graph labeling appear structure match every v solution component nest increase regularization within component formally address
integration seminal framework reasoning recall different base together ability bring knowledge adaptation address requirement reasoning possess active basis company institute technology mit prominent lexical maintain cognitive science laboratory university relation mit discover middle involve facilitate reasoning thereby illustrate reasoning briefly essence seminal resource point reveal want resource entity dash
entry reveal still vanish regret seem general loss outcome never one obtain batch arbitrary appear use assumption forecaster outline reader derivation forecaster actually refer prediction first forecaster minimize bad eq work start round deriving outcome reveal tp plug get q show repeat minimax recursively recursive exponentially expansion sequence bernoulli probability plug also constitute forecaster note equal definition
report ratio md dr k adjacency mean call local result sampler threshold identical local parent domain meaningful alternative pathway pathway contain table experimental cell pathway pass indirect mechanism exist widely indeed report enhance although unlikely may enhance dr landscape domain insight dominate global mode log md sampler reach mode contrary detect much negligible demonstrate md sampler burn mode mode dominant mode produce cc ten fold partition size nine calculate point subset learn repeat time dataset threshold performance md imply robustness tp md method compare panel average probability md sampler ratio utilize calculate regard use network store dr variability domain method probability calculate probability calculate md application new experimental datum nine validation cell cell one sampler dataset edge result ten experimental uncertainty determine
adversary response branch future precede e two construction tangent equivalence path allow term good rademacher therefore generalization rademacher handle range elsewhere machine know rademacher complexity furthermore rademacher class argument application determination
sequence function appropriately state give give priori number state technical gradient f generate update term tx claim sequence satisfy straightforward recall follow inequality jensen f claim consequently f f apply lemma imply turn simply probabilistic bound key take expectation randomness time lemma thus side eq fact g fix replace see remainder achieve probability rate term consequently notation consequently probability equip recall eq q lemma apply remain control complete non optimization develop provably oracle model knowledge technique smooth think least question remain necessary achieve convergence question smoothing whether corollary answer smoothing tight outline technique give provable method experiment show qualitatively develop stochastic anonymous feedback comment science engineering fellowship program support award dms
short bind site well tf outperform tf high median ic tf less utilize example tend tf high centroid factor bind fig ic ic centroid compare relatively ic property reveal content tf tf contribute median tf high median ic identify tf individual site tf bind site two scoring matrix use score assign search superior searching single cross validate couple centroid identification different similar embed bind clustered centroid centroid couple score denote embed centroid centroid binding assess centroid centroid counterpart without leave validation set tf nucleotide ic indicate weight nucleotide content impact centroid introduce centroid
conditional standard chain secondly markov produce briefly problem compose datum range per yet treat discrete variable interesting problem specifically logarithm correct enhance interaction covariate add explain construction dependency predictor multi modal short name business unit build weight distance radial full value proportion status second treat uci repository concrete strength strength compressive covariate add covariate table augment predictor explanation water aggregate aggregate age day analyze implement explain protein activity enhance interaction covariate detail predictor reason prior name ph buffer protein temperature want include interaction incorporate challenge exploration main prior feasible carlo implementation environment suppose fair sequential monte carlo stop chain carlo speed make comparison sequential carlo scheme markov effort evaluation hand however algorithm move analogue speed context chain detail report
converge therefore requirement fulfil brownian second exponential ern check computation xx xx remark gaussian consequently happen satisfied aim requirement accommodate kernel grateful anonymous approach let let validity examine role linear theorem coefficient observe positive z j form variable probability minimizer general predictor usually measure function respect precisely expect large approximation zero fast increase quantity error respectively satisfie immediately estimate start stick theorem replace relaxed point simplicity fix error accordingly factor keep advantage therefore competitive input similar result represent regularity decay still hope satisfactory
formula mode equality mode fix say median formula mode approximate
give situation incomplete subspace upper coherence vector j tu without observe wish chernoff size draw replacement second similarly p u unique implie use along j x union least denote column complete least residual subspace u allow entire since may prefer level simultaneously prove rank completion completion examine match draw span subspace incoherent generate span procedure seed software procedure implement completion result message provide rank complete term simulation
angle fan wu sis manual fan li journal american transaction pattern machine fu penalize regression bridge versus journal graphical p natural efficient inference thesis california berkeley asymptotic b monotone package manual adaptive penalization technical gamma prior lasso manual fu asymptotic statistics lee generalize biology bayesian variable journal american statistical partially adaptive american inference extreme globally act statistic gibbs sampler journal american e minimax mean r shrinkage machine machine learn r tune absolute mixture lin estimation model journal american association yu model journal
process invert reduce mostly learn successfully method lie stochastic representation obtain knot law approximation location go complex process model rich become understand difficulty approximation knot determine accuracy concept knot coverage intuitive need emphasis metric geometry help knot base knot offer face severe difficulty automatically geometry cause covariance enable geometry call predictive predictive add two knot candidate dynamic determine knot connection predictive approximation cholesky factorization well view
use generic eigenvalue hold enough complete central write prove tend divide integration part integral substituting since integral arbitrarily suffice integrable sufficiently consideration
krige use smooth classification importance derive meta converge impractical importance estimator refinement stop progress require factor acknowledgement grant engineering financial also importance simulation rare pt universit assessment system failure numerical probabilistic failure define event failure define follow introduce equal turn carlo read asymptotically unbiased variance
inner lower derive approximation packing number explicitly give pack local approximation term continuity know exist upper bind uniform h hence lemma consider topic distribution major depend shape back primary article regression idea might beyond parametrization invariant utility nonlinear clinical finding real nonlinear medical biological instance prior distribution remain external quantify example absence jeffreys variant pi
lag time lag influence influence nothing else effect quite lag fewer strong except play advance dynamic flow among price price generation source apply wind water reservoir estimate causal price relationship price well price water reservoir wind power similarity difference peak price peak peak due flexible innovation response causal link price market stand alone exchange rate price stand alone stand european may case price price adjust
involve bilinear general nonconvex bilinear np global particular program tractable show stochastic controller address discount stochastic blind controller incur sn via bellman denote np undirected loop restrict
methodology criterion analog rf incorporate empirically cf compare base cluster support cf manner formula mis perturbation insight relevant remainder paper organize detailed description relate spectral section mixture well spectral benchmark comprise phase one discuss cluster phase aggregate problem instance base feature vary vector govern quality cluster cluster square feature currently use denote iteratively expand consecutive attempt expand cluster initialize competition f bf datum space project get f setting reduce mode growth competition procedure feature competition motivated prevent noisy
xy mx x concatenation common represent string string pt minimum height input input let observation reward illustrate diagram receive new observation another goal agent discount reward consider environment action reward deterministic xy observation pair implicitly computable previous leave dependence action policy history action interact play tuple play kx define infinite reward discount geometric reason consistent discount discount horizon human grow old existence depend discount discount discount discount purpose discount prevent discount
everywhere query al adaptively pick sequence unlike accuracy requirement lower introduce realistic variation price setting agent interested establish almost tight upper low bind within lower information sample bind function learn show approximated polynomial conceptually highlight learn polynomial tree would immediate care novel result per number also low apply derive new low introduce observe agent agent interact interestingly query preserve theoretic function price query paper paper tree paper paper bind learn construction natural construct economic setting sum allocation focus set approximate learning paradigm submodular main class recent paradigm necessarily limit less class read toolbox polynomial specify price prefer bundle price contrast price bundle class distributional
qx jj x x jj introduce formulation usefulness classical formulation try dictionary follow norm control amount prevent arbitrarily belong norm become unconstrained formulation formulation remove replace solution normalize knowledge equivalent empirically behave prevent degenerate formulation constrain dictionary original equivalent unconstrained size
predict goal jacobian average execution time scale computation time estimate remarkably overhead especially evaluation estimation surprising twice score evaluation arise order author internal conduct computer yield result half time computation allow per local global optimization step depend evaluation per step number without new identity
need mechanism efficient physics machine probability carlo algorithms parameter tune often domain expert consuming manual adjust algorithm reader recent excellent review field various adaptive firstly theoretically chain update history markov ergodicity ergodicity sampler ergodic level vanish asymptotically space adaptive approximation practice limitation stochastic approximation approach rely know optimal disadvantage
evenly space simulate plot datum blue red expect present posterior focus exchangeable infer responsible parameter component hereafter early posterior analytical marginal develop draw challenging aspect countable develop cope way control thereby sampler posterior describe alternative finite base measure neither sampling successfully bayesian nonparametric contexts beta serve adaptive level stochastic proceed associate auxiliary positive sequence binomial represent gamma conjugacy component slice u k round represent conditional sample share process beta slice sample detail defer biased finite beta continuous converge may leverage approximate posterior sampler fidelity detail conditional gibbs behavior sampler toy process utility approximation component sampler gibbs slice sampler toy bar ten component beta weight exceed sampler underlie sampler attempt iteration practical unsupervised inferential task consider observation word vocabulary generate topic underlie intensity parameter vector mass binomial heuristic exact slice sampler let document tracking attack transform summary field add account seven vocabulary group document organization report claim list remove organization randomly aside next independent organization specific drawing classify document document confusion ten display slice sampler finite approximation nearly claim organization experiment
see induction take integral appear state entirely optimal utility assume subsequent calculate expect give restrict max policy simplicity policy temperature reward policy choose perform chain monte policy fig metropolis mh performing mh proposal rise describe alg enable two alg sample thus utility conjugate distribution case condition reward simple sufficient statistic continue hybrid
factor show setting moment satisfied kernel worse countable kernel interesting note result also oracle modify another scenario entropy wish thank van de ep international grant theorem conclusion theorem exercise uk regularize regularization lasso group multiple novel countable transformation
I ds fs ds gs gx justified independent polynomially hence also fact let mesh notation act help deeply c l p p bind backward z ds prove collect x assume center condition statement get claim course constraint derivative z brownian position copy lemma coefficient guarantee smoothness coefficient respective remainder derivative function respect suitable probability measure note prove c q solve existence growth solution sde ds generator sx ds
cost action parallel focus difficult adversarial arbitrarily round round leverage non bandit setup next section work connection give algorithm special small box specifically subgaussian uniqueness value point henceforth ci ci width ci separate cope unimodal unconstrained author variant rate non optima derivative vanish derivative optima yu study unimodal bandit setting al armed bandit function algorithm regret convex lipschitz grow exponentially case cost hard adversarial point good regret adversarial setup include function exploit setup bound attempt solve convex instead specify setting find feasible membership oracle convex fact noisy elegant solution random walks separation oracle membership oracle mostly noiseless much drawback often explore observe closely noisy whereby query domain produce minimizer
note generally multivariate terminology random tree structure v class consider hide hybrid combination model tree include take value variable take take vector represent relationship markov markovian addition assume recover th condition identifiability latent condition estimator rank matrix introduce additional limit rank redundancy hide furthermore hide node correlation correlate hide redundancy two node reduce soon direction ensure latent generalization condition consider remove subtree remove triple effective logarithmic achieve though variable though conditionally relationship neighbor effective depth therefore polynomially large second isotropic denote ax x determine topology subtree four possibility induce subtree return subtree output guarantee subtree topology subtree correct subtree integer respect four confidence tune apply empirical
enjoy high generalization respectively insufficient guarantee distinct setting proceed development intuition insight mix almost sample stationary et technique building condition far sequence idea combine theorem start hold first recall second use lipschitz part begin first inequality inequality expectation coefficient relate expect predictor stationary field small risk lipschitz assumption mix key test mix loss remainder complete proposition control respect apply confirm hypothesis random point remainder let denote expand sequence regret summation three summation boundedness guarantee note q use increase
become illustration use plant partition free landscape overlap assignment result marginalization permutation assignment generalizing difference produce asymptotically attractive illustrate varie observe overlap situation arise true co transition fig locate energy phase
parameter achieve wherein good bayesian consideration offer therefore decision advantage von number pilot symbol generally time deal course elegant computational synchronization prior model finite unique substantially expensive
compare show average grid performance run estimate contiguous two centre leave centre four region contiguous rectangular single three figure estimate outperform behavior consistently confirm experiment brevity cccc contiguous different centre contiguous centre centre four contiguous group underlie pixel foreground random plus noise due uniformity hard grid compressive instance image compressive besides sparse haar
scale multiplication state point demonstrate keyword path integral apply hand many intensive control purpose interpolation response slow simulation simplify account due fast produce
neighbourhood baseline relation like etc fully unsupervise upper row model full neighbourhood see experiment semi learn variant complex prior completely unsupervised posteriori complex priori model complex see belong segment generate priori reflect correct force model simple shape input image show baseline right complex well perform leave dna cell segment circular neighbourhood diameter neighbourhood grey two per segment supervise tree well complex complex figure see prior capture lead segmentation model
ac uk likelihood probability smoothing concave maximum estimator concavity estimating use justify use breast diagnosis classification smoothing constrain great deal tuning idea characteristic shape log exponential moreover drop convex smoothness drawback however circumstance attractive visual justify substantially hull rather section concave likelihood smoothing convolution density preserve concavity shape concave fully nature idea upon challenge convolution integral take figure
select representative sentence pairwise include redundancy similarity select summary pairwise summary sentence rely citation sentence give summary lot binary sentence dependency redundancy arise rank decision introduce dependency dependency sentence hmm hmm produce summary sentence expressive crf disadvantage sentence model range method closely use retrieval label available diversity coverage problem approach thus wide range add simplify furthermore closely
estimate approach aim vb population distinguish observation statistical put aside observation probability belong interest high mse evaluate average dataset mse performance achievable aim obtain weight oracle view negativity constant contain allow numerically take constraint account achieve vb oracle display different first vb good provide pe provide comment
redundancy observe variable multimodal principal easy conditional mixture model density form constrain keep reasonably ability density mixture approximation paper hereafter distribution gaussian mixture row find mode model joint course redundancy point constrain mapping absence information answer mapping turn case problem constrain obtain reconstruction redundancy nearby nearby accord distance depends continuously result continuous bounding numerically approximate derivative apply condition end sharp sharp near norm typically euclidean different contribute also form difference another often reconstruction problem forward map small candidate reconstruction effect reality reconstruction difference incorrect reconstruction spurious generally help discard incorrect reconstruction introduce involve whole define reconstruction break reconstruction analogous add constraint sequence eqs forward experimental sample length trajectory pass combination constraint global variable product candidate reconstruction combinatorial optimisation express short figure pointwise reconstruction total exponential order fortunately local bl layer define leave leave path highlight fully link associate reconstruct short case short acyclic programming dynamic optimality graph decision node layer keep figure give forward recursion dynamic programming global due symmetry backward definition set reconstruction minimal length layer il I I unlikely tie return programming link node adjacent layer achieve mm mm il mean concatenation select edge close point improve proceed edge undirected algorithm need path term
use k fact complete local iii record white wishart chi variable degree freedom value definition asymptotic iii show recall obtain integration thereby obtain since work expectation hand front give eq w local thm thm proposition university california berkeley hypothesis testing multivariate size classical working derive achieve great state roc curve simulated test parameter determine substantially situation cumulative occurrence sample become increasingly application molecular fmri analyze fmri fmri measure order hundred thousand investigate sample large several may applicable accordingly grow develop procedure well deal dimension b fundamental manner mean definite problem know define pool singular nearly seminal short study test suffer subsequent year bs sd
computationally intensive procedure reconstruction picture surprisingly vast majority image skip immediately impose nonsmooth object noisy mild assumption suitable highly work well method advantageous modern practitioner medical perform preliminary reconstruction drawback substantially study view object within algorithmic testing pixel top accuracy grow drive stop rule human stop image focus serious
wireless channel take negative network account secondary interference secondary primary user sense user platform cloud compute customer automatically load accord knowledge loading optimize derive chinese restaurant social phenomenon quality deal traditional strategy chinese customer restaurant deal price effort restaurant quality try restaurant come high restaurant quality low customer department electrical college md usa institute communication part two part chinese restaurant learn negative agent chinese restaurant influence agent study illustrate restaurant cloud social formulate chinese restaurant learn agent chinese restaurant well decision furthermore agent different decision order conclusion influence agent avoid make maximize various dynamic spectrum cognitive service
collective speed run likelihood train repeat range prediction two train validation datum prediction second precise set unable right bar trust determine final away properly parameter choose nn accept show quickly accurately likelihood thereby overhead choose network probability posterior sufficiently require hessian product slow large time calculation follow analysis obtains even likelihood large flat spectra calculation long regardless ccccc datum set
hereafter sense constrain grid grid sampling account grid square yield cause show satisfied grid propose bayesian case true bound gaussian idea value svd computational sensitivity noise power propose exceed music
weight location image order equivalent scan would read page repeat dark patch little local content dark like structure near main roughly mirror section fact anomaly g rapid etc patch detection extremely difficult patch surface anomalous usually one new parametrization patch anomalie rough patch anomaly see quantify study rough region suggest study parametrization provided introduce slow patch notice patch anomaly fast matrix entry patch follow extremely graph conversely lead concentrate diagonal walk slow patch slowly previously replace notion distance quantify time diffusion distance see section patch patch term remove evolve seminal equivalent walk characterize therein therefore patch distance notion proximity patch homogeneous markov patch evolve slow patch start g narrow bottleneck visible upper matrix diagonal take slow patch quantify computing start vertex eq I symmetric version walk fully eigenvalue label vertex emphasize transition diffusion vertex distance compute walk initialize decompose volume follow scale regime come patch agree euclidean original distance graph consequence idea manifold foundation indeed accept error first dimension patch embed
dimension guarantee count every tt x last equality follow fact class query database select accuracy bound constant database claim complete uniformly note obtain element random reconstruct occur symmetry recall private also find size usefulness extend axis align space clarity discretization query preserve proportional easily bound array bi repeatedly perform partition approximately fraction mass terminate round database mechanism differentially private query laplace query differentially sensitivity composition differentially mechanism differentially union database none laplace great every step error synthetic interval contribute contribute low axis align class nevertheless exponentially interval query mechanism possibly discretize
move center simulate prior likewise density birth death reversible ratio reason move center likewise acceptance show variance reader mean proposal acceptance choice center choose accepted move accept probability use proposal improve actually general model reversible iteration proposal implementation horizontal axis time number transition reversible respectively increase transition seed total assess reversible jump chain reversible proposal clearly efficient proposal reversible
n ix nx tracking capability active dynamic nature due weather condition change illumination adaptive strong nature keep track adapt accordingly adjust importance decision widely around detect surveillance surveillance area furthermore computer security circumstance leave capable automatic necessary security forest operate forest six feed system whenever alarm reject active fusion classifier function real slow every sub linearly weight adaptively red detecting system capture compare regular camera unless forest long distance
end last assume support lasso notice term let sufficiently deduce second must develop x replacing tx tx hand tx g case support different satisfied
independence quality equivalently implement advantage motivate far sound increasingly open expect technology discuss series markov gs use gs greedy outcome discard inconsistent indicate gs cascade error produce error approach optimize generative compute interestingly quality quality advantage avoid cascade assigning learn independence execution massive test contingency record strategy decompose markov insufficient edge add mistake edge comparison experimental publish experimental quality publish quality score versus fitting independence adaptation recently improve quality gs pc mb fast mb section develop improve accuracy axiom basis detail independently black box independence mutually large test test determine test become dependent useful example show insufficient relate axiom improve data analysis technology independence network low quality learn however motivate independence sound availability grow increasingly expect solution pose open
path unfold overlap break unfold pathway pathway mind unfold trajectory trajectory question analogous reversible p break trajectory causal unfold mechanism contact obtain subtract contact map contact reveal break pathway colored circle division number average list every unfold ensemble iii pathway representative rna description panel contact green circle secondary region b note pathway unfold pathway table respectively ms ms unfold pressure unfold obtain pathway sf cccc unfold ms ns sf ns unfold pathway sf cccc unfold ms sf first unfold event predict path average e step mean time
discrepancy accord express discrepancy size alternative statistic conjugate bayes base distribution respective therefore sufficient computing power close bring approximation realistic informative population experiment per per tree per scenario national core processor experiment approximation assess fig approximation hard assess though furthermore universit de paris universit france abc tool model al rely inter property show wide model choice information induce insufficient summary statistic effort spend necessary representative bayesian
home show player run cover accurately create central define player position start double walk walk generate run effectiveness measurement run per game team possess player base measuring depend reach advance cumulative statistic compute six outcome single triple home number average sequence performance game consist position take digit position denote base ignore markovian recurrence represent outcome state also use probability percentage express take expectation expression differ mean classify triple home home home run home use require player require result home three base score one assumption triple require walk base calculation sense statistic drawback statistic wide scenario average wide average well player average calculation show fluctuation run affect player scoring attempt quantify player production entire allow wide change rule average contain plus walk triple home technique analysis
low cluster block diagonal one course order rank exactly low rank look block one idea depict propose perform convex drop structural low entry unchanged state observe entry know surrogate k norm singular objective surrogate objective semidefinite sdp cluster tradeoff determine exactly relaxation add constraint entry constraint definite even guarantee automatically imply relaxation formulation reason importantly recovery cf section important regime tight relaxation seem benefit
criterion scoring localize characterize pattern insight heterogeneity exhibit strong regularity visually pattern regardless cancer cell type breast marker g etc capture texture image patch construct mask relevant scoring report salient contribute intensity etc description image toy indicate occur three value panel right matrix cell correspond illustrate bar successful image matrix frequency pixel intensity across neighboring pixel particular spatial relationship define direction distance distance interaction pixel involve interaction pixel choice extend involve relationship appear sufficient spatial common summary employ classification typically thousand however capability random forest subset entry order mask scoring realize choose image patch consist
communication moment enable computationally term posteriori variational robust student essentially approximate model determination kl result latent gamma parameter maximize likelihood expectation current hyperparameter drawback determine minimize reverse kl divergence uncertainty previously mcmc laplace introduce gp find form bind marginal student likelihood parameter ep hyperparameter optimize package definition ep general py un normalize iii cavity site refined approximation match marginal finding equivalent matching moment finally change repeat site normalization df refer recently scheme ep site calculate site although cost ep computation cholesky number require roughly scheme easy hyperparameter enable efficient map initialize non likelihood arise discuss improved ep robust loop site I follow controlling amount view small saddle converge find double double loop fail model consideration parameter cavity site approximate distribution respectively min condition consistent ec free energy view ec ep bethe energie double loop fix outer inner affect current
conditional price asset deviation price expect deviation uncertainty management course distribution price look design process process function technique temporal attractive view index
agree classifier eliminate classifier eliminate bad excess classifier round essentially conclusion request label request growth final feasibility set implicitly update perform calculation constrain include algorithm elegant additionally improvement algorithm well modification certainly reasonable another meta agnostic thesis natural advantage aspect yield agnostic meta way complexity achieve algorithm often publish literature extend include finite output cm implication appropriate setting follow algebra represent disagreement coefficient f improvement learning expect often significant improvement unlike author even maintain guarantee sometimes small key exploit label disagreement learn shrink proof passive break furthermore disagreement algorithm improvement passive leading region even first p n thus wish achievable term two distinct refer characterize xy xy way least rate must occur interestingly set query behavior infer agree classifier phenomenon threshold uniform satisfy scenario never focus query improve passive sufficiently point either two implication dominate vote infer agree label agree label quickly converge neighborhood request converge attempt satisfy convergence entirely general providing however early meta behavior set thesis thesis explore unlike label never fact region limit whether lead noise xy diameter bayes fraction learn difficult infer fact particular versa get rate purpose explore note specifically interested label achieve well handle one part detail follow fix achieve learn main theorem extension entirely label xy nontrivial active noise case agnostic sense work raise contribution fundamental capability active establish vc make possess favorable property passive passive label dimensionality space margin similarly whose dimensionality wang preserves dimension passive preserve passive active passive input preserve request phase reduce passive guarantee fed algorithm conditionally dependence among fed clear whether slightly request might allow passive clear passive example improvement theorem algorithm possess vast majority publish style passive subroutine construct execution learn label whose passive heuristic method passive example whose expect type passive passive satisfying early base trade nontrivial scenario ease allow factor scenario make proof careful inspection proof reveal converge remain open nontrivial definition underlie passive trivial algorithm definition implication complexity problem nontrivial label passive various select among trivial may counterpart nontrivial many label rather require nonetheless question label complexity label complexity relax specifically relax nontrivial still maintain existence class least replace say nontrivial relax achieve via reduction coin via label xy restrict single relax nontrivial possibility outcome conjecture issue achieve acceptable simply nontrivial alternative definition interesting concern consistent sequence run largely issue adapting whether efficient bind corollary request universal reduce construct label disagreement batch nontrivial universal space batch meta question whether increase universal request every might disagreement base another search sign interesting specialized execute two instance component usual guess bias via response query might require regularization construct universal space infinite vc usual condition run algorithm round especially whether continue strict improvement passive achieve specific label van improvement passive question
discrete seek relate measure fourier boolean boolean entropy fourier boolean square fourier view distribution entropy intuitively influence replace unchanged influence computer mathematical physics social theory etc survey express fourier fourier normalization coefficient
consistently indicate satisfy tune nonzero component lemma eq submatrix corresponding imply tune result establish avoid illustration dataset macro indicator cover production price start order stock exchange rate apply express special real activity economic price rate lag regressor series nd logarithm raw forecast forecast year table column unlike robust primarily lag know lag begin flexibility ahead outperform behavior universal forecasting performance purpose figure summarize classic estimator scenario able lag lag lag confirm come typical come business cycle economic development spirit motivate normal equation extend consider much difficult consider
vx related condition page page old p condition define page ergodicity ergodicity I acknowledgement helpful comment author advance fellowship capital support al foundation centre dynamic research condition remark box stochastic occur diverse unstable expand focus stability general condition incorporate possibility sequence setting theory sa find measurable measurable numerous behave suggest recursion find integral need approximate numerically focus rely exactly possible instead monte probability distribution sa dynamic obvious dedicated study section occur ergodicity term make vanish hereafter situation fail mapping zero priori projection induce difficulty sequence form cover x stable markovian noise x
redundancy support conclusion inductive set learner generalization benefit number understanding loose learning definition common sense enable fundamental literature compute match give rank amount relevance synthetic discrete feature illustrate prevent available point principled algorithms assessment subset survey main categorization feature comment tool cover experimental end conclusion work selection comparative study demonstrate datum typical lack wide fully control environment possibility redundancy availability another issue assess normally encounter feature take place leave aside aspect namely subset affect
require heart limitation protocol provide picture factor association group heart disease heart prior child homogeneous total describe family information characterize child factor I characterize child post quality ease one adjacent type factor outcome interest score cognitive development assess year group miss birth birth birth family birth family birth birth length family birth birth school education economic status birth number risk max low low head weight year length year post head year year post post score group percentage value percent completely observe observation score child head birth child seem infeasible result loss use cognitive development heart birth
give leave extra leave sufficiently allow log distance q symmetric denote corresponding eigenvalue orthonormal note symmetric real use eigenvalue eigenvector somewhat large enough second eigenvalue provide large enough uniquely classical e show eigenvalue possibly pass eigenvector deviation enough infinity pair small threshold pick estimating could take empirical indeed distribution build everything else leave node state build recursively child root reconstruct construct estimator convention form estimator u turn assume priori eigenvalue decay recursively presence
move frequentist depend review statistical mention approach inferential interpretation fact publication analysis frequentist two sometimes fire intensity time frequentist incorporate bar force inconsistent interpretation statistical general approach model statement advantage allow interpretation split framework seem I inference statement partly eliminate bayesian eliminate jeffreys subsequent hand really matter frequently focus challenge high capture theoretical figure text path break excellent
tag connect document relation relation tag author subset circle mine author intersection subset circle focus combined vision mining possibility totally biology three exclude fig quite union multiple tag distinguish tag separately document content tag prior effort topic rarely specific topic dependency tag topic mrf relations generic allow theoretically mrf topic modeling problem mrf model develop classic belief hypergraph bipartite fig pairwise important modeling structural many estimation object segmentation infer mrf structural labeling property co document tend high configuration smoothness higher equally need topic labeling encode structural design encode representative smoothness make efficient parameter rest follow introduce mrf bp inference estimation propose focus model among extensive several mining broad interest conclusion lda circle probabilistic topic text mining art lda basic label hyperparameter
unweighted weight
g uniquely determined marginal lebesgue simplex z derive direct imply let p inverse uniquely normalize almost applicable construction continuous measure convenient example projective cumulative interval nk ni ng ni ni represent way indexing index unique node pass child binary representation arbitrary associate beta measure suppose particle tree reach recursively beta ny applicability set b np satisfies b hold continuity property tree unique p tree measure measure absolutely difficultie recognize indexing space ingredient draw dirichlet recognize construction measurable excellent survey construction approach
scalar side towards word dimension compare factor estimate admit constant take eigenvalue produce constant offset illustrate explain give eigenvector maximize specifically h mis follow theory relate scalar eigenvalue negligible factor sample performance modify correction satisfy narrow range pointing well see quantify factor model bias relation analogous em equivalent requirement behave explain intuitive understanding phenomenon magnitude residual variance impose additional b straightforward scale scale outperform question tm believe
test three undirected indicate circle respectively around node weak tie community highly density gibbs distribution assign stage initial heat chain average step produce marginal show least threshold stage explore five correctly label node belong aa achieving explore nine classify perfect perhaps algorithm choose sort bar node stage central node argue uncertain learning
classifier point class discriminative learning margin learn label force away great margin slack variable formulation additionally directly metric classification classification nearest neighbor result unchanged show invariant transformation limit rate near infinity scalar laplacian trace hessian choice glm optimize local rate approach solve optimum compose eigenvalue matrix eq constant determinant unity neighbor distribution bias equal metric glm bias rise note close attain minimum minimizer identify semidefinite stand eigenvector stand diagonal negative optimizer scale determinant attempt rate generative discriminative neighbor solve result metric depend specific metric specialized
unique addition control complexity indicate set topic label approximately count example token give topic add topic approximately lda increase consistent document keep hyperparameter term sum two power prior benchmark total benchmark dataset short document length chain seed burn iteration take assignment sample average run single estimate multiple chain unsupervised set store test model lda run lag reduce autocorrelation dependency lda incorporate ten estimate chain across set run compute change sample flat lda average prior add estimate distribution conjugacy multinomial combine weight I choose use word prediction would mostly influence assignment document word assignment derivation document token need compute remain factor assignment label beta numerator abuse denote argument cd beta function function range change equation analogous lda analogous word estimate comparison highly benchmark alternative area put classification publish publish literature evaluation metric version compare able publish value least evaluation metric utilize lda goal put area demonstrate svm similar tune svm elsewhere demonstrate competitive state comparison score rank good publish publish version yahoo publish use set train split yahoo numerous discriminative additionally label least introduce account label demonstrate use competitive present consider lda worse good discriminative yahoo dependency macro performance frequent lda discriminative micro score yahoo bad method yahoo label dataset discriminative dependency lda macro micro generally bad yahoo provide power strength macro score type depend dataset lda competitive even advanced discriminative publish benchmark numerous us wide consider consider style macro micro score prediction prediction thresholded micro optimize macro single macro optimize micro tie macro micro lda dependency lda perform svms however outperform macro additionally micro
trivial save sparse complete detailed moreover sparse use dynamic distribution overhead interval argue final sure guess exactly learn sparse succeed output try sparse close hence overall routine poisson form unknown tp place work correctness routine eliminate value define note ensure proper intuition binomial distribution future reference case satisfie tp distance certainly lemma observe exceed claim claim hold first last assertion cauchy rearrange follow inequality last binomial valid bind distance lemma notice fact learn ready describe similar proper modification translate poisson hypothesis choose hypothesis convert close tune analysis guarantee overall unknown conclude generalization problem unknown throughout know learn sample th simply independent time bernoulli efficient draw bit time sort result independent value aware arbitrary theorem show test subroutine target cover
estimator need numerical applicable demonstrate operator idea base contain derivative gradient local reduce desire one datum difficult iterative accurate reproduce space rkh value definite kernel dense k reproduce hilbert f definite boundedness positive kernel rkhs yx k xy vector reproduce derive operator statistical definite property
verify obviously q expression analyze design assumption p uniform excess lemma relate excess risk measure introduce x c c symmetric measure know proceed apply lemma variable analogously kl kl n p divergence bernoulli verify dp p j p n
rescale offer rate class rescale counterpart ensure class article explore utility exercise independent draw assign rescale old rate without sense coordinate information cast fast coordinate would cast gp perfectly dependence explore extend gp variable low question equip rescale offer
research receive receive award speech student international conference speech conference well song r award communication high estimation dr rao member technical processing communication process signal remark rao edu nsf grant zhang address nonzero correlate consider correlation performance correlation propose block derive bayesian superior recovery especially temporal highly extensive interesting motivate sparse recovery compressed sense correlation recovery compressed processing basic mn unique available noise source measurement wide arrival estimation imaging eeg localization estimation measurement measurement l unknown solution row possible source often application orient model index identical row solution nonzero compare case rao advantage relax mild exponentially rao analyze
wavelet coefficient reduce splitting complement orthonormal basis q parent child dependent necessary fine encode large variation approximation correspond first correction advantage tangent correspond geometric disadvantage really encode precision claim equation wavelet point lie exactly outlier lie tree close point unique enough scale calculate encode multi compute two set coefficient fast associate advantageous construct wavelet generate approximation technique process thresholding multiple haar smooth wavelet tree understand suitable suitably tree try tree metric euclidean operation independently tree balance suited purpose detailed investigation publication cloud store cloud cloud cost resolution manifold sample point fix interested theorem even encode cloud large expensive cost geometric count wavelet geometric take independently datum geometric wavelet wavelet basis add correction term geometric cost encoding wavelet encoding
fast optimization perform artificial database artificial separate point gaussian circle former consist two equal one axis show increase identification artificial database classic semi density unit define produce define three configuration artificial database dataset breast cancer compose fine breast database compose traditional method tendency spherical method work rate
particular display summary part informative exploratory representation exhibit provide brief phenotype section concern sequential algorithm simulate deal display plot predictor vector phenotype adopt coefficient density family distribution arise double exponential central independent across coefficient simplify freedom refer double prefer think generalize early context also understand direct derive density prior hierarchical prior whereby coefficient priori tail generalize light tail tend clear infer posterior exploratory examine map estimate generalize problem associate prior local posterior
alternatively area risk accurate suffice explanatory risk plot reach well overall optimize partition simple treat merge block finally partition far modularity parameter measure dynamical definition n next assign suggest interpretation successive training accumulate error see grow decrease cumulative risk cumulative risk select modularity overall gain value total indicate simple significantly early cumulative modularity network cumulative risk equivalently total cumulative risk modularity
reason high incidence far hand predict characteristic diabetes highly health service level distribution state differ quite function incidence fulfil calculation
able reproduce weight
probabilistic name dependency combination markov probability component dependency bayesian set parent cyclic distribution px dependency network collection separately specific would regression network sampler apply dependency joint dependency causal author network basic module gene gene module function employ module reduce similar module gene level coefficient genetic network description like class entity associate attribute fix relational describe entity consist dag attribute conditional special world characterize edge neighbor connect let denote average reflect block link degree decay compare power neighbor property study structure appear network behave connectivity small network group free major approach method approach attempt set attempt drawback identify structure base explicit try fit probabilistic score score indicate fit space intractable evaluate structure heuristic find popularity year essentially
definition belong population choose b ta transformation
inconsistent respectively factor imply description consistent investigate whether inconsistent rest bayes calculation use probability set rv datum inconsistent rest regardless three analysis conclude bias together bias rv residual analyse amplitude roughly correspond period despite analysis rv old set contain signal remove need purpose calculate say calculate correspond inconsistent give combine reliably rv rv restricted probability
h b place shall assumption satisfy consistent accord variable case wiener integral brownian motion condition integral simplify hold denote n pn immediately residual lemma assumption strongly proof theorem condition satisfie bound shall look hold unbiased separate consistent function estimate separate integrable martingale
time occurred discover end censor occur remove patient lose follow cancer could location understand fact early censor individual partially j survival analysis ci call tie tie thus goal formalize order way build scoring pair put survival analysis class supervise ranking problem quality medical longitudinal study call censor degree unknown factor clinical feature put emphasis survival commonly treat usually event occur
inaccurate believe largely condition spread widely round variation would unique atom would slower algorithm despite learn initialize underlie dominate inference since use process beta consequence sampler stick break offer flexible representation superposition countable poisson may generalization appendix f df lead exponential simplify third equal dd n nk
alternate possibility hypothesis graphical either consider instrumental applie model usefulness correlation social commonly hmms find variable goal one acknowledgment thank discussion national science fa quantum see write obviously sec rigorous measurement choice possibly ax pr ax without generality hide hide share pa r pr pa pa form combination write graphical minimum special
easy ess mathematically prefer criterion essential weighted reduced density usually reasonable choice correspond see anneal efficiency density determine neighborhood metropolis hasting mcmc sensitive depend scale parameter determine satisfie potentially minimize scaling consist adjust scale whether optimization finally coincide intuition intermediate concentrated ergodic aim discussion anneal involve delta verification need first prove describe generate explain reasonably unique limiting annealing aim anneal follow condition generality trivial case second account prove aim probability example chain distribution irreducible positive assign uniform etc aim generate irreducible therefore denote recall simulation course measurable surely
discovery discovery information differentially two microarray dataset mention breast classification summarize small identifie identify figure finding level validate accurately figure breast cancer study gene connection breast cancer gene expression level validate light surprising fail mark blue bar height bar posterior include dot z breast breast cancer gene expression compare identify tail histogram figure identifiable nonparametric empirical bayes testing include oracle particularly differ finding distribute work behave normality identification
x l eigen smooth eigen subspace fix q conclusion boundary smooth conclude sense invertible next norm expansion thus together equality come complete proof eq module haar laplacian operator adjoint uniform operator eigenvalue dense topology g hermitian differential module count copy inside frobenius law branch irreducible form form note calculate module call frobenius k copy irreducible calculate irreducible dual lattice unify fundamental power standard representation irreducible representation high weight power irreducible representation split irreducible high please irreducible irreducible module irreducible module run integer irreducible module run simultaneously irreducible module high weight please base g precise side split k high weight please step relate representation consider endow bi invariant semi metric enjoy relationship highest irreducible high positive root induce inner please note combine irreducible representation particular decompose irreducible module eigenvalue inside character formula calculate highest irreducible I pm irreducible pp n please original odd os nn odd otherwise eigen put eigen irreducible list list happen eigenvalue em eigenvector conclusion see compatible thm example wu vector massive image shape generalization dimensionality heat heat low space field diffusion near dimensional embed laplacian operator dimensionality heat alignment connect point quantify past decade reduction locally
program transform semantic preserve make program semantic preserve program generate prior program generate prior program frame program alpha program search towards program mean size program body frame abstraction abstraction abstraction abstraction abstraction body body abstraction size track particular produce important quantitative large make furthermore give point account exactly describe computation fact likelihood mean case program estimate sequential generate discrete extend selective point tree frame tree evaluate program single correspond generating selective averaging frame program tree replace gaussian program program topology parameter score topology score trees topology tree map compute tree tree inf topology inf apply score score fact function program mean code syntactic implementation replace return list replace abstraction abstraction make name abstraction abstraction return abstraction pattern abstraction abstraction convert replace abstraction program body body convert convert body define list choice uniform rest test list replacement replacement abstraction abstraction name abstraction name abstraction transform abstraction pattern abstraction abstraction let convert map abstraction convert transform body make convert converted body force program detail beyond short
likelihood parameter posterior formulation quantify involve nested purpose might variance empirical uncertainty induce uncertainty propagate sometimes though krige provide interval refinement krige surrogate refinement interest introduce krige technique krige probabilistic krige prediction refinement consist reduce optimum sequentially achieve reliability interest limit subscript clarity krige denote might onto state surface correspond uncertain spread achieve locate margin express finding maximize quantity bring well krige decide criterion optimization paper propose refinement criterion include easy exist platform fall multiply weighting consider weight original pdf confidence region pdf reliability equivalent gaussian reader discussion maximal justified numerically indicator q
explicitly neuron topology connect follow denote embed stand number build mind energy lyapunov neuron determine overlap cosine pattern term appear hand equation contribution whereas concentrate comparison extend classical quantum rewrite local stand
weight add importantly time vanish critical network otherwise stop track constant show agreement noise present attain agreement tend square error mse bind sufficiently agent behave agreement allow individual give node flexibility operate end enhanced proceed detailed diffusion strategy important consensus decay set combination adaptation evaluate incorporate neighbor diffusion actually mean square diffusion achieve rate consensus dynamic algorithm vector gradient gradient component denote symbol expectation randomness node gradient express diffusion q observe quantity highlight algorithm arrive convenient carry assess estimate minimizer square mse sense necessary force agent acceptable flexibility beneficial biological network adapt forced requirement agree small behavior
new paper think think weight h directly free parameter inequality choice therefore average allow relate average law reference sample know closely moment moment proof hoeffde variational basis introduce pac high set expectation set expect define randomize randomized round apply next pac generalization guarantee influential support machine cluster name extend hoeffding et al tighter apply pac domain average inequality
quantile nonparametric regression posterior computation automatically incorporate remainder quantile focus efficient partially collapse integrated drawing explain detail conclude discussion section th quantile asymmetric assume q skewness parameter direct likelihood scale mixture satisfy link distribution zero hyperparameter write constant identifiable exactly
much covariance matrix variate kronecker delta rule array variate structure parameter describe simulation dimensional gene study array element say array array array principle technique multi result correspondingly
optimization potential bound strictly feasible program lagrangian saddle saddle similarly saddle v optimal u e saddle lagrangian nash equilibria pass equilibrium equilibria complete within game literature player control single player two consider player player game uncertainty author already analyze sensitivity message study
dataset experiment handwritten digit recognition task dataset pixel range due dataset set pick validation consist test repeat experiment plot section give rank experiment report describe overall counterpart suggest kernel range width measure characterize versus four mnist htbp suggest mnist experiment choose cr affect change however finite improve outside small base well base kernel set expect c nd cr da du rank mnist letter dataset mnist letter experiment evaluate uci letter recognition include capital letter validation misclassification measure mnist kernel however experiment dimensional available table build examine learn target supplementary material achieve task experiment high alignment translate accuracy aside mnist see cr outperform repository median
following let positive scalar invoke diagonal ii ii important positive eigenvalue sort decrease order likewise result proof hermitian finally definite satisfie triangle symmetry obvious show prop triangle arbitrary preserve prop triangle consider prop z immediately computer show scalar hilbert may ask previously embed different kind admit embed negative definite suffice eq arbitrary unfortunately quick although answer positive characterize necessary essentially treat matrix definite prove map follow product whenever cover invoke
people idea science greatly via people never work year way use process mathematic science lead
ab side equation q outside interior equal center radius singular note integral second say eigenvector eigenvalue outside everything loss strongly penalize minima let convex ss w ii could path use since equal zero term minimum
denominator find definition remark theorem parametrize dimensional long need complexity mutual apply interaction lower dense match qualitative behavior bound achieve stochastic differential sde dimensional brownian px unknown dimension
subset range integrate entire second refer efficiency discrete finite useful order arrange increase index network unweighted discussion describe percentile rank tie order knowledge specify exclude nan cost purpose irrelevant population mass treat level discrete element probability occurrence every weight kk define every computational generalization open one due opposed case handle particular topological cost interest become integration treat integral probabilistic treatment integrate particularly helpful consider estimate monte scheme readily integrate level order answer question briefly consider fix cutoff fix iii four example datum extension idea several simple topology correspond association specific discrete topology naive network proportional association weight v association denote association simply proportional illustrated element standardize cost solely interestingly easy use equation since naive topology say straightforward quantity show additional efficiency cost metric increase thresholding graph fix cutoff weight tend level cutoff point separate cost adopt author condition topological thereby calculate quantity small since topological arise consequence transform integrate fail distinct functional connectivity parametric transform produce regular simply choice
object interest shape vary negative boundary impose shape unknown base applicability via achieve prove object noisy problem image quick noisy mathematical object noisy basic look ask primary diverse automate detection application random picture make intensive procedure surprisingly majority skip shape permit object interior aforementione automate material detection regular method advantageous object problem extend tail give implementation program statistical program language statistical object seem spatial triangular periodic case complexity pixel
expansion estimate advance sampling toolbox estimate well implementation adaptive constitute analyze measurement identifiability sampling call restrict isometry show bernoulli uniform toeplitz appear bind deal rip rip scale extend contribution rip regression involve tool utilize rip bound counterpart sparse regression scale also applicability simulation aware estimator cope curse dimensionality yield parsimonious record work letter denote stand density covariance pp expansion notion multivariate problem nonlinear respect curse inherent involve motivate highlight input mapping truncation practice yield condition capture term module module reference goal expansion memory extensively present vector pn np tn ph arrange accordingly tn linear generalize filter aim approximate nonlinear tn several multilinear prediction problem natural science economic choose carry setup henceforth easily
wise suggestion presentation acknowledge foundation nsf grant dms randomness institute arc proposition continuous q lipschitz twice satisfie cube minimal compact possibility second dyadic follow vanishing require regular gradient denote subgradient representative possible lipschitz remark dyadic cube note subgradient assumption choose independent hyperplane contain integral remain boundary center cube eq q impossible exist dyadic edge
candidate datum equivalent amount adopt previous provide quantify theory issue possibly noisy discuss extension next objective nonempty convex objective function search I multi fx loss estimate represent e additional constraint cost observe almost orthogonal closely related purely unknown observation try maximize amount experiment worth note exploration ensure balance robust objective visually depict htp l exploitation ex ex robustness multiple trade list exploration versus exploitation put exploitation e trying achieve observation computation capture risk exploitation present utilize framework address present discuss shannon information infer machine constitute involve error function alternative follow spirit try try estimation point
market long confirm major impact occur close region plot km take use multivariate multivariate already derive sake completeness additional moment auto causality test process causal effect present specific bivariate monte carlo maximum behave magnitude confirm close contiguous short period hour perhaps subsection medium arrival sum assume consecutive day natural might poisson regression derive autoregressive denote series imagine operator eq values compound binomial autoregressive process q markov chain integer moment interest high dimension provide
mean ern smc approach variance mat ern covariance jeffreys endow prior log thank
occur every fx z iteration fista theorem lipschitz although leave stay follow essentially backward substitution following iterate alm fista give backtrack line start fista inexact linearize quadratic fista version fista iteration surprising fista gradient handle label scale impractical fista serve subroutine involve section example group solve notational subproblem ball norm simplex time et euclidean onto euclidean follow replace problem absolute take projection onto take test fista fista framework interface provide core matlab alm find loop experience show slow even system perform pc intel ghz processor gb framework subroutine termination criterion criterion primal dual residual criterion appendix outer tolerance outer tolerance
model desirable necessity evaluation functional bound dimension differential equation generalization shannon rkhs possess reproducing reproduce able save birth kernel trick rkh underlying application field reproduce kernel base theoretical issue devote inclusion rkh reproduce interested rkhs rkhs understand reproduce relation multi rkh study reproduce kernel critical success avoid overfitte principle overfitte large relation inclusion establish reproduce rkh application map past purpose
measure association gene phenotype observe association phenotype disease phenotype association indicate rank phenotype disease phenotype association roc roc c c pt report disease phenotype phenotype cross phenotype associate phenotype retrieve phenotype gene phenotype disease phenotype experiment disease phenotype near reduce report roc leave validation algorithm outperformed achieve well sp score global rank disease phenotype rank rank clearly achieve ranking query case ranking phenotype phenotype query target disease highly good fig compare query disease phenotype phenotype fold threshold suggest short distinguish phenotype neighborhood utilize interestingly detect association high set include gene disease dense disease gene module neighbor tend dense association phenotype query gene fail
norm employ high constrain trace norm small achieve precisely scale large application multiclass fast acknowledgement discussion schwarz science need generalize let assume denote hold smoothness tell e eq multiplying side support combine rearrange equip ready let vector value thus operation monotonically increase addition lemma imply
case section outline method picture neuron shape advantageous situation picture level pixel algorithm picture typical picture relatively fig algorithm simulate neuron detect detection describe experiment h pixel value black grey picture detection choose see consideration reasonable thresholded symmetry least noise practice consistently strong numerically phase nan picture noise
process contain stable extend model addition four parameter n property parameter lose truncation support implication general lose truncate exception family illustrate result discussion numerical likelihood software implement numerical procedure perspective stable context practical take restriction ensure skew result satisfy closed expression function distribution illustrate loss increase number software stable website american stable c c already practically number heavy even fitting heavy tailed represent tailed demonstrate therefore stable sensible select stable admit sub model present expression consider assess
spectral transition approach eigenfunction calculate exact tight relatively approach difficulty low bound without calculate different comparison positivity application area call augmentation da procedure expand see exact comparison mcmc prevent use traditional theory theoretical comparison among index tend hold consistency
separate conditioning though line sum doubly past past stationary simplify stationarity equality dr neither dominate doubly estimator whereas dm require similarly dr section argue compare dm look fast treat past policy suffice moment variance derivation write sum let policy doubly eq account reward second variance weighting
onto suppose coordinate equal coordinate zero feasible p continue obey coefficient rhs suffice pp desired sense x straightforward extension vector suppose order claim become without none vector vector follow inequality obey furthermore use yield q ready
distinct look month observation year show user month look look day week month conduct focus correlation query volume trading volume amazon com netflix make whole day month whole year fig month volume trading volume observe significant stock trading activity typical user suggest profile people financial expert remarkable trend order confirm analysis yahoo yahoo yahoo yahoo visit user yahoo complementary standard engine include page query search yahoo yahoo statistics limited line yahoo user yahoo gender country seek behave differently yahoo user set least half whole property aforementione respectively report worth observe user small representative yahoo concern gender include slightly country get find come california new aforementioned states united acknowledgment support grant cm cm yahoo research diagonal le south uk institute advanced study mail live leave
none call action dominate dominate action action none theorem two context allow generality action vector keep action remove keep remove action correspond keep equal dominate dominate action outcome game draw vector action lie bottom boundary hull action rise connect non dominate action convention order accord loss outcome introduce chain dominate action consecutive degenerate degenerate soon separation distinguish game merely technical proof exclude minimax game outcome monitor finite monitoring outcome satisfy separation outcome satisfy er dominate action regret satisfy otherwise respectively prove trivial minimax dominate four dominate action trivial appendix prove entry change change achievable
rigorous consideration statement definition complete note q site lattice side vertex maximal vertex rectangle exist constant inequality close closed left denote cluster contain obviously proof help pp everything product apply thresholde operation step standard step operation save position position white exponential false iii prove object picture assumption thresholde
limit finite capacity assignment place assume mask variable correspond create coin bias determine whether show ibp appendix package available package implement cm p crp mixture crp crp matlab ibp matlab david ibp latent factor matlab key choose appropriate appear choose nonparametric method side step determine introduction nonparametric use question model select one metric usually include fit second term simple one provide different rather grow observe traditional clustering analyzing nonparametric previously mainly random chain monte nearly later tool nonparametric central formal mathematical using hide model make posterior hide likely bayesian complexity advance survey bayesian bayesian mention mixture
discrete countable cutting cutting becomes cut iy z ny ii nz subject control group assignment fix exist assume consider cut estimate cumulative however cut end say denote extend observation identically ij auc converge density follow dominate variance use calculate gold require computation one diagnostic variable motivate maximize auc study good multivariate aim usually involve issue since combination give
result still room improvement usage variance distribution exp tc action hold hold however fraction suffice direction pac combination pac bernstein inequality enable finite exploration exploitation selection simultaneously direction result evolve possibly
environmental sciences national fellowship contain show pointwise let define assumption b design metric denote assume satisfied assumption every b put break truncate positivity mesh intersect parameter make mesh subset covariate region hyperplane b constant loss generality take bound theorem respectively random design equivalent case require care rely let condition nf nd nf nf f f f quantity cover supremum norm iv meet work linearly transform suppose b
break guarantee know construction validity let show approximate reveal choose possibly constraint quadratic hilbert densely selector computation subproblem classical choose weight subproblem orthonormal rather saddle particular situation slightly different decomposition characteristic function admm projection step program quadratic algorithm q set close onto intersection accord set create intersection successively onto individual projection ht primal find constitute explicit project decrease estimate note application appeal even set rectangular image detailed geometric interpretation mr projection problem however enter slightly sophisticated present q onto index set property continue necessarily illustrate capability regression yet different denoise apply aforementioned henceforth moreover forward define amount implicit diffusion time solve
maintain objective master computation researcher delay asynchronous synchronization issue certainly optimization decade least seminal minimize prior assume throughout however al stochastic possible aggregate low modern stochastic et network possible asynchronous subgradient incremental take date illustration asynchronous subgradient suffer rate delay procedure elegant play delay delay convergence centralize attain asymptotic objective approach provable asynchronous delay parallelization delay update provable cyclic worker compute gradient pass date master master diagram gradient build processor compute stochastic common convergence processor appropriate hold independently remain show discovery update suffer delay delay show different range range either centralized delay paper
pricing achieve demand regular respectively note regular pricing detail pricing strategy detail regret demand distribution constant choose construction reduction case pricing sequence strictly increase item fix instance sale agent correspond instance agent round remain interaction agent since happen demand sale problem pricing instance achieve realize instance let ok ok kn part roughly bind become pricing monotone hazard price p np sp pricing proceed consider price price pricing essentially close parameter minimize mechanism approximation regret offline benchmark demand satisfy monotone hazard rest devote far appear difficult prove else lemma lemma additive benchmark pick multiplicative compare turn offline demand regular let price kp technique also detail least easy go observe hold examine stop choose pricing encounter sp get satisfy exploration claim
analytically minor lemma adapt assertion proof fact z otherwise rearrange together fact sequentially hoeffding martingale pac martingale sequentially long contribution possibility sequentially encounter apply positively limit feedback exploitation although art yet bandit whole reinforcement bandit contextual reinforcement domain incorporation beneficial
may genomic coordinate secondly gene segment throughout consistent l relation exist allow relation specify closure closure denote denote set element also alignment alignment end align read distribute compatibility parameter read start distinct generative seq poisson count distribution multinomial equivalent likelihood maximized discuss seq clarity detail nz priori arbitrary obtain combinatorial useful summing suitably leave maximize independently right unconstrained maximum equation eq interpretation poisson interpretation formulation adjustment bias overlap genomic coordinate equivalence distinguished overlap genomic whereas map present rna seq includes alignment location relative restrictive simplify description select site within depend abundance site relative determined content position length
return try every every corollary tree finitely corollary subgraph adapt dimension integer integer edge color complete subgraph color call call sake every ordinal positive lm l positive assertion km vertex red otherwise assume u jj u u il j j jj l li n apply n rx v show inequality I condition theorem system distinct obviously fa great verification gb gb n side rs side linearization
propagation equation set ordinary differential ode uncertainty condition polynomial handle quadrature nonlinearity model moment condition give low moment depend thus infinite hierarchy equation moment one closure interested first order moment closure variant transform closure interested find induce knowledge independent ds describe uncertainty use convolution di di
rectangle rectangle rectangle rectangle rectangle rectangle rectangle piece edge become rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle generalize close introduce notational straightforward adjacency independently independent observe ij effort kronecker exploit independent bernoulli approximately one follow recursive initially proportion expect north west north east south west south east via integer proportional repeating time reduce probability time reject code v vx nk start kt start end kt end ks start length digit integer edge value multiplicative attribute bit vector even bernoulli ki additional reveal representation call sequel ease analysis initially assumption relax counterpart sampling attribute
locally indeed fill several propose strategy interpret improper local cluster mode performance computing platform check optimality criterion model design experiment l build learn onto context leave krige build minimal reason refinement krige predictor mean costly state factor contrary evaluate estimate get classification counterpart consequently instrumental towards counterpart propose stop refinement stop problem limited refinement variation happen problematic define x denominator cause argue never exactly krige equal zero precision figure divide three independent step require coefficient estimate number
auxiliary polynomially optimal nonzero element stand constructive principal component polynomially identify principal introduce spherical variable exist index argument polynomially develop degree well study tool pc computationally modify lasso elaborate nonconvex combine tackle nonconvex technique program present
x I guarantee give high produce w I adaptively hyperplane small iteration combine hyperplane gaussian obtain good set na n di iw ji ji main loop terminate iw iw j j elementary calculation ji terminate know end eq last get claim output w least fail markov fail proof hyperplane separate analog well motivation perspective machine hyperplane correctly point separate separate exist provably hard hardness admit np
one n objective regularity great consequently locally lipschitz meet recall q strict exceed curvature r meet sum argument quadratic continuous meet minimizer converge finitely descent minimization finitely iterate connect iterate accomplish iterate guarantee differentiable coordinate elastic net penalize change continuous differentiable define prove apart subdifferential differentiable expect function iterative solver give pair symbol hadamard active speed computation initial change active tucker meet kkt violate repeat sn compare regression rapidly warm start regularization start iterative subsequent regularization apart sufficiently intercept come small regression q grid equally spaced
shift recover related discrepancy explore term try prior work around vector take base set purpose description closely report pre coverage interesting epidemic understand report complete reveal inter infection feature infection frequently locally persistence couple average reporting take analysis use represent period pre standard therein birth period area rough infection part birth effect conclusion biological want answer transmission school period versus non period period movement infect people time period question period infer hence key almost indicate change infection period n kt interpret amount movement represent amount infect leave city th infect people come city
wrong whiten whiten correlation problematic solution whitening case whiten covariance section without take root computationally whiten way pre whiten sparsity undesirable mathematically way naive whiten property solution corollary global nan right value unique addition rank value factorization comment operator semi definite global non obtained immediately result stem relation svd decomposition subspace rank possibly solution still great flexibility conceptually computation massive obtain storing permit massive structured k solution global problem variation calculate high dimensional set approach discuss name vice versa problem follow scale allow high quadratic operator wish decomposition recall finding combination linear combination account structure transform norm arrive loading orthogonal right factor proportion find one massive section outline method relate methodology help quadratic operator class quadratic statistical structured leave quadratic operator connection interpretation well correspondence connection variate covariance interpretation separable
theoretical low bound fig estimate noiseless capacity capacity dimensional constraint eq noiseless child channel width horizontal dotted shannon capacity
strong version require symbol follow directly theorem sufficient theorem condition remark may lemma v total decompose rv ii verify decompose cv property h n yield finally choose eq q conclude bind follow xt path view space iv lemma j ba b k lemma exist theorem rx rx r pa self adjoint adjoint therefore decompose argument n jt jj w n w w brownian motion hand side zero boundedness approximate constant apply replace course brownian motion w p moreover similar finish done cut h pt q replace assumption assumption easily may term right uniformly imply r show thus finally check embed space symbol differential finite proof subtle appear differential universal denote change p establish ap p integral space
share segmentation token build token instead short token apply way mean token token string token mle token token token token example segmentation mle follow unknown cope string token possible generate certain follow count token give time string q
reward play challenge problem work unknown develop original applicable bayesian setting propose bayesian meta treat different develop channel achieve logarithmic average first literature generalization arm bandit mab decision uncertain arm player seek time maximize reward obtain challenge bandit arm evolve bayesian reward model mab update parameter probabilistic objective know regret
case focus state theoretic extension presentation replace polytope present terminal robustness property terminal important prove terminal feasibility maintain variable model mark uncertainty state nature reason science input know gradient yet mathematical recall fact first steady suitable matrix k k fact terminal reach admissible control constraint input property formalize eq q component parametrization arbitrarily maintain invariance scheme even appropriately use feasible nominal nominal trajectory feedback grow initial nominal initial horizon width nominal lie trajectory lie imply control keep optimization define q feedback
c forward central arise third table twice difference central difference arise central arise net laplacian fig grid map original net forward city even centroid dot line good solution multiple periodic nonzero anneal fig city many cyclic permutation number show shift sign produce semidefinite want power general infinitely must agree modulus fix dft eigenvalue correspond complex power several thank dft dft long dft divide inverse state symmetric semidefinite even eigenvalue matrix sign prove matrix decompose associate first matrix cosine diagonal summary always eigenvalue obtain inverse dft verify forward central difference family next imply symmetric p even real pd pd formula centroid confirm q consequently even original number nonzero already k number sign proposition odd even table central element centre high plot order coincide odd strictly though carry analogous sign c spectra section define odd decay finite pattern difference even index odd shall statement another different option combine net generality combination differential accept high order forward central result convolution fourier combine domain spectra since consecutive forward equation power negative row square pass matrix maps forward n mn mn express square mn relevant taylor vanish constraint derive horizontal pass net heavily direction contour around align circular however basically corner would column interesting show behaviour methodology property periodic c detail additional way perhaps use idea particularly domain rectangular adjacent domain investigate simulation b elastic nonzero fall net centroid discard size net dimension toeplitz g eq hard formula eigenvalue plane become contain analysis many extension derivative periodic grid think derivative area even derivative linear centroid net
large benefit curve trend fashion curve uninformative additional yield substantial relevant ht neighborhood reconstruction regular node neighborhood one vertex vertex may adjacent estimation neighborhood neighbor node group term approximate approximate sum remain necessarily adjacent ij ij list neighborhood repetition concatenation paragraph pick frequent appear list exactly frequent error contain pick neighbor appear node appear incorrectly node
zero lp recover binary give sufficient signal link uniqueness still open unique solution lp use unique lp eq denote
string name early convenient treatment due require relate far statement input proceed negative lemma perform third subsequent create necessary link section letter alphabet find irreducible coincide conclusion algebraic process light process imply string process process compatibility model def call rank string alphabet finite string example process let process take characterization process submatrix ask agree rank string generalize equivalent trivially arbitrary argument proof know argument
solution precisely try eq similarity produce refer entry order constant compute stein expect risk less formally minimax maximum bayes minimize risk bayes risk bayes favorable prior minimax estimator bayes specify set estimator reduce minimax follow extension minimax work simulation draw matrix different minimax q since visual summary set green many know estimator stein variant fall prove green show generalize pool proposition familiar regularize choice choice classic section entry diagonal non entry set task follow pool entry choice pattern towards write appendix use similarity nothing initially however asymmetric regularizer
taylor evy x keep return return posterior rejection rejection sampling expansion expansion axiom theorem conjecture exercise theorem theorem paper propose estimation center clinical heavily gamma relevant augmentation first gibb need analytic traditionally come logistic example allow incorporate technique bridge compute regularize odd augmentation naturally suggest default jeffreys method case analysis center fit way contingency table augmentation focus table common meta analysis center practical approach problem nonlinearity augmentation address issue need analytic approximation hasting describe involve rely upon mixed topic table trial efficacy arm treatment suggest treatment
option meaningful display scale visual comparison informative question reasonably rare human intuition poor look call option imply tail look experience tell inspection tail reason formal payoff likelihood provide unbounded paradigm outcome g research belong implie bound infinity neutral use indistinguishable growth view research law probability say product choice derivative huge market product good thing substitution see extreme view expression view instability expectation reflect market place extreme limiting end
state state suppose positive set b xu b definition originally convex concave however counterpart risk contraction hold rv v rv c add change side inequality vx c proposition iii first inequality rx vb monotonicity replace let tv tv tv b tu u tv b rv f tv suppose f tv tu tv f due f v w tt tv tu map b theorem relation endowed induce equivalent fix conversely point exist th ht h due hold risk axiom rv risk measure map control mcp ax
cdf may adequate rest article organize variety application mle unknown test goodness suitable statistic statistic study exponential model obtain example either take logarithm denote natural logarithm show
mab unknown arm arise short path span dominate unknown design generality performance though offers show inferior classic policy target specific whether one finite performance generality edu armed mab problem arm unknown select play sequence exploration exploitation arm policy tail reward achieve total ideal reward heavy exist th knowledge moment tail offer logarithmic mab general parameter cardinality variation mab include mab objective player observation mab dependent arm arise span dominate unknown multi armed decentralized armed bandit combinatorial armed mab sequential arm choose arm obtain different maximize reward mab range
heuristic level extension interaction framework matter scalability error particular mode grow construct give language bias experiment department college gate uk mail ic ac uk ba e mail uk principles division national mail case drive iterative design framework call virtual use open semantic logic programming one desire property comprise event specification sharing discuss system section describe principle action bind member proper acceptable state matter cx terminate term state context outline formal framework adopt mapping answer set event bring state concept capture evolution
standardized intensity characterize dependency latter resemble joint deriving illustrate high htp x represent density trivial great integral denominator variable positive mean proper denominator order denominator use gibb random
solid amplitude sensitivity observe statement specification alarm define uk measure statement limit frequentist notable trial look elsewhere probabilistic statement one property illustration simple signal encounter nuisance
nonempty x gs verify property gaussian entry decompose finally vi cone except loose surprising dependency among group signal group much large signal measurement refine block see theorem group pay extra measure group signal noisy observation noisy solve atomic relaxed constraint take
choice furthermore weight candidate general powerful illustrate among scheme accord proposal performance reduce produce would version manuscript work project ref ref ref project ref corollary es algorithm advance normalize technique weight specify analytic arbitrarily great mcmc also give
bivariate underlie copula belong page inequality identifiability e van page satisfy condition iterate algebra family already check page sample rescale df correspond df q compute solve iterate q estimation estimation whose q absolute hand copula absolutely density base consistent root main copula moment improve score l moment statistic fact interpretable special one
power method engineering discussion phase similarity multiplication experiment explore reveal matrix final objective guide choose design adapt conduct purpose implementation parallelization evaluation give runtime th parallel generator non zero index width height demonstrate relationship generating compute load locality requirement compare experiment list multiplication square dominate particularly actual complexity multiplication square remain figure generally assumption plot show summation beyond storage multiplication line linearly plot theoretical benchmark illustrate theoretical prediction section curve curve prediction gap explain store total running equally stage discover cell plot vs whole picture
first rewrite invert sum sake follow ti q recursive simplify expression weight difference deduce prove consequence proposition minimize constrain almost everywhere tail pareto estimator mild illustrate finite index move tail denote survival zero refer comprehensive extreme various framework numerous dedicate
theoretical notation logarithmic conjunction continue bring acknowledgment nsf finding recommendation material necessarily view mark huber college edu high relative often large make nest count percentage fall bernoulli shrink balancing consideration provably ideally
solely static brain call two main conclusion brain
effectiveness seven retrieval future area ability gradient objective enable wide possibility retrieval document particular recent advance gradient discrimination speech general broadly gradient operator correspondence finally linearity expand prediction representation acknowledgement rank ranking present difficulty fact permutation include precision cumulative expectation characterize marginal expectation
one zero prior combine element another vary useful analyze area finance environmental science investigate platform biology finance environmental sciences inference conjugate wishart place element wishart parameter definite matrix normalize cone matrix constrained indicator wishart successfully limitation wishart prior element computationally challenge recent advance model rely completion intensive decomposable graphical expression unstable graphical determination determination place prior prior non decomposable fit jump metropolis hasting unclear essential absolutely continuous shrinkage subset element represent flexible develop
behavior incomplete gamma modify integral behave eq satisfy relation call dataset field evolution observable area anomalous diffusion model find system include inside period also process might
consequence constrain satisfie certain condition result theorem quantity nonconvex obvious costly concern rigorous iterate run project descent controlling assume previously surrogate surrogate aid follow deviation satisfied version sometimes easier derive tight inequality second corollary verify various form quantity change result global surrogate apply claim probabilistic enter analyze statistical satisfied lasso choice past allow surrogate example arise noisy dependent satisfied probability miss nontrivial inequality principle optima separate optima triangle inequality guarantee similarly program scaling optima ball whose nonconvex additive plot rescale align rescaled plot addition specific simulation additive state
sx sx p p dy sx sx dy inequality page assumption normal mixture explore rate class densitie construction offer study break keyword lie non mixture density construction part derive class main construction desire scalability
benchmark ise correlate homogeneous fundamental physics finance natural correlation direct ise physic constrain capable reproduce average configuration infer ise model mathematical implicit equation interaction average boltzmann various inverse ise bm learn message method likelihood
chernoff twice claim small apply cover q proof assume triangle inequality corollary exponential sum dependence explicit matrix dimension chernoff inequality analysis analogous dimension quantity small bernstein moment appear regard
datum embed embedding set training integer fig illustrate fig embed quality quality embedding improves increase validate visual manifold learn wide first new motion coordinate effectively normalize embedding topology set validate propose improvement future state meanwhile accelerate manifold densely datum lie manifold outlier efficiency solution implement denoise manifold work different acknowledgement partly china china grant cb zhang current natural quantitative measure assess learn greatly limit assessment normalize embedding le euclidean meanwhile scale quality name scale efficiently configuration
bound kl ucb significant kl index low reward ucb kl ucb solution adapt ucb law simply change definition build great elementary independent interest sense concentrate well bound significant advantage ucb ucb even ucb various compare bernoulli organize follow section kl kl ucb case kl bound optimality extensive experiment show superiority kl devote elementary main interest action expectation step action observation possibly randomization get
configuration implement logical function first set neuron equivalently percent logic absolutely occurrence output neuron compute put become matrix connect fuzzy inference input neuron example become occurrence neuron neuron neuron belief fire neuron fire fire neuron neuron high neuron locate neuron put thresholding network two output thresholding neuron eliminate value affect rest neuron thresholding value neural determine process fuzzy logic know necessity thresholding function use aforementioned instead common determined threshold modify create hardware implementation fuzzy way exclusive gate show version note base system base rule base way utilize
correspond weight nonparametric maximum maximize goal fit seek consideration another modification typical whose choice penalty bic penalty discuss sophisticated scad context factor practical posterior distribution mix draw dirichlet dirichlet imply surely well quantitie flexible structure algorithm fit dirichlet unknown approximation approximation efficiency suitably search essentially parametric problem anneal novel annealing call asymptotic property asymptotically subset measure kullback divergence act like distance
cluster expression profile bioinformatic genome article material li david term denote indicator gamma
hypothesis procedure fisher reject level calibrate ensure collection introduce throughout section successively nan level accept note reject q hypothesis reject collection accordance collection whereas final consist successively negative upper theorem notation model family testing accord hold imply derive procedure follow inequality accord matrix appear orthonormal orthonormal projection thus explicit namely coefficient hand reject point family x order define successively accept consider
separate separate triple satisfy may lead possible sep set step oracle version conservative use finally step algorithm triple result pt triple structure triple version conservative step use proof oracle version obtain well version modification give oracle let prove output maximally orient tail orient result oracle orientation orientation really causal modification give graph additional path size graph consistency weak section level independence find initial set list update step step initial skeleton definition triple orient base lem triple rule ix jx jx ik x ik detail step list store triple find condition triple conditionally check step subset conditional kx ix k j removal triple add moreover triple remove testing removal edge find set example show
execute use present problem since four near neighbor anomaly compare hypercube hypercube four nominal one dimension draw nominal nominal anomalous anomalous distribution experiment measure square euclidean curve auc simulation grid six select require select choice search ccc method best nn nn thousand approximated spline seven represent use trajectory dissimilarity instantaneous along histogram manner dissimilarity trajectorie speed histogram trajectory uniformly along dissimilarity euclidean roc k training along experiment consist consist auc compare spaced combination auc criterion worst show perform false weight auc find
assume binary generalization normalizing ensure illustration tractable ph strategy minimize average achieve necessary cd parameter note cd neither performance use objective refer hybrid discriminative perform separate rate wish layer expect contain predictive vary layer activity large issue generalization copy layer approach make present different nature input believe nature
discard discount assume daily cover area prior write describe uncertainty true specify model specify ik lead treat specify term approach nuisance round mm challenge mode input log effect consist point intuitively residual residual optimisation objective provide robustness show dotted sample marginally ccc transform indicate observed value summary joint posterior implement adjustment use kernel scale individually margin select estimate understand adjustment prior informative statistic similar incidence change complicate moment difficulty adjustment marginal wrong place inferential purpose cross validation usual predictive involve simulate posterior high nuisance abc raw output available
model expectation accumulate initial storage subsequent analysis perform simulation interest hour option suboptimal completely avoid describe simulation bit sample operate proposal suffice preserve complete influence reject mh view channel bit operation necessary writing bit also help
discrete nonnegative limited range present flexible beta completely model discussion binomial correspond binomial create evy aa assign infinitely borel set sign generally evy compact evy notation nonnegative l evy additive bp space define product evy concentration parameter infinity satisfy measure atom atomic mixed sum appear location atom unit atom construction attractive distribution diverse modeling binomial
discriminant combination path lda change basis change often extreme good believe difference propose diagonal matrix mean implicitly explicitly look decision suit remove decision suit generally regression interaction main hence relatively accuracy condition convenience subject psd recall one semi definite program
interior hypercube interior cone condition nf l sf h distance rkhs norm mu fill adaptive mmse difficult problem dimension address course suboptimal
well observable covariate discuss property detail appendix various empirical focus population thus estimator use size population heterogeneous biased latent approach circumstance interested heterogeneity explicitly individual thought variation size age put finite population collection method parameter covariate observable therefore classifying avoid thus covariate thompson employ comprehensive comparison specify despite justified problematic capture conceptually parameter nuisance capture
agree member determine ensemble rest grain compute assume distribution accomplish processing phase learn aggregate together final implement could framework distribute file define map pair reduce execute assign file map case correspond pair pass node function receive key associate produce map partitioning scheduling failure file assign show train ensemble local random show thus receive approximately output file step use importance sample build repeatedly tree induction produce comprised base incorrectly repeat bag get prediction ensemble see training outside base training differ receive equal ensemble simplify merging ensemble show contain incorrectly avoid possibility draw ensemble mb
possible state transition time may update subsequent regret time suboptimal infinitely information state bounding play introduce remainder denote know way ensure belong subset indeed exploit continuity start introduce state lag state play define state single infinitely partition set state contain infinitely belief belief stationary distribution belief although depend arm set infinitely belief lie outside belief increase element partition rewrite contain l infinitely information define extension belief belief diameter diameter decrease enough action follow l gap define diameter hull infinitely state exist diameter small extension hull conclude optimal action player matter close respective action suboptimal know policy suboptimal grow challenge lead first almost always optimality control horizon choose false state arm set state n g simplicity analysis true action optimal I draw player play begin probability zero subset draw bandit f hold
becoming infect per recovery patient approximately treatment recovery average duration infection become individual equal average development become transmission transmission probability average period infection duration infection duration dim scale degree age non linear linear prior c period median infection median infection h ccccc cccc c cccc h age ccccc activity university south south grant bb g university methodology forward carlo calibration system ordinary differential epidemic type involve age gender relevant infection type two paper extend population formulation suggest mix matrix allow unknown mix matrix involve epidemic describe member population bayesian allow duration treatment duration gender age group unknown parameter relate trajectory markov markov chain recently human far classify potential risk basis association transmission infection causal factor majority cancer head attribute develop clinical infection subset type high publicly national national costly decision health economic infection small resolve cancer many decade acquisition employ incidence benefit calculate relevance transmission population force infection infection individual infection reduce benefit order course
trend need well trend fluctuation series trend state fluctuation trend identify procedure smooth many evolve low may old become try point model little reason take opposed transition nothing evolve totally fashion period predictive history lead forecast e contrast forecasting collection matter path process prove forecast well hard see face arbitrary adaptively combine
set hausdorff compact nonempty completeness compact nonempty grant subsequence limit nonempty subsequence contradiction note exist minimax state minor applicability space semi continuous semi continuous apply semi iff monotonic grant monotonicity analogous continuity low semi continuity monotonic scenario concavity iff affine since affine repeat author thank discussion manuscript relationship stochastic establish set minimax theory grant spirit opponent reveal opponent repeat select play game game value single play essential need precisely second case
scale temporal behavior common autoregressive dynamical series exhibit extent unique alternatively use generalize transition emission class broadly address attention treat combinatorial parametric variety align interact univariate use approach series independent factorial hmm underlie widely ibp evolve markovian instead focus modeling series series align problem detect anomaly share among time hmms trace state represent dynamic bp ar hmm discover series ar eq fig truncate ibp sequence color mode one autoregressive white dynamical order autoregressive ar cc estimate average mcmc series autoregressive indicate define tail ar dynamical mode positive occur define ibp estimate mode sequence map mode map maintain infer dynamical series discover cause commonly dynamical green making nonparametric incorporation additional dynamical bp ar hmm hierarchical ar hmms tie together share parameter hdp ar mode hdp prior hdp hmm ar dynamical third hmm assume share matrix match
nonparametric condition positivity linear eigenvector equation frobenius matrix functional analog uniqueness discount factor let pdf q matrix analogy clear eigenvector condition conclusion nonparametric iterate inclusion equation identification discount factor identification utility consumption homogeneou positivity marginal utility support risk use positivity correction formation utility analog stationary compact extensively long compact general cite paragraph apply consider generally identification return equation restriction instrumental additional restriction singleton chen provide sufficient condition identification restriction important model usefulness primitive quantile iv consumption base asset pricing appendix square h e h open assumption hold open ball invertible banach theorem map conclusion list suppose operator follow step select basis perform step sequence
average speedup achieved serial delay measure get nearly linear speedup computation raise rr ever slow gradient question run simulate figure c time achieve delay take long able million gradient per hour gradient intensive rr method take advantage sparsity machine enable linear variety implementation instance problem moreover speedup even computationally intensive scheme occur certain particularly add update computation sgd memory processor recent propose bias avoid processor enable acknowledgement support nsf award cr support air laboratory contract nsf award award
scenario explain satisfied triple goal point quantile qr ensemble particular highlight potential benefit triple research note qr respectively estimator consistently plug therefore parameter readily obtain dispersion ensemble model chapter highlight limitation spatial non spatial observed range exponentially parameter aspect calibrate accord moreover note poorly compound symmetric rank unit choose control skew asymmetric deal skew assign low value ultimately applicability automate preliminary conclusion chapter superiority car car quantile spatial car laplace quantile qr normal car yield plug finally analysis quantile concentrate mean therefore car dispersion specification laplace car variate distribution fair car comparative spatially structure qr parameter however empirical tend skewed quantile regret formulate plug point could joint quantile differ rr change produce preference another need triple plug function indicate estimate specific wide rank risk gr heterogeneity triple estimate constitute area specific risk report summary heterogeneity interest inferential objective however gr highlight function cm chapter parameter ensemble different rank total amount answer threshold classification threshold total specify element review research rank adopt concerned case spatial spatial unweighted decision originally mean plug necessarily base decision surveillance datum uk good classification false problem point several statistical classification datum search pre class essential ingredient classification dissimilarity classification dissimilarity theoretic arise accord area posterior particular wish classify mass necessary allocate accord attract france study cancer breast cancer plot table cycle north anchor west dp p consider great general classification choice apart dependence therefore ultimately particular inform risk surface statistically problematic sensitivity specificity encounter frequentist adopt note discrimination undesirable consequence cut optimal threshold decision vary choice rule different mix choose chapter ii discuss calibration consider mix spatial draw extensively rank substantially differ one easily similar oppose lin et al despite inherent similarity classification substantially quasi theoretic advantage classification require optimal specific classification simulation plug estimator spatially section illustrated uk chapter discuss broad light theory point give decision framework estimator estimator link concept sensitivity specificity advance cut terminology express positive negative concept misclassification I false misclassification parameter denote reverse base specific parameter interest threshold classification loss unweighted threshold advantage quantify ensemble normalise pc p correct none element loss precede obvious context section threshold quantile equation distinction quantile quantile chapter function chapter quantile proof estimator fact solely consequence respectively indeed follow proposition weight unweighted note early immediately proposition describe identical trivial former whereas chapter unweighted relationship supplement simulation evaluate plug rest chapter focus unweighted except otherwise unweighted page expect formulae ci classification correct incorrect penalty vary whether classification distinguish false diagram former cut positive threshold threshold observation diagram attain expect great zero proposition justification gray corner anchor cycle anchor north east scale north plot cycle anchor west dp scale anchor north node anchor north north east node north anchor north anchor anchor west dp representing rank interested plug rank concern proportion estimate ten percent condition classification percentile cut denote high wish percentile false positive function percentile percentile percentile refer percentile except otherwise notational distinction fp percentile base unweighted analog unweighte describe estimator report refer reader original proof
target conventional system mode model eq velocity observation random process observation mode summarize mode cause ground move trajectory tracking target mode finite compute typically art interact multiple variable move road type markovian sequence motivate level abstraction call suppose surveillance interested pattern track infer possible loop circle exhibit construct characterize various linguistic main devise polynomial syntactic parse spatial pattern conventional track perform syntactic filtering abstraction track compatible design conventional tracking four form production stochastic regular sec regular regular polynomial practical hmms dependency embed syntactic filter system several formal production human easily pattern building permit high trajectory ability encode important lack data trajectory model difficult handle consider model hide branching regular hide parameter regular multi branching rewrite calculation target indicator platform syntactic filtering etc vast amount motivation develop yield interpretation track main result describe track formulate trajectory arc describe syntactic fit complete
ground result overlap superior determining pose support part da nsf thank support program content
coefficient specifically wavelet coefficient certain location scale small neighboring fine child magnitude accomplish overlap function child depict child group leaf coefficient orientation vertical orientation fine many option grouping
algebraic calculus base evidence function assumption probability inference algebraic additional justification allow consider positive integral inductive updating oppose aim lattice relation evaluation transformation algebra particular form inductive dynamic determine inference logical inductive justification impossible practically convenient axiom constrain relative candidate inductive inference evidence appeal expectation bregman entropy metric expansion ar entropy iv frequentist paris kullback leibler relative updating
decay generate fast detector plus energy event event take ref expect surface simply curve indicate dot induce integrate several region sensitivity low mass prior note curve fact show infinite inference choose carefully sensitive prior manner irrelevant break small expect count property define prior parameter beyond count numerous test argue interpretation test prediction risk reach conclusion example region complete opposite access result consistent systematic address prior
define base sum generalize precise dimension hilbert space continuous space act semi sequel minor statement main illustration consequence space finally cone semidefinite square g vector hilbert consist sequence equip inner begin background hilbert space example hilbert basis positive decrease converge k k k choice inner function namely element equal converge special symmetric function open subset semidefinite
rely framework provide variational hyperparameter consider work offer move section establish adapt gps applicability advance approximation enable thank university interference alignment author science innovation grant collaborative effort provide gps association source propose mixture novel characteristic gps mixture global clustered space variational recover show problem problem arise position association surveillance tracking htb human observer effort
knot locate posteriori consider location fact knot correspond surprising inference location
measure place nn implicitly calculation together give minimax rate minimax bound homology inference various possible ambient implementation homology become thank advance topology homology several sound drive volume lemma tangent ball collection minimal contain manifold dimensional piece add smoothly join inner piece two construction volume curvature evenly manifold guarantee otherwise construct outside manifold differ remain spread evenly density disjoint clear bind calculation main tell except volume homology least notice minimax straightforward case noise identical noiseless factor manifold identical step clean datum eliminate point radius homology probability
field mat ern unfortunately equation arrive function triangular mesh easy computed ij ds sparse therefore ds essentially numerically increase q zero matrix correspond ern field define sense piecewise nature vertex centre triangle advantage piecewise basis lot work leverage show functional derivative functional length edge mesh functional random space square fs ds fs ds q constant large triangle
covariance expand taylor third moreover expansion operator functional equation decomposition split three part estimate main quadratic equation build approximation conversely term procedure give suggest build size choose onto lead f x term negligible asymptotic
acc toy quality performance yield algorithm require illustrate criterion small smc european project simulate scenario job evolution activity virtual individual central france model second intel ghz proximity service initially census economic job job outside drive four parameter simulation census matching criterion census discrepancy observe eight infinity generate prior hypercube select move parameterize twice weight previous parameter study particle
interest specify non restriction let individual vector stacking alternatively contain table subject contingency table response configuration ignore either approach stack span complement fit constraint computation addition n almost infeasible canonical g contribution likelihood level log become
reduce believe long predictor getting want dr manuscript want li reading asymptotically time derive exist readily convergence sgd
object collect ten stimulus throughout object illustrate evolution illustrate ten development universal stimulus illustrate collection object begin network level empty grow intra module object add threshold intra inter take area spike collection feature map update implement previously dash coding cause reciprocal feature aspect change cause intra inter connection schema area inter level system principle organization recognition spike structure feature repository model single neuron grow feature storage
rademacher ball view predictor entry maximal singular zero elsewhere amount analyzing expect meaningful generalization indeed rademacher complexity know regularization fail focus uniform complexity ns magnitude argument analysis reduce logarithmic recent bounding tp combine various rank guarantee regularizer receive regularizer superior art uniformly correspond also assume predictor sample nm compare three avoid dimensionality slightly bad make incoherence guarantee trace trace magnitude uniformly show assumption regardless use focus optimality condition work ball trace mostly year denote frobenius surrogate give max row norm optimization involve constraint trace norm possibly
parameter good vc query column query boolean clause thm combination run join query size tuple combination join dim vc dim different due limitation increment number store list estimate histogram impact intra uniformity satisfied help estimation combination join create query predicate involve clause operator clause wide compute histogram query histogram method predict especially independence default histogram well prediction situation percent predict analyze quantity query deviation percentage decrease technique predict histogram hold fig histogram soon method prediction keep improve sample grow interesting deviation prediction histogram suggest high prediction fig marginal impact estimate sometimes give prediction less explain histogram reason line
kernel apply extensively effectively machine tool linear datum space high dimensional computation feature possible pca leading eigenvector take become match identity perfect pca whether simulated signal signal ec choice depend kernel ec knowledge kernel kernel manually kernel lead eigenvector detection case subspace know similarity work national
detector outperform aim measure difference realistic set compute within mean absence thus probability occurrence relevant improvement relevance advantage availability relevance make depict always every vertical meaning word depict idea relevant relevant query word topic indistinguishable close event explicit lack event although compute latter valid estimation depict depend depend discrimination query word confirm increase decrease large nevertheless either become small topic
five phase transition model regularizer sign exceed interpolation close show phase low transition regularizer sp sn complexity sharing validate handwritten digit illustrate accord handwritten extract collection dataset handwritten handwritten digit totally digit image provide digit provide total class provide record shape transform profile integer regardless learn ten digit make comparable range appropriate number perhaps large feature sure dynamic division care divide table divide coefficient sample digit sample row digit setup block handwritten ideally solve tune result cross digit get error support average theorem notation proof prove theorem rigorously norm
rhs fashion exactly analogous derivation upper rhs since recall set take ks find proposition corollary partly support department electrical engineering ia usa email edu regularize modified pursuit mod bp condition weak compressive discussion available partially sparse obtain condition mod similar another wang iteratively improve obtain support repeat cs cs residual recover signal early cs cs filter cs kf residual size result
relationship true distribute estimation interpret accuracy design contrast response treat need simplify standard design primary concern initial population population already concern unseen detailed random setting quantify essential design reveal approximate regression effect design soon enough approximation vanish ridge moment particular choice covariate enter except immediately square aware characteristic optimize appropriately theoretical modular probabilistic bound outline discuss excess ordinary discuss design computation inner restriction
restrict integral interval sophisticated equation proceed integrate back substitute follow number appear banach modulus convexity let reduce covering follow complex ball ball method covering banach consist norm measurement basis fouri recovery measurement generalization sparse state quantum physical describe density rank measurement carry measurement previously lead statement successful show sampling operator obey rip isometry acting say sampling distortion rip low work gaussian application rip hold open rip result
player dominant network strict nash payoff dominate utility economic context transmission apply wireless practically constitute measure connectivity utility path formation dominant empty agent pick agent become agent action establish agent become less payoff continue way satisfy game restrictive cf payoff dominant distribute payoff dominant network pool game refer interaction need resource rather resource share agent game pool pool view finite analog pool game common game player call action player failure pool game desirable profile example action correspond action payoff dominate game game moreover equilibrium profile pick increase either utility action
propose method knn svm knn describe semi supervise contain medium time cell cycle phase classify al sample synchronization discard consider discretized realization datum use use
sufficient table distribution see derivation sample reference implement hasting chain uniform sign uniform move chain stay markov basis ensure proportion table great analyze contingency table evidence conclude asymptotic value chi monte dramatically see eq differ eq less package example one procedure run cell case computation basis symbolic computation independence symmetry quasi independence basis theoretically numerical consider markov form move connection variety vanishing
sparsity pattern enough zero stay iterate numerical reference zero advantage indeed randomly eventually proportion spend non zero visible much htp blue iterate grow value start reach stay increase reach perhaps look thing via eq indeed suggest policy accelerate let adaptively change uniformly variant fix change particular illustrate effectiveness shrink apply start origin shrink htp notice nonzero element shrink shrink shrink modification iterate plot exception plot choice point initial choice b reasonable intuitively former preferable start shrink true numerically huge machine block coordinate relaxation alternate support necessarily measure prohibitive space gradient execute usual
pn replace jointly variance normally first central en n pn pn k w k k pn pn n pn normally distribute lyapunov neighborhood fy n summation take limit state let k take value origin p x x lyapunov lyapunov exponent x k exist contradiction period j e k j g x e contradict unique complete skeleton w nonlinear pt k k wu taylor expansion exponential fan page w w k mix mix mix sequence exponentially decrease fan page eq hand pt x e x g ty k mx mx x mx k mx martingale lyapunov complete acknowledgment support grant management institute university anonymous constructive grateful mathematical national nonlinear section width pc school
reference text current style preferable possible guide reference author al three correspond book page chapter necessary reference book first page article refer state report available include reference accept publication citation passive citation environment give place proof argument restriction conclude paper need repeat acknowledgement body information contract reader exclude material material technical code example appropriate please file material title acknowledgement material sentence give paper material style reference contain remark division mathematical science engineering university chemical co engineering mail mail com com u ac select parameter remark without response center change transformation predictor te penalize square square solution g penalty tuning
take model conditional ise exponential parametric vector example estimation longitudinal stochastic material let longitudinal explicitly parametric compute h dd enumeration rely vast majority involve transition metropolis hasting accept walk often mix mixing self avoid potential encourage family accommodate combination correlation background prove correlation connect exist
couple time coupling coordinate x always set show proportional time couple along edge loss since I imply analogous kn independent simplex immediate p certainly give check less prove possible fact event equality time hold part partition assume equality mark successful x equality k st coupling occur inequality
analogy interaction research question cross product advantageous computationally plausible course simulate feed sigmoid activation however abundance matching provide dynamically variable likewise variable product contrast common ica machine auto network interaction interact value weighted product closely model think multiplicative interaction energy rotation predict product mapping unit energy model receive cross model type multiplicative model motion relate frame vision relate relate projection encode independently content energy cross entirely practically field square somewhat fourier component analyse transformation case encode phase variation model know invariance polynomial compose unit sometimes pi unit pi stand sigma multiplicative interaction make build symbolic pdf triangle value three sum filter represent early subspace put sum compute share filter parallel bi linear
label boolean case occurrence embed leaf label pt implication right assume claim give follow query sample combine query path query path boolean convenience boolean query see boolean consist path query representation semantic characteristic also infer boolean point keeping contain redundant path query label query boolean query mention previously query conjunction infer path set path root query need reduce path boolean restriction relax unary unary expression path unary path root select unary iff unary root query begin path unary query class belong practice advantage set store leave leave pattern equality text value rarely path query construct essential result show restriction finding setting inspire extend stand begin universal consider select node sample construct stage htb xx xx xx xx ss r decrease replace let l edge find stage attempt every path refine invariant mutually every stage take last occurrence symbol create copy replace unchanged third attempt execution xshift n edge
abstract formalism exploit formalism author modify probability relevance intend interference traditionally concentrated extract evidence accurately belief ir superiority document ir ranking accordance vector classical available measure vector ir relevant threshold minimize definition use introduce aspect relevance subsequent explain relevance poisson observable introduce observable rank optimal observable confirm observable make remark actual refer l ir concept observable color state recall alarm threshold observable frequency color relevance
establishe independent decomposition matrix leave vector value
conditioning suggest estimate tt xt yy yy py xt estimate tn zero st possible lead mention asymptotic irrelevant least irrelevant measurement difficult censor truncation provide might time range limit large complete ty disease vary converge h h replace propose yy tt
e whereas absence quantum term mathematically information unit suggest customer unit index represent informative interpretation implement indexing time internal physical number answer correspond occurrence index index represent occurrence occurrence something geometrically vector superposition interact place vector observer move subset assertion element universe event belong subspace subset belong span basis relationship g plane span information give whole x maximize eq note connection even interpretation tie together criterion assign whether algorithm implement perform follow read occurrence e feature include relevance etc reject precede ranking
separate concave satisfie know digit trend logistic crf multinomial value discrete naturally multinomial scenario assignment tie q configuration write involve straight logistic simultaneously predict correlate chain crf define normalize computed variant forward backward hide multinomial see crf noun share word combination group plot regularization sequential dependency order use edge variable wish pt c original case know runtime lattice tree minimum structure degenerate structure optimize pseudo conditional form logistic compute use define function denoising different illustrate method
envelope desire follow factorization box monotonicity real multiplication simplicity stick three case along probabilistic arithmetic yy box number dependence unknown fr fr explicit formulae arithmetic operation provide inference marginal box box save reasoning cumulative result interval induce usual strictly rely bit induce cycle cycle cycle cycle achieve still strictly without notation conservative extension dominate natural valid coincide similar hold cumulative arithmetic operation probabilistic constitute example concern harmonic toy concern height issue simple harmonic determine mean fast engineering complete determine actual design account expectation uncertainty box seem quantile necessary rescale take supremum simplify reasonable distance calculation modelling fill node cm cm rectangular xshift cm leave
outcome information gain give time bandit horizon always grow give infinity increase intuition growing bound expand h simple action input markovian determine probability agent try learn corresponding observe perform estimate environment array gain observation accord precise infinite recursion discount initial investigate gain expect
process penalty necessary interpretation regard condition characterize play central condition example viewpoint insensitive state ensure pose smooth control complexity second part kalman formulation operation execution finite value degeneracy support low subspace computational computational practitioner powerful believe model development number application
somewhat entirely estimate stationary time suffer major analogue histogram increasingly cause choose increasingly integration numerical become increasingly kernel however question certainly drawback derive principle violate seed literature series asymptotic generating mix mix rate single stationary independence active series
appear information variable add basic gate three fact interaction indicate gate information focus despite fact gate gate interaction variable redundancy return indicate cccc c default state node receive drive note connection network pr correspond topology c ex ex represent node example partial indicate entirely redundant counter node act independently structure partial information redundant return result magnitude redundancy note interaction exception several correlation amount result total correlation dual reflect scale interestingly unique example return redundant interaction intuitive driving driving note less redundancy interaction result redundancy unlike solely partial return redundant provide accurately drive
picture optimal make detailed describe name swap swap swap reversible minimize rejection configuration iii w jj j start use swap operation construction therefore whole
assume covariance latent specifically graphical space cca advantage low model allow optimize consider come produce pose summarize pose representation software choose similar probabilistic noise latent dimensionality vary give pose computed variance cca square
result algorithm majority language universe denote query description circuit restrict bn call run tn note learn circuit say majority depth boost learner exhaustive negligible harmonic membership extension argument one harmonic harmonic majority depth circuit evaluation result privacy language parameter work threshold reduction release improvement release new improvement central open question differentially private release work database run way release way provide rgb definition question release database contain sensitive answer computationally release seek sub exponential primary reduction private release query thresholded predicate general learning threshold release differentially private way conjunction counting way contingency release bound ignore dependence database database bound database ignore run database answer way run run add answer way distribution threshold sum relevant predicate polynomial threshold overview fouri database value new polynomial release counting query depth counting query harmonic circuit elaborate preserve release threshold section loose release simple reduction query release threshold
du x du final law j type law eqs feature success z tell data exhibit exponential q three note finite subsequent beta bernoulli hold law atom choice discount keep total mass concentration carry extension stick break generate feature bernoulli empirical growth right number derive line theoretical theoretical line leave power law number law type quantity number show classic black case black point asymptotic quadrature plot compare asymptotic asymptotic ultimately prove respective show curve straight behavior describe growth expect feature represent line normalization grow expectation many first change connect middle green eqs index assignment great type iii law type law somewhat easy visualize iii law lack difficult visualize
goal gap problem computational distinguishing enough bottleneck life line research joint stochastic descent potentially demonstrate via importantly advantage example supervise learner past target spam email spam target x natural loss classification otherwise algorithm I unknown distribution domain training assumption prediction goal risk tell risk agnostic pac learner expect risk return denote upper example main propose
aim paper sampling fairly functional theoretic treating procedure treat truncation unify function hilbert subspace functional covariance estimation square penalty tensor rkh rate argue logarithmic factor eigenfunction resemble principal line treat various smoothness functional component translate ellipsoid principal significant substantially sampling difference explain lack lead identifiability issue difficult order exploit empirical well concentration exploit localize reproduce hilbert technique norm finite bound remainder organize material reproduce space study functional statement discussion consequence section result technical defer material supplementary devote lower sharp hilbert schmidt inner e ij introduce
latent quantify associate processing bring idea powerful describe image bayesian model probable posterior latent uncertainty practical compressed sense framework allow principled employ prior heavy filter probabilistic signal compute heavy tail sample gaussian close computation adjust parameter tight mostly quantify quality give rise bound ep relations variational sort integration contrast base order around algorithm require variance sparse point routine approximation capture
tag give situation hypercube possible large meaning shape underlie influence reliably identify look class work panel answer adopt membership reflect estimation near sparsity recognize accurate qualitative influence number covariate binary positive negative medical test outcome determine diagnosis hypercube series therein expansion reason various length short central coefficient widely boolean harmonic hypercube sparsity hypercube describe thresholding recursive thresholded analyze prove illustrative technical denote string use concatenation string kx return empty index arrange order real number denote generic whose change absolute throughout function inner reference introduction instead generalize
nash large search support game say clearly surprisingly exploit technique game exist subset trivially subset small game game equilibrium assign strategy least constant broad equilibrium game ny yx show strong theorem provide game random define pair nash e game small equilibrium universe size form take uniform nash equilibrium go uniform argue polynomial nash define player similarly define nash equilibrium justification lower bind contain indeed least belong moreover evenly satisfy appendix uniformly random succeed equilibrium inverse complete strategy enumeration sampling come game mix
result consistency asymptotic normality leave asymptotically side second asymptotic normality leave er drive let variable true bivariate mean algebra behave therefore establish side right hand direct addition know p p theorem effective size pt theorem program award award foundation science david model year researcher various remain pose address yet question illustrative exponential random answer question depend rate estimation magnitude practical sharing normality random dramatically sciences biology bioinformatic economic mathematic physics contribution come spectrum see perspective economic
role using even find likely table since customer utility choose table choose customer e accord belief construct available restaurant combine game learn chinese restaurant game new analyze predict social recursive achieve tradeoff decision decision table customer advantage playing dominate learn contrary table play later well thm department electrical college md usa communication decision either decision agent learn make decision play role social still limit restaurant random structure decision chinese restaurant process chinese restaurant formulate analyze chinese restaurant game derive achieve network setting learn field communication device adaptation cognitive decision achieve may need may external market limitation expand experience help mostly case decision enhance take account belief belief maximize social impact reward impact e one action reward state state becomes learn agent nevertheless reveal preserve partially reveal network subsequent agent former
version martingale probability poorly handwritten labeling image point decide label tend computationally change pool unlabele sample budget constraint allow query informative pool al overcome query discover query necessary pool al margin minimize provably estimator risk unbiased stream algorithm importance speak round put pool pool query learner minimize risk concern e weighted forecaster expert weight loss pool forecaster prune soft manner place determined pool true linear
j case q set event occur recover clearly pair n k union bind argument noiseless r rs respectively bind subset r sa monotonically second bounding ensure binomial monotonically entropy far exponent apply bind stochastic third bound pmf symbol note low main proof proposition typical delta chebyshev bind law zero arrive bind continuous second exponent prove denote say furthermore upper hamming partition hamming distance loose distance optimize free later use fact monotonically decrease k cardinality differ term cardinality hamming weight position upper rank lemma large equality objective sufficient exponent sufficiently small denominator justify nc negative obtain particular sufficiently differ choice also remark restriction serious claim complete direct circular convolution pmf fourier dft dft probability evaluate whose linearity dft power yield decompose q complete code single linearity since complete appeal pairwise lemma sum eq integer define corollary field element draw pmf constant version also integer imply sufficiently dominate satisfied apply inequality since borel lemma receive electrical engineering college ph electrical
lead eq mean common covariance mixture dirac avoid enforce optimisation lead updating importance replace student robust alternative freedom degree freedom choice among cast section thank shape student distribution know gaussian case moreover covariance pooling way prove clear optimal optimisation sufficient nevertheless express em root use approximation specifically decrease size importance estimate set currently fit associated form note size define lack role latter constant target recursion obviously normalise unity effect kernel usual thus sum unity measure execute iteration decide user parameter instrumental work formulate
list ranking list response gene ranking attract considerable collect position gene ranking vector sample use estimation adapt rank list behind want exchangeability appear ranking gene exchangeable always gene situation exchangeability pair new exchangeability random rx pr exchangeability variant maximal exchangeability distance analogous introduce exchangeability introduce attain exchangeability gene exchangeability score random hypothesis uniformly cardinality normalization score analogously side exchangeability uniformly exchangeability gene exchangeability leave irrespective number gene distance exchangeability panel synthetic framework list list position give rank list dissimilarity comparison specifically certain list determine important list order weight rank universal gene microarray order subset characteristic list create list order denote position list characteristic exchangeability global list use similarity angle
unweighted unweighted since pathway preprocesse keep gene standard group typically select group gene involve since enforce reasonable table select learn version mention coefficient allow remove lead support suggest min fold signature fold choice weight weight fold microarray drug target identify disease form densely information interaction biological perform edge group breast datum present pathway reduce almost suggest informative fold select isolated graph connect table large average deviation might increase connectivity merely cause overall select select gene connectivity gene gain connectivity new drug target cc generalization group sparsity group group connect graph give condition recovery decomposition induce parameter highlight weight correctly empirical characterize collection group group efficiently encode collection case analyse consistency comparison lasso formulation important continuity order prove start section notably theorem ingredient lemma several lemma technical lemma correspondence resp correspondence h c correspondence l u check c moreover product state maximum continuous real correspondence consequence indeed definition apply subdifferential view direct consequence maximum correspondence correspondence easy verify value need compact therefore first ia md ik u c classical f paris france norm sparsity predefine overlap call group latent
explain high order se perform due hyperparameter become performance benefit integrate hyperparameter gaussian process generalize class additive structure indicate additive dataset allow gp recover arbitrarily flexible modeling efficacy interpretability procedure hyperparameter additive gps state regression acknowledgment like helpful repository follow wishart construct fx x sum add well variable ignore
preferable contamination demonstrate contrary suffer ht db background display change estimating point present iteratively change point determined stage apply value decide segment stop large result
reduction many use experiment competitive performance explore increase random projection accelerate small gap imbalance multi several rank accomplish projection minimum row mapping label representation via predict label nonzero representation coefficient concentrate find label correlation preserve mapping explore label robust imbalance relative mapping feature knn cpu ml knn
discrimination affine much intra digits service handwritten digit pca coefficient kernel validation c tangent human texture class texture pose illumination surface variation classification illustrate large intra across always well optimize markov field useful texture algorithm digit exist rate optimize polynomial
density distance prove estimator principle crucially old bandwidth furthermore proposal reasonably least principle choose kernel like risk distribution distribution call order monotone kernel density distribution depend however functional aim minimize empirical discrepancy bandwidth choose depend take kolmogorov generalize distance metric suggestion differ multiple discrepancy principle introduce widely choose recognize problem discrepancy chapter chapter detailed account
work incorporate ordinary least mle variation exceed word observation establish regressor iterative evolutionary accord trait comparative regression cm give tx correspond incorporate step evolutionary curve different regressor search conduct exhaustive search space high search notion evolutionary regression curve explanatory
transformation integration properly isotropic contribution transformation fig illustrate analytically compute mutual influence capacity marker correct htb limit svd perform approximate evidence probabilistic pca score exhibit principle cross learn svd compute
arrange lattice periodic boundary connect near keep randomly cost repeat cost reach follow path procedure appear exponent distribute fig system far example chinese truncate degree show usa obeys capture introduce degree heterogeneity
herein author interpret represent policy temporal evolve change node many static propose framework relational evolve vary predict time relational exploit temporal outperform sophisticated ensemble issue temporal relational evaluate temporal classifier outperform compete ignore necessity relational ensemble dataset evolve temporal relational representation temporal relational ensemble temporal temporal relational internet citation social network arbitrary temporal apply construct temporal relational attribute select repeat step relational relational tree provide possible temporal model special node attribute email attribute attribute attribute remain existence add refer tw refer assign c pt c pt pt traditionally classifier conversely classification three attribute
objective belong category sensitive prove algorithm player pure equilibrium consider share access access represent restricted level also wireless optimization main traffic capacity coverage operator capacity area macro able serve demand mobile operator solution reasonable cell answer problematic sharing access mobile operator customer problem model service customer actor indeed amount bandwidth bandwidth pricing determine agent interest actor especially spc equilibria share bandwidth pricing et al game bandwidth require existence primary article profile potential game classical pure
additionally write start simplify n generate probability numerator dominate furthermore dominate receive value k mean point close distance great start assignment must move amount cluster gaussian posterior assign posterior c asymptotic go mass limit penalty put everything behave exception cluster form whenever away exist centroid initialize cluster specify dp algorithm depend cluster dp datum process one future consider adaptive method choose ask objective loop mean simply objective number threshold control
plug density accurate around extensively nonparametric condition various level third turn contour formalize modified exponent efficiency prediction class nonparametric inferences negative taylor expansion degree old differentiable function require essentially require roughly contour fast contour exponent condition modify exponent density condition level constant eq differ interval contour contour level measure constant unless cut indicate contour allow enough contour contour smooth away
eqn explain crucial conditional give prior map estimate sdp give eqn support set eqn relation suppose sdp show solution estimate satisfy eqn regularize laplacian truth observe importantly family qualitative feature varied procedure wishart density eqn misspecification world generate ground truth equivalently graph width height lattice necessary edge
interactive require evaluate point interactive interactive interactive follow differentially private release mechanism interactive run per adaptively non interactive bind release mechanism interactive set eq theorems notable universe sparsity give inefficient mechanism query since typically view polynomially database improvement fact work universe entry string attribute specify length universe element deal infinite universe would length universe element encounter run universe query may still possibly infinite run resource interactive match would perturb require trivial trivial sparse course query super exponentially advantage interactive interactive answering query preserve differential give answer algorithm answer problem
assume parameterize cell proportional mle solve straightforward variance relate hashing matrix burden numerically first probability intensive cell rest include cell bit hashing
covariance marginal method table cell partition cell free near free value contingency determine upper sample algorithm replace distribution variance bound unfortunately guarantee identify successful line bound condition work successfully justification strictly lack table define induced marginal rely calculation algorithm produce slightly modify fix random free cell present substitute rigorous applicability expect seem calculate way table subject linear markov algorithm sample context algorithm discrete proposal chain tie target moreover end importance identify present reference set feasible free value bound translate low upper receive permutation probability could candidate vary considerably theoretical small mix state stationary
infinite consistent thm thm conjecture infinitely subset natural number poisson measure monotone undirected show infinitely projective permutation mapping infinitely system subset restriction show connection machine bayesian inference infinitely exchangeable point set number infinitely exchangeable area analysis previous ng
q n n one n x n n eq end quantify independent current independence lipschitz term satisfy define highly concentrated sense nx z last nn positive also I conclusion drift drift limit drift assumption drift acceptance equation express x nx prove suffice let prove remark lipschitz choose independently prove converge lemma x nx dominate prove follow zero nevertheless quantify bernoulli independent position prove satisfie equation give proof independently definition readily nx q jx nx n sx j j x martingale error introduce approximation hold pair satisfy eq martingale proof equation essentially identical easy globally sd x equation consequently suffice nx suffice show equation
pt university department university college place bt uk department mathematics ny usa xu mathematics ny usa machine obtain lasso total method regularizer proximity advance approach practical admit appeal proximity choice work group fuse regularizer solution certain proximity efficient finite accelerated method order technique class argue simulation efficient state art overlap match fuse structure lasso
firstly elegant nonparametric gp discussion concept drawback computational heavy apply already limit overview value assume zero variance number choose characterize since noise well continuous function accordance outside follow scalar mean variance key mean used provide subsection observation discuss subsection address quantify provide first important provide amount search gp shannon necessary mathematical measuring quantify information quantify avoid conceptual communication ignore conceptual shannon
site neighbor site call cluster thus notion determine configuration I probability give site active site probabilitie graph influence via adjust active alone fix random consider simultaneously permutation site visit consist cluster denote coincide permutation hence active site active site site first permutation estimator active permutation storage consume run permutation check site merging already check site large site belong active belong label update site
distribute tend lie direct look normalize define definition q enumeration subtract model prove consistent related capture stationary uniform prove depend appendix evidence graph model source even output markov chain stationary iid non illustrate text would find hope markov source situation count similarly require prohibitive appear situation stationary source illustrate language length hold definition two case behaviour range example mention bioinformatics finance sensor write style security approach work alphabet suit bioinformatics analysis style review
optimal show gain multiple system shown see observe dimension consider communication multiple recent exploit optimize design wireless network concept irrespective directly depend neighborhood thus power scheduling jointly schedule scalable even centralize receiver receive diversity reduce thus wireless bandwidth feasible
continuous reach imputation validate possible number imputation hour available white four miss order encode error compare significance encode hash miss capability categorical deal categorical see datum emission set pair dna domain patient different record mainly nominal fashion nine level always miss namely decrease data decrease range hand except point primarily tailor
make predict analytically example effort eliminate lead improve exception change need implement weighted fit currently partition fold imagine partitioning carlo term recursively fold could confidence vary spatially despite well mc fitting point transition number average mc know never sample close post processing trying estimate integral david uncertainty quantification engineering calculation mc suffer sample stack post exist
subset small one reconstruct fraction noise need preserve subsequently provide privacy privacy see add bound avoid barrier exploit formalize privacy preserve online privacy privacy programming underlie gradient ascent private online class improve regret exploiting include differentially online offline practically problem online online section convex online player adversary select pay hence tf player minimize cost fix select move incur offline solution provide let select convex regret select random convex regret notion privacy context privacy convex differentially give hold change amount dataset much reveal extra privacy affect td
framework prior induce covariance continuously combine computation inference specify dictionary cope require seek control propose stochastically shrink toward flexibility zero shrinkage explore property nonparametric focus regression support property nonparametric constraint choose aim condition specify satisfie statement row towards enough induce place prior function arise dictionary function weight specify assumption property condition wishart variate wishart specify satisfie provide condition prior choose assumption prior inverse denote induce element conclude limit predictor towards decay go zero infinity diagonal equation length property regression wide sense solely order stationarity follow stationarity recall solely imply truncation propose derivation update conditional conditioning associate hyperparameter gaussian specify define follow examine standard hyperparameter shrinkage update h incorporate induce dependent sampling sampling x ik eq condition
cross distribution belong relative entropy exponential family separate term f bregman generator divergence leave deduce explicitly constant kx fx e kx f kx e fx close expression enyi divergence
corrupt online missing arise comparison round extend hypothesis relation value miss bound several uci baseline algorithm example suffer life corruption every adversarial situation arise medical diagnosis array medical constraint incomplete non response task deal image setup accord distribution minimize h scenario imputation imputation unobserve feature value x make elaborate I rademacher thereby also hypothesis reason imputation analyze general hypothesis definition regret control rich linear predictor
object unknown figure toy symmetry depend try pair present dataset edge b pair side show relation cover framework relation become clear example start notation relation moreover relation mapping define inverse value relation value kernel rkhs rkhs transpose cardinality formally follow select appropriate quadratic rkh accord admit dual representation form dual correspond alternate edge graph establish every consist couple node norm q mention make different case loss hinge opt function function result type regression loss like insensitive case specification offer couple restriction relation specify kronecker individual kronecker
ard display solid run deviation firstly ard recover hyperparameter datum use correct db data top bottom robust recovering secondly assume however statistic observe estimation significantly ard bottom much right initialization deviation zero demonstrate section report introduce invariant value body show fig truth position background form associate four equally arrange adjusted black small large ard hyperparameter initialization return ard number basis iteration ard see perfectly recover adjust log scale log ard show return ard manual correct low indicate presence minima consistently ard inspection learn dictionary learn ard decomposition despite residual noise inspection reveal despite fully relevance evolution spurious component
well nm maximum block give x norm block block maximum unitary invariant block diagonal unitary mn mn stochastic q block complete hermitian size radius submatrix decomposition unitary u u eq hermitian matrix lower complete nm nm mr stable matrix satisfy diagonal lemma express complete replace submatrix assumption evaluate expression evaluate term rhs knowledge proceed invoke principle approximate q arrive theorem tu h adaptive agent enhance estimation reason level help reveal process important get
tangent ready term integral radius big approximation product determine high mean enough ball radius around change ball radius radius replace integral ball interior function take order odd therefore vanish term inside integral order next interior near boundary normal term normal normal near boundary direction show local result generalize walk interior l u limit easily maximum variable bound high region region maximum analysis apply inequality notice come equation
subset entry denote submatrix index whose entry index critical quantify incomplete span give incoherence control paper detail vector influence demonstrate quantify fix classic absolute deviation admm alternate multiplier therein accord introduce constrain manifold neither estimate could triple admm refine approach detail evolve problem relate corrupt let index rank matrix span version rewrite offer efficient differ major column time greater say discuss piece variable adaptively choose estimate observe optimal find lagrangian equation admm alternate quantity invertible guarantee soft discuss admm detail
dp dp procedure converse everywhere unknown whereas require hold everywhere show compose give constant composition allow rapid allow interactive eq union combine composition release datum release together result composition first histogram privacy differentially count private histogram partition sx mechanism satisfie demonstrate take output neighboring meet either complement partition fall
vector markovian member integer exactly truncate sample correspond truncate signal radial assess correspond calculate apply receive radial velocity mixture estimate calculate marginal use reliably suitable selection amount parameter purpose extract several search nearby star method therein radial velocity reference therein therein search rotation reference current determine model assess ability explain also assess set term statistical different number star relative
location calculation processor core avoid pre operation iteratively implement problem logistic r accelerate quasi newton bfgs factor exploit operation r much precise conclusion nearly iteratively least square converge reflect quadratic likelihood poor iteration sometimes severe cause unless trust basic version expectation maximization robust combining quasi equally robust fast pareto penalty simulate least grey line
fact nearly perfect unnecessary removal impose penalty removal figure visible end optimisation method concern notice detect cpu significantly two cpu average cpu start fail correctly cpu rmse detection classical least criterion regression sensitive investigate find various derivative method dynamical optimisation factor variable ten criterion removal ann backpropagation undesirable feature loss impose remove criterion accuracy outlier reliably ph bold indicate r ph value bold r outlier ph noise dimension ph mx mx mx bold r r ph bold
immediate given obtain valid qualitatively generalization case statistic separable infinite vector machine statistical special vector rbf kernel svms space rkhs
section matrix map depict panel notice quantity information estimate associate obtain image subspace svd whole hyperplane locally image comment make svd panel reflect contain conversely svd bottom panel large linearity summarize evaluate estimator linearity quantify property exploit hyperspectral follow hyperspectral acquire field panel scene black area compose green pixel abundance estimation pixel live dataset refine locally subspace bottom compute svd panel local unable non scene bottom panel
block graphical lasso paper original glasso operate dual directly box use interior operate quadratic problem every row dense qp lead attractive kkt derivation entry optimality box qp consequence w update qp equivalent require column last implicitly solve solve return though problem sparse use solve boundary glasso primal glasso glasso glasso operate regularize appear choose use coordinate use warm available work copy way appear generic descent algorithm easily detect stay zero get pass solution zero hand procedure effective coordinate update gradient column experimental glasso glasso zero per glasso dp small qp particularly attractive
along line planning city distance effective heuristic planning module guess heuristic heuristic action follow e heuristic initialize terminal execute ss terminal term search short state typical simple ignore need position immediate plan advance play gray position neighboring depend situation another state depend figure planning figure plan planning planning planning step initial action select tie break episode run randomly constant experiment free
mean analytically ex ac ac uk combine structural bayesian neural flexibility correlation response heavy monte variational multiple volatility model demonstrate multiple multi volatility expression popular expressive interpretable avoid fit predictive thorough machine learn network show bayesian hide value aim correlation particular assume could input imagine represent heavy laplace something would vector latent additive kronecker gaussian network random label square function label vector gps adaptive show generative hyperparameter
discuss lasso solution maintain accuracy initialize intercept minimize eq squared iteratively calculate st calculate iteration evaluate scale choose coefficient single enter iteration avoid effect eq parameter update expensive accelerate fast effective particularly compare used formulation technique encourage little I tolerance binary goal three process individual original statistic attribute dataset object dataset total negative binary use select use time cross assess overall misclassifie total misclassifie observation misclassifie experiment show solver overall ensemble numerical ensemble bag three repository fold validation test dataset equally proportion class testing bag take test name breast ghz processor rule ensemble many contain multiple class identify applicable ensemble easily
event see gamble x ax generalize describe link unconditional must classical theory generally subject example gamble maximal decision comparison gamble gamble conditional think operator gamble subset note social event function option gamble choose relative option necessarily single preferred option choice choose gamble present partial coherent low gamble whenever rise function gamble eq recall envelope maximize gamble admissible maximize utility least admissible maximal strict partial non event gamble write order call empty gamble event select gamble maximize gamble usually worst depict coherent example contamination observation q find gamble apply suitable list gamble reward determine shorthand notation l gamble induction require gamble obviously utility
must gate validation group see split calibration remain problem perform without analyze understanding plan experimental expensive investigate may able experiment challenge newly split increase capability paper demonstrate systematic assess extend reproduce experimental datum use ultimately capable gate fail quantity partition constraint allow determination ii requirement edu
continuous mapping lemma imply every path u expression gets maximize argument show follow iterated logarithm walk satisfy hypothesis show c apply notation w application lemma expand n numbers z class vc subgraph integrable envelope notation four converge since measurable assertion immediately family almost surely see density equation show satisfie lemma n know corollary n om let note n om bootstrap distribution poisson h u denote great bn bn sufficiently dx dx another borel bl l easily turn take prove bc characteristic prove permutation page subsequence converge z sf surely subsequence proposition nh b increase write n n kt k kt kt h ks kkt h k k permutation random I h sure choice subsequence proceed analogously compound distribution
across diagonal node elsewhere describe role across homogeneous value initially membership around around membership center third compatibility matrix give diagonal variational original ground trajectory mix show simplex three verification dynamic membership stochastic though simplicity trial evaluate actual infer membership method membership capture
setting specify loss allow configuration variable find refer optimisation establish extensive regard exist equivalent solve lar algorithm cone solve approximation amongst high excess used model prefer sparsity give weakly specify laplace loss laplace contrast optimisation parameter latent positively constrain exponential distribution nk sparse explore specification place rate shape respectively model accomplish carlo base
heuristic surrogate offer full three order magnitude test moreover accurate reveal correct requirement promise mean adaptively stochastic optimization offer considerable gain incorporate experiment hierarchical structural successful design resolve current surrogate provide thorough bayesian natural extension rigorous incorporate sequential quite effective research energy office office advance research rv r htb different reaction r species body specie explanation delay peak release characteristic peak characteristic peak occur characteristic release peak value peak peak htb bind prior expansion heat release validate htb htb htb outer iteration surrogate utility course terminate expect utility output factorial htb pc htb surrogate f htb htb different size line outer loop red dash bar show deviation gain information gain individually fix perform conditionally design marginalization obtain differential individually use fact equality remain expect experiment never gain equality conditionally equality information two equation bias bias reduce evaluation result source list question nonlinear information estimator one outer loop blue marked marked bar many
recover sign sign arbitrarily fix sign sign moreover recovery hard refer reader proceed easier construct four entry quite standard literature decomposition probability provide notice generalization involve random utilize sub condition follow define imply uv inequality schwarz notice p f hold assumption theorem strict must assumption existence variation idea approximate te tw error uv te consider observe clean also small uv te largely still natural subtract remain one set repeat correct geometrically
analyze analyze large classic r enyi node uniformly tractable assign interval idea graph discover perform graph path form node depth traversal present discover origin study visit correction procedure correction sample small methodology consequently bias correction independently two paper fundamentally final heuristic generation base generation implement situation generation expensive generation capability reduce graph sampling unfortunately none applicable problem speak use either sample iii average previous work change successful correction scale life internet topology procedure property complementary ready use implementation stress base differ without graph except seed exploration seed recursively visit technique walk walk walk technique start choose
wikipedia behave surprising wikipedia cluster cc build find page metric platform validate score presence long tail page among score solve issue highly score identify interesting wikipedia phenomena level medium page score single aggregate shift less due role work alone score highlight group improvement language processing aspect platform collective interested thank nsf award laboratory nf
set belong capability rand rand range measurement partition indicate agree partition compute rand schema partition latter partition report rand repetition l rand reflect appear high ht result since discover highlight figure highlight institute mathematic report approximately locate us period test series consider
site consequence lattice threshold infinite lattice infinity site cs cs translation invariance infinite thus site call leave analogous pixel black exponential factor mean error correctly site leave right connect hence imply addition prove yield cluster infinite lattice possibly cs cs cs infinite lattice hence essentially pf en en en enough therefore eq help convergence remarkable tend exponentially fast object exceed ii
small conditional entropy integrate dynamic centrality affect principal see fig topology value sis directly also find remarkably present peak similar variable sis topology dynamic feature investigate relationship complex report critical concept dynamic level along allow effect approach despite uniformity er topology dynamic uniformity article investigate
challenge interval fundamental find monotone operation interval computation due dependency effect optimization robust introduce numerical computer interested method propagate function contrast define irreducible derive incomplete increase information great importance uncertainty model separately probability assign evidence conjunction combination
describe maximum posteriori yshift rectangle yshift xshift lda document topic term feature topic replace hilbert space linear softmax simplex generalize replace expressive semantic rich dynamic belief tractable address dirichlet basis extend effect admit efficient implementation kernel topic model generalise price increase modelling flexibility computational cubic number document cm pz pz south pi edge pi
element unconstraine project perform iteration reduction large fairly invariant seem neighbourhood fix quite closely sequential manner accuracy produce work fixing despite parameter estimate likelihood essentially identical seem advantage arise along move symmetry maxima explain responsible remove symmetry range standard deviation log estimate corrected value likelihood numerical multiply method choose six third intercept design intercept result perform smc gibbs step step case optimisation despite marginally comparison smc gibbs perform mean estimate one find standard value gibbs agreement distance value sampler automatically provide likelihood likelihood smc marginally comparison standard integration option still smc em probit monte truncate normal build student freedom produce smc sampler seem typical ideal smc evolve update avoid greatly particular performance example six fast since monte provide truncate student clear extend variable extension multinomial course complete em result generalise full examine identifiability model
share strength conversely fail bootstrap edge motivated development bootstrap subsampling setting problematic interesting investigate subsampling maintain robustness averaging implicitly error optimize estimator combine way average proof hz hz suggest notation view outer define set behave asymptotically though directly empirical subscript brownian furthermore pf almost surely desire state sequence process converge brownian noting class behave eq functional delta delta conjunction assumption b b j assume proof desire first support sample simply use computer yield consistent upon align capacity computer immediately method two trend attention regard become increasingly resource toward architecture provide hundred thousand processor distribute present capability storage inferential clear bring issue include issue exploratory visualization remain inferential uncertainty remain dataset frequently fit accurate assess efficient processing necessary achieve bring notably wide inferential bias uncertainty risk
dimensional spectra genome face novel approach fit unsupervised framework dimensional manifold relate base near regression optimize variable reconstruction problem neighborhood solve allow topology appropriate sort iterative variant experimentally dimensionality important play understanding
modeling rule provide formal framework p normalize parameter ignore distribution empirical evidence aspect make inference unobserved reasoning uncertainty become lack explicit characterize observation publication interest considerably simplify nuisance variable value paradigm principle statistical complete process contain expectation relative central challenge often include analytically demand use approach include mcmc sufficiently pool random underlie characterize intensive slow variational convenient potentially accurate analysis description uncertainty summary posterior sufficient obtain instance ignore interpretable summary point sufficient cost crucial outcome discussion learn thesis formal inference set incorporate paradigm view rational concept average analysis learn highlight often focus optimize estimate bayesian prior distinction comprehensive demand analytically often approximated analytical procedure mcmc variational summary summarize uncertainty result convenient available learn procedure challenge complex topology global optimum exhaustive intractable contain ultimately resource central thesis standard representation identically p fit maximize maximization likelihood estimate model operate function optima additional less posteriori additionally ml map ml case assume sample dominate become point ml role prior task needs take principled accelerate order stochastic mcmc exact solution computational potential acceptable give gain efficiency characterize uncertainty tractable distribution approximate potentially approximate easily maximize subsequently datum yield divergence q equal minimization analytically tractable kl maximize low variable factorize approximation infinite model marginal density variable optimize value expectation step evaluate analytically variational optimize expectation determine density update contrast marginal maximization step belong suitable convergence identification optimum guarantee incorporate prior avoid framework publication learn learning identify either globally optimize value method require accuracy objective assume topology set derivative optima estimate gradient direction towards desire gradually improve ascent characterize optimize manifold appropriate function bfgs method newton approach thesis probabilistic probability unobserved event characteristic generate observation characterize overfitte refer avoid overfitte overfitte avoid collect investigate accurately new learn test performance assess testing publication thesis widely probability underlying simulate variability observation space event resemble density multiple bootstrap estimate variation bootstrap assess publication flexible therefore simplicity impose soft penalty prefer solution publication theoretically principle
characterize number graph weak depend graph define expert reward reveal multi armed select reveal assume mean reward action change round case establish achievable combinatorial large clique partition number clique node deal constraint regret minimal clique find np hard clique handle graph change round versa regret direct graph efficiently graph regardless demonstrate approach analysis armed action alternative endow rich however paper armed bandit include action reward assume draw action property approach combinatorial bandit consider choose observe crucially reward separation obtain value partial
turn q co vector physical value physical phase time choose least disadvantage approach freedom location section suggest possible good belief belief panel relatively coarse exploration right exploitation blue bottom panel roughly forward cone bottom leave system section radial function initially equally large scale cover trivial I sample region intuitively belief suffice belief evaluation comprehensive discount control control
kl divergence permutation increase demonstrate kl place place c c kde anomaly pure false false test would receiver operate contaminate robustness check anomaly contaminate detection alarm pair well yield aim contour kde tb htb c kde employ psd view rkhs kde correspond quadratic employ robust robustness contamination training kernel estimate iteratively weight influence problem make assumption percentage contamination nominal problem interpretation influence
discuss disease prominent role non disease transition state sometimes disease depend duration influence medical decade state suffer disease specific duration show disease incidence high rate historical reason number disease henceforth rate depend time
sufficiently subgraph order near neighbor hierarchy subgraph approximate hierarchy form perhaps concrete approach context guarantee prune cluster carefully work pruning prune spurious guarantee set tree salient still estimator underlie cluster recover interestingly prune tie base rely identify remain along dense region cluster clean formalism propose seminal linkage empirical cluster unfortunately lead
tangent local vary analysis accurately track subspace bind yield behavior demonstrate trivial introduce principle quantify perturbation tangent matter need manifold noise concern indeed perturbation convolution clean point geometrically manifold help center manifold volume clean correction idea volume clean issue concern determination estimate tangent plane taylor series generalize dimensional exist represent principal direction system align point coordinate quadratic manifold first tangent inside ball sample subspace capture inside volume ball standard sampling linear subspace sample contaminate noise I store allow decomposition plane component vector q wish pca direct access work proxy span recover invariant refer local perform pca give density quantify define local neighborhood note align coordinate axis principal direction quantify invariant frobenius rotation axis align direction embedding rotation indicate proceed configuration position span note population covariance enforce mild density usually extra achieve implicitly consequence eigenvalue create instability prevent notation account center required realization realization copy version problem pose prevent directly perturb dominant proceeding review perturbation relevant reader familiar skip subspace respective orthogonal distance equal large angle angle use onto subspace may version present purpose arrange partition entry eigenvalue span possible tangent span column onto subspace entry variance space fact hermitian frobenius measure reader refer f u u hold quantify exist I map span invariant angle zero restriction normal tangent numerator measure quantify effect tangent component tangent contain tangent measure
compete finite interest among via particular analyze theorem class model computationally prohibitive crucial ingredient selection multivariate describe efficient combine select computable compare method one select illustrate analysis set machine cognitive collect discuss overview adaptive present reduce rank immediate component canonical cca perhaps literature recently refer historical reference application extension propose penalize nuclear penalty achieve adaptively minimax see calculation note remove side response regression model row transform univariate later reference optimal despite advance base practically ease comparison achievable estimation
predictor general abstraction actual implementation method pattern output region partition extract expert result must validate assess particular whole generalization achieve gate expert find gate resolve mlp network optical band also improve overall improve statistical refinement determination conclusion set compose respectively forward neural network represent mapping input variable output one mlp variable neuron arbitrary layer feed forward layer adaptive represent neuron feed network activation th layer feed forward input hide single layer kind define counting connection instead number perceptron weight connection th node activation sum include bias unit activation activation activation function set hidden layer output respectively th unit activation activation mapping logistic mlp connection good network kb method determination minimization back propagation bp simple important role view update paragraph task aim define way good regressor stand asymptotically regressor uncertainty sample directly extract approximate noise case convolution abstraction regressor reproduce accurately mlp learn space space correspond evaluate case involve determination mlp also throughout color turn since color depend statistically population kb vary region space kb define learn region network well
thank author thank david pointing associate comment recall fact chain chain obvious accord form factor draw rewrite second backward detailed known filter forward time backward sum message fig kx mx px
problem performance generalize principle pursuit literature requirement reader consider comparison show logarithmic mc emphasize close shall isometry however isometry theorem elegant david phrase detail notation also keep zero restrict isometry rip define rip infimum x suppose f rip check constraint show absolute coefficient define hand moreover result cs eq singular let whose constant xx ax lemma
chain ergodic rely provide necessary verify reflect beta construction bivariate chain evolve split x sx sx appropriate sx sx paper split
path total path length bin bin passing equally contain particular average neighboring symmetry path let length path pass first cauchy path note path number path edge eq thesis take least algebraic graph stand product q analogously matrix one last cc let without generality label j ij dm low notice q bind quantity ij z ts diagonal z z z da ij union variable apply hoeffding q eqs use eqs union dr j decompositions definition dm n let laplacian establish property spectral exist claim sum equivalently write lemma position node k manifold geometry base random graph precise noiseless reconstruct measurement distance
satisfy assumption wu infinity var u var u nu j end proof proposition nu nu n inequality note also ap n give statistic proposition n proposition eq supplementary proof result notion wu theorem wu wu wu depend j drop subscript jx qx cx cauchy schwarz k decrease lee lee critical sake brevity wu carry suffice j n cauchy inequality te observe n give assumption tu n sake brevity j proposition consequence hold p equality nr cauchy schwarz pr np markov four maxima np n nn pr three focus step approximate martingale counterpart lemma deal deviation lemma conclude explicitly deal residual proposition sigma define u q nd jt shall dependent version lemma hold jt cp give q j k u j wu substituting give jt independent suppose provide moment absolute third
quality fundamentally impact subsequent datum continue work much underlying aim gain insight nature datum contamination particular phenomenon mis highly desire formal give primary treat arrive contamination arbitrary though joint thus contamination attribute image contaminate mis pixel extent capture essence relationship amount mis shift certain impact highly scene e drastically consist forest form amount mis classification accuracy phenomenon mis contamination error error occur contamination contamination consistent contamination contamination amount discussion
similarity experiment multiple graph tune detection modularity maximization fast greedy graph layer truth adopt three criterion angle define intersection cluster respective entropy binary decision negative false negative show algorithm three result highlight bold font indeed easier much lead improve independently achieve perform present benchmark baseline combine result imagine regularize particularly come way compare maintain competitive computational significantly original modify namely term show difference two truth dataset unbalance cluster nature much attention characteristic work improvement confusion mit illustrative column row intend reveal data performance mit literature method exist widely powerful modeling pairwise object highly major
jk anti symmetry desire scenario e perfectly anti gp preference call readily pairwise number closely active perhaps design gps use eqn eqn calculate marginal observe decrease efficiently covariance motivate objective fundamentally carry update point yield update update filtering indicating procedure entropy model eqn use regression uncertainty confident perform classification repeatedly boundary observation uncertainty domain show mutual
mle abc mle great abc abc establish property procedure asymptotically hmm hmm transition original hmm likelihood immediately perturb perturb mapping boundedness convergence demand immediately hold hmms hmms next consider normality mle fisher invertible asymptotically asymptotic equal proof perturb dominate perturb invertible abc fisher general quantitative qualitative see strict sufficiently small mle tell estimator relative mle grow choose noisy mle ignore provide fisher see appendix doubly sequence variable difference law law law law asymptotic hmms perturb additive immediate corollary assume hold defer comment similar summary abc form fisher fisher hmms manner inherent lack smoothness estimator ball problematic try smc due abc likelihood smooth lebesgue estimate via
adopted argue often convergence mcmc bayesian optimization advantage exploration objective method exploit result optima consist phase sample learn policy explore phase procedure uniformly adaptation ergodicity adaptation increase shall soon detail require process adaptation storage need reason step consequence back conclude remark strategy detail mcmc auto value run chain entire observation function prior distribution noisy statistic select next refer reader review statistic acquisition function zero automatic determination ard eq surrogate intractable expectation respect invariant measurement hyper hyper typically proceed parameter experiment however good alternative quadrature integrate obtain extra indicate make
sensitivity specificity accuracy hc normalise ratio instead estimate parameter network nod one network trend size verification hc consistent sensitivity well consistency moreover substantial structure correctly hc successfully recover alarm successfully recover alarm attribute effect test specificity sensitivity combination network reach overall rapidly converge slow rate compare two likely consequence edge per alarm slow convergence also inherent limitation dense
show exhibit derive encourage result order second transition self enable consistent multiplier transition symmetry separation shannon contract statistics principle entropy additive regard analysis blind
km ff leibl trivial constraint interpretation observation pr need common assumption surely light continuity rate stochastic preliminary sequence assumption martingale assumption converge surely complete eigenvalue calculus reveal matrix write f f ff definite transpose jacobian irreducible reversible distribution unclear lyapunov main rate
column norm internal via parametrize often take thresholding independently function drive occur node activity whole function output case separable ensure guarantee specifically convergent fast cost differentiable activation invertible norm likely invertible thresholding relaxed calculate used appendix barrier solve know soft similarly program zero activation work focus interior include ls gradient projection homotopy entirely problem decrease tradeoff utilize recovery signal leverage hardware architecture burden unit processor gpu utilize perform fast improvement favorable property unclear sized architecture embed digital similar iteratively minimize apply thresholding enforce euler approximation analog basically digital incremental rather iteration digital linearize iteration show section demonstrate analog cs benefits analog architecture
powerful reservoir internal reservoir reservoir dynamic neuron architecture reservoir able reservoir depend past reservoir must operate threshold dynamical ensures gradually forget reservoir compute flexible detail reservoir regime threshold instability tune parameter flexibility reservoir computing amenable large realize water analog digital implementation report delay difference type non linearity period delay highlight isolated channel comparable digital implementation reservoir experiment almost magnitude order speed flexibility reservoir promise computation physical digital provide solution reservoir road build analog computer discussion reservoir computer key detailed treatment refer reader supplementary material task discrete internal reservoir reservoir mask note use mechanism subset state use build output layer combination state reservoir target training mse easily typical run reservoir fix minimum reservoir non delay feedback system widely nonlinear loop internal store delay loop entire internal process implementation instantaneous absence system simple nonlinearity implementation system evolve way convert discrete hold procedure loop regime delay act storing delay nonlinearity state
meaningful position pearson paradigm classifier solve q function series ii first identify classifier enforce counterpart positive fix give lipschitz satisfy q error check strong ii extremely would certainly achievable also subsection suffer degradation well achievable binary desirable result proposition fix surrogate far exist nonempty ensure surrogate continuous optimal mention proof uniformly consequence chance programming classifier candidate
nan ready decode almost property subspace hold suppose nonzero every vector range matrix satisfy kk solution apply triangle obtain eq property error decode error phenomenon isometry lagrange lagrange optimizer appearance energy clear later explicitly get direct estimate almost subspace subspace subspace satisfy almost property intersect mesh mesh subset distribute h follow half absolute standard lagrange duality find objective restrict still bound plug minimize minimize serve upper I complementary go infinity probability
difference define generality vector da result lem invertible define position section since favor fuse pt aggregate square combine difference use imply invertible aggregate invertible essentially technique lasso treat one deal present advantage lead impose partition overlap k among order sparsity small pattern assign cf outside sparsity obtain estimator define obtain group aggregate satisfie support overlap illustrate power oracle setup namely combine remark class estimator
n element wise diag
model quite lda dt low transaction one thus stable affected mainly rely fair target historical scenario may nevertheless whenever demonstrate cm cm cm trust management derive base past sufficient behavior reliably scale setting experience resort indirect experience assess agent history assess form trust relationship trust always drawback wrong recommendation affect derive trust iii trust nontrivial develop trust determine eigenvector trust rely design impose drawback trust group like member belong author control calculate author eigen eigen trust transaction management aggregate score stop reach advantage system trust service give rate rating value shape determine assess however cope may influence trust virtual beta compute attention issue evaluate potential I knowledge trust probability fall certain estimation rely experience trust ensure consider address inaccurate perform opinion represent successful adjust feedback inaccurate work initial trust profile accord aggregate member trust agent belong additionally
estimator function consistency simple simultaneous band weight compare achieve end future finite suppose real belong size probability unit pair draw simplicity subscript convention curve available discretization measurement center index necessarily temporal also random unit smoothing eq reconstruction perform use fan statistical convenience support finite constant classical thompson curve membership otherwise hold unbiased establish framework discuss study infinity recall notation constant whose may place study nk x ks kt normal deal design must fraction inclusion far fulfilled design old continuity mild regularity design essentially negligible compare
possible specification increase magnitude entry property example return volatility process separate vector infer outline account output account fix correlation introduce use gp process correlate prediction spatial set methodology multivariate volatility interest evaluate dynamic make prediction inference relie solely fix limit correlation structure matrix quality focus contrast scale need dimension conjecture wishart mainly limited dimension cholesky operation general operation
cover song negative network ideal centrality node sub threshold apply short mathematically th song index prototype community cover song second span previously closeness centrality result percentage music hypothesis song tb algorithm closeness centrality centrality nan p communities accuracy centrality discriminate become statistical significance assess test song tend centrality discriminate centrality valid dissimilarity measure represent think incorporate aspect audio content accuracy complicated song enhance coherence furthermore organization cover tendency song could query task audio song identification detecting piece extract raw audio important modern organization song nevertheless song music effort
tradeoff try avoid enough hand affect approximation available lemma remark di transfer reinforcement source rl adapt source target illustrative reinforcement rl speed rl underlie assumption source combination task useful transfer rl thorough focus transfer trajectory mdps use target particularly suited involve interaction sample available source case arbitrary reward finally index I available sample generate index available target source target e notice action generate l build lp access return case form iteration propagate transition kernel mdps weight determine I average bellman
n ordinary equation sequence tend ik ng ng ng ng ij exist away zero n expansion n fx pt nx nx nx compact converge atom rescale coefficient coefficient vanish j ij w vanish vanish contradict tend infinity conclusion converge converge vanish entail hand display must almost identifiability contradiction true subsequence equation contradiction term strategy triangular g w k two convolution sphere fourier function support fourier transform inverse fourier transform bound f g wasserstein distance thus note dx bound lemma line sign denote variation g constant
q give denominator component ic x iy ignore dependence approximate boost specialized leave tree forest thousand real uci market specialized classifier prediction probability moreover experimental uci artificial market recently implicit observe market never outperform market market outcome contract market win contract contract market possible outcome outcome exist price simplex r instance make market consist market participant budget importance tell market price price know describe price classifier feature show represent artificial prediction dot probability example base function logistic play potential feature market entropy contract price amount win mm mm set participant govern price achieve example involve market outcome example correct htb
size configuration extract used force identically vanish sparse datum inspire natural mean analytical formulae efficiently cross configuration equilibrium exist prohibitive ergodicity breaking affect quality question acknowledgment thank leibler thank center system contract pattern rewrite partition eq temperature considerably entropy partition configuration role dual entropy give entropy system measure expect measurement respect assume configuration pattern derivative calculate logarithm replica introduce expand I enforce definition lagrange multiplier give symmetric elementary gaussian saddle fact role pattern define self entropy center biology nj sup paris paris sup paris problem infer set frequency inference case ise space much negligible compare mechanic correction stress include attractive infer interaction criterion many attractive pattern noise configuration require good illustrate understanding fundamental various scientific model application small component pattern identify mechanic originally intend average power pattern linear correlation validate discuss issue configuration amplitude pattern plan define bayesian correction decide bar synthetic devoted real biological alignment domain reader interested
discrete dft field spectrum covariance via exponential expand spectrum truncation correspond adapt formal expand yield z feature coefficient summation field spectrum coefficient multiplying quantify complete orthonormal impossible value produce model mirror symmetry coefficient specification describe straightforward discretization utilize control mesh without generality fill refer indexing indexing map grid frequency wish grid maximal lag entry k h provide formula derive write rather produce take large produce compute determine exact important likelihood reverse device causal skew field skew field precisely causal causal may define move
already walk case find current vertex occur fix nearly discover unable continue self avoid past find total ht component class vertex walk account candidate vertex probability way drawback degree consume walk roughly since self class st see table st classes self singleton order observe walk
ordinal order set countable ordinal resp length resp quasi order naturally equivalence try explicitly p write suppose rx vx rx rx v v difficult quasi order ax rx vx aa accord lemma quasi order strong
next attempt physical application discuss ref reconstruct individual model procedure response direction reconstruct filter estimate angular power ref discuss sec bayesian diagonal facilitate via parameter reconstruct way coarse grain conduct ten middle still bayesian right iteration region dependence show pure estimator ten evident bayesian boost marginal time consumption main strength lie reach
compatible semi indicate duality mapping banach uniqueness product duality mapping difficulty uniformly differentiable banach convex uniform convexity duality uniqueness close uniformly fr differentiable ball uniformly unique compatible useful characterization banach space banach fr analogue representation banach space banach compatible semi duality exist unique compatible banach space compatible inner lemma homogeneous semi product however reduce variable namely ready value banach sometimes prescribe banach function norm sense banach consideration evaluation usually refer call
draw distribution eq define likelihood remark distinguish distribution eq maximum draw figure specify specify parameter via draw distinguish actual estimating maximum specifie sort great draw contain great draw bin contain great remaining find draw plot number draw require via maximum sort specifie mean distinguish integer estimate sort bin great draw bin remain contain great bin draw distinguish actual define sort specifie distinguish plot draw distinguish distinguish thank fan wolfe also thank many include point identity show hellinger version confidence significance significance set positive negative whereas figure probability false false negative remark figure rejection figure investigate draw say surely limit actual fast consider significance converge fast draw obtain accord simulation bar top bottom level remark estimate please draw increase plot trace converge straight line
leave path update parametric form indicate path involve unary potential potential cut variable define max product depend bottleneck capacity recover cut notably alpha location bottleneck bottleneck pairwise remain since valid expand beyond chain know potential cut expect example choice potential cut graph cut never unary potential modify mass cut fair back potential normalize leave thought view tree ascent block choice see however chain coordinate dual analyze chain subgraph optimum bottleneck optimum subgraph dual objective optimality way path setting conjecture mass cut side min marginal compute second recursively root mass differently pass theoretically analyze two guarantee assume converge guarantee update bind whether notable exist yield submodular belief optimize solution convergence message guarantee converge suboptimal possible via update somewhat temperature guarantee relaxation numerically implement even submodular energy belief propagation reveal message guarantee assignment submodular short path strong potential strength nonzero unary convergence could reach fix modify unary potential take interestingly modification mp become cut solution map notable closely relate block ascent dual interestingly optimal
discriminate action current state principle intuitive description formalize decision mdp find reinforcement definition necessary explanation decision formalism sequential process action state reflect relative use agent state apply agent obtain reward continue reach rl cumulative cumulative reward system goal reinforcement maximize reward
correspondence logical appear logical logical par inversion elimination product simplify sum true false another insight large identical member identify consider assignment argument truth assignment elimination truth group count truth efficiently count elimination par constrain choice counting elimination context rv fact eliminate par elimination exchangeability par par rather section may par appear par rv aggregate variables aggregation par extension consider alternative setting recognize computation first implicit specification presence second case identify share input value discover construct without rv graph later carry computation speed improve perform approximate extend algorithm two last computation simulate rv effect distant influence place bin close bp stage first graph compressed template represent group factor nod message super edge member weight super equal edge template bp modification message send super node weight super node counterpart except next algorithm target evidence group factor variable node par lack ensure type identical induction bp message evidence mostly useful learn bp template graph proceed stage bp determine evidence affect send initially variable super contain subsequent super factor node separate type factor super super refined type super process guarantee minimal e template obtain simplify description factor logical
newly add contribute active terminate step sequel greedy greedy node conditional negative loss likelihood threshold backward step factor inverse ji break k ji j j row symmetric zero positive follow appendix convexity notice estimation converge always suppose per
cc enyi penalize cc cycle enyi penalize logistic ise positive potential graph topology attractive nonzero uniformly entry uniformly ise gibbs knowledge employ criterion since truth possible measure good figure model model potential trend attractive note distance decay increase method decay graph regime rate alternatively fast theoretically regard run fast graph consideration global threshold expensive large unified paradigm ise estimation complexity condition succeed ensemble world separation acknowledgment author berkeley berkeley wu discussion graph author anonymous comment notation observation paper appear support uci award star fa dimensional ise propose structure independence conditional derive complexity consistent third novel require learn model
svm standard original raw pixel dimension pool anchor per anchor round anchor require anchor predictor state art thank dedicated dimension class multiclass hypothesis rely prove establish smoothness property first taylor hessian point hessian rewrite h maximum value form last imply eq equipped identify feature greedy iteration progress use denote index zero two rearrange inequality convexity hand conclude conclude claim ex multiclass classify multiclass predictor use linearly share multiclass predictor predictor fast set benefit success art classify relevant surface
probability side express intersection divide length due box simple straight line summation break show dash frame note sect projection complex two close position along calculate probability intersect box intersection line except come reference scan narrow box length well difference node position scan corresponding point correspond line come corner define measure intersect analytically evaluate break connect break corner easily change aligned corner calculate break adjacent break pair
discard remove early development game keep sophisticated preprocessing unnecessary reduction conventional focus reproduce rather dimensionality reduction preserve essential illustrate analyze dynamic complete construct win probability describe associated complex phenomena game dynamic play introduction complex high reaction bad coordinate fast configuration exhibit dynamic project close reaction propose base protein atom sample represent form optimize coordinate htbp
classification library
data fashion point clustering aggregate dimensional projection plane fig distance graph graph figure correspond variation clustering dark clustering highly band row order clustering close clustering significantly lack unsupervised clustering apply idea meta represent clustering connect inverse exist coupled clustering differ vertex meta dark around meta block corner large meta result depend expect enable handling topic averaged meta cluster cluster average clustering cluster traditional clustering return representative reduce clustering representative partitioning combine clustering equal co clustering never connect cluster representative clustering cluster representative clustering identify original nine clustering identify perfectly clustering individually clustering point set information representative nine clustering reconstruct two choose fourth clustering representative clustering clustering
nc ff domain optimization block implement local assignment carry could regularize iterate expectation nc c strategy time subsequently centroid update minimizer yield keep parameter fix eq solve solve simplify n scale update c interestingly show outlier outli hence step solve numerically interior root input cluster initialize update via robust spherical operation per large convergent minimum q n minimization proper semi imply empty non unique block block unique wise iteration term different coordinate wise converge cluster whereby assignment posteriori I outlier cluster sequence warm expect solve warm efficient suffice paradigm convex argue offer tight enhance
norm mu space far transpose stand norm vector magnitude fully tractable integer triple satisfy u u mn kn n km n positive integer regular recovery associate nothing triple augment appropriately speak triple well efficient synthesis extension choice consequently case achievable b h ta vanish complicated maintain validity intractable know imply gap system admissible computationally tractable necessity necessity assume norm assume choice implication nearly routine satisfying form v nd proposition condition nearly nd proposition
arbitrarily neighborhood uniformity corresponding assumption present section thresholding coincide adaptive soft thresholding coincide natural distributional adaptive thresholding insight simulation sample distribution respective simulate depend histogram ii adaptive lasso scaling component least normally partition block block equal factorization respectively comprise row three correlation correlation number either parameter adaptive imply estimator irrelevant outcome correspond proportion value simulate variable purpose scale derive red present compare adaptive soft thresholding remarkable agreement respective design design condition difference marginal turn counterpart situation somewhat strong figure ill matrix figure qualitatively h h h I subsequence may converge constant maintain assumption must since prove converse subsequence without hold choose opposite sign bound zero maintain contradiction complete part obvious n assumption eq first claim immediately claim q handle analogously subsequence suffice converge result support p n supremum prove prove observe p p ic select pointwise converge I se p uniform se se ps p ps complete eventually se se I
summary abc opposite value summary within three block use abc tolerance quantile previous experiment quite satisfactory highlight ability summary statistic statistic inconsistent choice thus bring answer abc true statistic wrong place fundamental choice practical setting statistic abc comparison mean statistic distinction differ classical summary addition testing summary range expectation summary statistic simulation model setting production pseudo quite preliminary neither final toolbox handling motivation description core feature particularly bayesian rely statistic thus result distance tolerance choice ideal carlo factor
policy class see relaxation could case extensive discover addition anonymous helpful comment support I fp ip sequential decision extend reinforcement exist none tight show near bayes current obtain robust reinforcement acting change obtain act instantaneous markov action equip suitable algebra
sphere cumulative lie proportion spherical total surface case mr r b xt b shannon calculus definition chain derivative encode sphere spherical gray bilinear encode exploit orthogonality intersection unit sphere subspace quantization
tree structure markov message unique bp r surely unique bp node potential message upper convergence vector guarantee conservative may element grow exponentially show conservative behave follow graph conservative address bp converge ordinary work log related technique condition message message oppose message recall normalized compatibility minimum row feasible involve graph suppose contraction consistent converge surely bp claim tree non mean error component dominant norm establish norm theorem expectation hand order include logarithmic factor part probability guarantee satisfy sample high exhibit bp contraction message geometrically quickly mean iteration accurate early bp message put piece ordinary operation compute comparison long desire computational become comparable graphical represent approximation reality theorem analyze guarantee convergence require message oppose relate message
check contain embed ask fix come figure suggest width line eps graphic graphic graphic http www ps width arise set demonstrate cox problem influence way small cause large bias bias variance
q lie unit eq right root determine stationary solution hull basically return ar q proof random attain vertex g calculate complexity interested diameter stability give tight prediction g care predict constant rate st daily
partially claim notice lemma replicate kernel center decompose influence note slope kp j correspond see small geometrically decrease use order geometrically finally discrepancy theorem discrepancy discrepancy choice simultaneously discrepancy range space accomplish vc distinct discrepancy change use slope size kernel kx kx nz ignore dimensional volume exist net slope
first suit perfectly comparable randomized initialization repeat time pca diag gmm gmm accord full performance full em dimension fact discriminate em matrix gmm diag gmm trend low diag gmm little bit increase tendency dimension performance fall poor performance explain fact subspace simulate fisher increase furthermore suggest succeed latent initialization model accord transformation simulate comparative purpose bic gmm diag gmm com gmm observe link different secondly select simulate right present datum leave axis separate accordance point bic conversely fisher discriminative gmm com gmm diag gmm pca bic leave dataset recent hand compare full gmm diag gmm gmm model mean version mean lda benchmark uci repository split specie describe dataset dataset characterize dataset describe detail family discriminate divide class accuracy deviation experiment
extra reward core converge priori keyword lyapunov tu von communication community essence tu comprise player thought distribute allocation benefit characteristic bound bound characteristic value mean expect coincide long dynamic core tu continuously player instantaneous subject know process bound generating find extra receive budget core drive priori convergence despite uncertain time vary tu arise value time vary see robustness latter work generate unknown line concept formalize worth mention common recent game interval similar generally optimistic
type mixture normal lie normal mix distribution arise exponentially alpha variable close recognize approximate issue overcome way package implement triangular density normalize mix conjunction work context proposal early implementation point complicated excellent supplement appear sampling exhibit novel turn mixture normal design nearly specifically representation result monotone triangle posterior mixing explain detail lead deal alpha joint difficulty work bridge gain show many likelihood penalty proper discuss facilitate bridge mixture normal completely distribution q exponential laplace alpha stable exploit conditionally conjugate mcmc
random minimal information determine estimation facebook graph show greatly outperform art future future plan problem review related technique remark california usa uci grant nsf award usa sample metric walk appropriately begin find weight sampler impractical equilibrium balance weight achieve weighted achieve collect facebook college simple random walk rw accuracy metric population weight c graph independence node blue node white naive approach rw irrelevant network web www measure desirable small representative hard information base rarely publicly available measurement element importance example depict world population china blue node assume median china take come china even sample china much budget people draw category sample much
r unnormalized probability condition candidate momentum position detailed select element must choose preserve use add jacobian unit r restrict treat unnormalized say require state slice must moment satisfie follow invariant r leave uniform must resample constitute gibbs sampling joint allow conditional density conditional condition r u imply free leave invariant specific r conceptually generative process building repeatedly size tree leave momentum tree single node initial direction take step height half tree balance binary height leaf position momentum visit trajectory illustrate accord likely reconstruct tree height leaf node leaf expand tree course expand make begin visit visit want expand simulation extremely discover large distribution density go easy stop leaf state nonnegative recommend moderately rule sampler neither detailed balance overhead whether stop height height stop leave subtree backward
row decoder lead expectation rip isometry property gaussian constant numerically decoder conventional known checking np rip measure quick concentration next proposition signal decoder expectation draw map rip lead mse hold vector distribution assume kb nk second equality rip expectation last since nan check subsampling decoder reconstruct different realization signal since coherence simulation dct basis carlo simulation rip figure typical increase small ratio fix slightly plot bernoulli vary little subsampling slowly linearly typical cc bernoulli sense approximation calculate decoder signal follow gaussian eigenvalue average upper bounded term distribution follow description lead inverse real score among introduce
normalize depend start connected component apply success value start point several point parametric point explicit preserving constant orthogonal orthogonal viewpoint expansion constant differential existence recurrence ode recurrence since require huge resource expansion preferable derive series integral invariance expansion change see unless odd change sign unless come odd express jacobian hence express integral find right laplace give gradient derivative hessian
recover base contaminate essential focus approximately low structure regularizer include sparse application gauss random field asymptotic frobenius relaxation decomposable regularizer specialized extension behave closely completion recent probability frobenius decomposable complementary remain explore acknowledgement three partially sn partially microsoft acknowledge international decomposition prove part satisfy property norm decomposable regularizer require parameter r special parameter sample regularizer nuclear regularizer separate also compute gradient property norm suffice suffice condition begin former suffice upper application triangle put piece since equality equation yield eq q since substitute equation substitute early appendix form corollary yield constant significantly scale corollary specialized norm careful term throughout choose split first function remain ii thereby remainder claim rate alternatively high see examine condition sufficient hold require analysis bind variable around conclude use extend interest constant since adequate careful make analysis expectation cauchy schwarz function gaussians consequently eq
linearity ensure reader detail consequence kernel kernel distance dimensionality space projection easier space neighbor tool collection indeed complicated shape represent rkh popularity machine context many consensus easily exploit linear distance reduce significantly within significant improvement motivation
area brain show micro property neuron example analyze neuron fire implicitly send clearly micro neuron concatenation sometimes superposition neuron hence successfully rate spike noise neuron unknown record neuron neighborhood constitute core problem signal consequently noise audio speech classify spike shape wave neuron still neuron depend neuron wave relation fire neuron wave vary neuron spike commonly know sort perform pca spike cluster signal spike observation eigen vector matrix capture since select variance neuron
main player round behind forecast auxiliary strategy give assess strategy group round round play mixed inclusion inclusion follow probability inclusion use boundedness simply repeat value game follow therein respective denote payoff player player action possibly accord end round player mapping h player monitoring game problem constraint armed notation useful finite ingredient unchanged ensure value payoff respect player random signal sufficient prove indicate contribution lie constructive soon efficiently auxiliary refined lead past essence use able rate third short layer notation internal monitoring auxiliary strategy condition component convex define set value condition sequel partial monitoring refer vector payoff satisfy reformulate subsequent constructive structure clear follow non concave strategy illustrate linearity efficiency game column force dark action payoff respective
know trivial many illustrate fractional diffusion type let let consequence definition property self every increment however positively exhibit sense older admit martingale term process integrable martingale quadratic hc gamma recently refer chapter add evy recently establish say iii iv old path variation sense
p b similarly b b b b b u u calculation result previous proof lack hence implicitly every go conclusion proof bound spend time define admit let spend result value impose fix proof follow z time spend case clearly similar reasoning low lead state begin update convergence proportion flat histogram
bring limitation discuss conclusion subgraph count programming develop package gmm fortunately package highly dynamic function network exist code capable attribute ideally suit computational implementation gmm essential structure special termination gmm simulate evolution base structure class place verify model gmm store store simulation dependency structure belief generate belief pmf previous third scientific computing require therefore package compound require evaluate subgraph include calculation rich illustrate quality scientific package unit test code website test software list index description view gmm simple gmm specify gmm base termination rule simulate study termination rule whereby terminate network growth implement graph implement pseudo gmm rule graph connect mechanism chain illustrate right panel direction cause growth several classic model method model exercise gmm highlight strength recover classic gmm model growth I specification attempt recover classic model early excellent gmm utility classic attempt network growth
variable drop constant use nc nc ij quantum spin acting represent spin spin interaction field act spin encodes ij ix jx sc iy nh jx calculate weak involve implementation device wave processor alternative weight problem methodology develop weak classifier correctness implement dependent interpolation hamiltonian eigenvector identity act interpolation satisfy condition time sufficiently slow manner minimize energy find optimal weight measure opt q opt opt classifier appear perform assign optimize group classifier together time ground assumption provide fidelity state quantum respect te instantaneous small quality overlap difference equal eigenvalue integer depend value depend boundary smoothness ref crucial gap shrink numerator mild grow problem exponential gap speedup speedup classical prescribe relaxed software amenable quantum speedup gap hamiltonian beyond speedup numerically fewer comparable give generalization error overlap adaboost quantum hard promising shall theoretically perfect majority vote strong weak construction shall weak classifier classifier relaxation shall weak vector eq accurate set three meet classify accurate construct entire correctly classifier every classification specification classifier agree correct correctness classifier stand weakly correct weakly receive vote weak classifier comprise classifier weak comprise majority
regard degree freedom second moment infer posterior stationary rescale degree drift assume transition sampler satisfy quantity fact get assumption bold line dot freedom illustrate proposition respect check attain ap decrease ap enough inequality computation show fulfil iff hand eq compare reveal consequently mean sequence martingale difference obviously examine bound fix bound gray solid assume obvious dash correspond equal actual root conclusion analogous table focus
match link pca square euclidean al multi relational relational couple analysis include couple tensor problem propose square summarize relative relative prescribed converge factor unit solve couple mode fitting I true convergence word cp simultaneously extend handle miss deal issue opt couple heterogeneous gradient analysis common tensor factorize factorize extract couple matrix objective solve optimization gradient first gradient rewrite I partial component combine matrix
basis depend polynomial well n fourier spline close embedding represent center give closely approximate kernel construct unary b spline corresponding follow b spline turn classifier spline b spline consist uniformly spaced knot truncate polynomial basis polynomial spline knot space repeat matrix p order regularization experiment cubic
parameter interval sensitive choose cluster experience reasonably change implicitly reliability estimate cluster hence limitation mean always extend eliminate issue extension isotropic cluster raise explore explore particularly configuration explore explore algorithm long overcome available account edge algorithm bound point outside induce measure window create integrate sample outside model account bias create around complete analysis note possibility include surface account example case tangent investigate e introduce spatial cluster along system curve galaxy finding facilitate aid line estimate permit various perform structure cluster often around lie boundary lie around family understand technique c pattern describe stanford introduce describe curve assign undirected orientation window produce
apply appropriately choose vector convexity linearity figure correspond exponentially weight apply dt tt coordinate last lie length upper conclude square adaptive introduction tune first cumulative regret indicate regime algorithm original regret b main however contrary alternatively without self confident weighting already adaptive algorithm limit initial square x setting get bound conjunction function section use adaptive modification lipschitz continuous refined bound corollary address loss high curvature slightly real forecaster e square high address proceed
use proposition pattern recover last component imply low satisfy optimality condition repeat sake ty ty ty c mean virtue support occur imply plan iv sec isometry plan result tx ty ty use orthogonality relation x tx ty tx tx ty root quadratic ty tx strategy confidence interval x x hence tx cauchy schwarz z tx z property begin tx ty tx z thus c r ty ty ty combine last equation tx ty x x obtain c ty support sign clear exactly strategy strong note imply satisfy optimality exactly sub sign mention conclusion
short path different delay show improve accuracy scenario message along short short respect delay exchange along short goal discover use delay topology equally well delay participant inherent ambiguity discovery point context latent tree eps minimal eps detail topology node end delay redundant hide graph minimal characterize minimal give participant hide graph iteration remove redundant initialize nh nh nn accordingly v ng topology delay random graph suffice reconstruct representation small participant delay edge variance le g discovery desirable original structure reconstruct unlabeled graph distance graph adjacency vertex graph keep permutation node unlabele goal small representation obtain delay graph asymptotically recover minimal definition assume infinity challenging set large sample topology large structure size network formalize set say goal require number discuss simple concept incorporate discovery set topology discovery base delay delay variance topology
replacement follow error support h h ta ta inequality hand ta ta yield desire clique ta hold corollary note ta cs j j contradict j clique clique satisfy solve clique kb ax prove exactly identify eq intersection satisfie column column disjoint need prove side n j eq need prove n verify writing exact wish construct respect intractable infer network describe time primal lagrangian assume n n multiplier polynomial constraint yield dual submatrix extract column original relaxed polynomially many index interior duality gap upper th sequentially call analytic new incorporating violate interior method
simply yield lot zero theorem least row index correct correctly show subspace indeed load loading matrix nonzero big coefficient eigenvector magnitude include magnitude fail pick medium obtaining include along achieve focus paper subspace interest eigenvalue separate spectrum theorem satisfy j hand connect sparse j restriction subspace ball display logarithmic near estimator adaptive minimax sense choose define set intend lead principal subspace gap note incorporate could step suppose hold satisfied sufficiently hold restrict jointly spike turn pt justification estimate consistently important need theoretical algorithm section conclusion use consider multiplication element come vector
subset j jt I lie possible ranking fix hoeffde less subset replace exponent query consider object thus r time query thus query determine rank index rank rank rank line mark around mark around ball center object object ball probability object object none position take jensen fourth conclude pm pm rank amongst label correspond vector assume fourth corollary university ranking comparison ranking sorting method pairwise comparison interested object may comparison euclidean ranking reflect preference human subject could certainly support embed could determine similarity audio interest comparison reference two
ratio evaluate biased coin eq q show nearly large interval little hard cumulative equal confidence interval obtain solution
top advance solve problem frequent suffer scalability mining frequency previous collection guarantee quality case provide contribution contain frequent sample result rarely transaction article convergence apply field field advance vc encounter field application database management aggregate vc related field database privacy show private concept database vc privacy bind vc query dataset although present goal technique inspire present detect failure balance short way prominent tight new time proof conference version numerous improve theoretical connection far experimental concern comment evident ar platform describe definition later transaction transaction transaction transaction transaction find decrease tie break arbitrarily discovery ratio intuitively transaction confidence association interpret transaction dataset transaction minimum resp resp triplet extracting
accord communication bundle online minibatch minibatch ns per hybrid maximize possible save machine involve small approach specifically bundle use implementation accurate per demonstrate thank second distinction take execution slow hand control carefully scheduling decision robustness slow batch elsewhere iterative machine evidence project logistic regression substantial also approach base run average empirically effective create pass briefly argue theoretically partition carry separate display site recognition datum example undesirable node node transfer require create predict constant body recent year focus predictor function method exploit node environment perhaps reduce dataset subsampling really good argue problem
regression estimate hypothesis assume differentiable obtain elsewhere density series expansion aspect c lx lx consider dx deviation result set ii iii see give chernoff estimate upon function constant whenever condition satisfy function ask condition satisfied proposition assume function bound state
guarantee constant allow finally tree marginal factorization forward compute marginal factorization message message message part message bp compute factorization follow algorithm property section es communications department mail com research visit university grant grant national mobile project intel mobile communication technology foundation wireless hybrid enhance mobile contract combine analysis method equation region free energy convergent message passing provide underlie addition factor correspond division message iterative decode variational decade physic mf area e visible pdf mf pdf e g factorize leibler minimize message pass factorize pdf additional
application solve solution previous consistently encode laplacian non zero encode bernoulli parameter mr zero encode video image bilinear identical laplacian encode sequence predict previous value residual encode code method simple part discretize laplacian look quantization negligible minimize encode magnitude element keep produce algorithm two b video left image sample contain remove video eigenvalue scale represent eigenvector bottom recover particular eigenvector need background correction leave people pass present base inherent use metric automatic per basis collaborative variant learn dictionary theoretic lead robust dictionary regularization formulation prior spatial modification box result user dictionary fit characterize model application work address establish theoretic result derive virtue principle allow incorporate markovian include area
structure index distance equal final distance consider component indeed read cluster indeed constraint computed eqs refer htbp cluster cluster figure mark symbol belong cluster area china version propose homogeneity component close object component cluster homogeneity similar contribution close component result component reach allow homogeneity medium table dispersion globally allow homogeneity cluster pressure mean wind speed relative cluster cluster cluster cluster cluster fig well partition two histogram datum adaptive wasserstein clustering optimize use wasserstein algorithm square wasserstein dispersion advantage ability standard mean find reach square minimum give
number uniformly integer assign else cell cell integer bivariate verify sis far enough count exact level failure success write statistic via sis table h l option reject table polynomial sample table marginal via sis rejection general sis may rejection rate table sis rejection whose
move play adversary constrain game play adversary adversary slowly change decision adversary move strongly sequence euclidean move game adversary conclude decision making played conclude restrict move throughout restriction notice force restriction distribution indeed force value formulation make information explicit minimization observe batch supervised come relevant strategy adversary call blind I blind analogue value general supervise proceed relationship online blind batch eq pass strategy proof learnable adversary sense know learnable specifically exist strategy use give rate rate decay convergent sequence learnable batch q thus learnable learnable distribution player formulation restriction learn supervise game pick adversary player suffer restriction define write define value blind supervise shorthand classical rademacher define setting say supervise learnable learnable blind sense
angle converge fix correspond prove smooth although stochastic gradient find identification filter constant process algorithm work decade stochastic online identification parameter set appropriate lead new deal general riemannian manifold introduce cast general framework algorithm intrinsic embed convergence almost establish never prove hold extend four online convergence immediately pca randomize intrinsic illustrate third detail concerned semi definite mahalanobis distance allow application g novel algorithm meaningful guarantee algorithm simulation indicate riemannian fast usual natural b brief preliminary z w differentiable cost measurable stochastic cost compute explicitly unknown instead access compute perform approximated cost I application hope almost g attract lot machine community filter matrix preference rating item book way overcome ambiguity constrain
netflix set propose max compare theoretical conceptual justification moderate ccc ccc test sampling estimate location marginal like via trace bound smoothed marginal theoretical disadvantage learn smoothed provide suggest marginal introduce well reconstruction smooth empirical smoothed distribution sx although smoothed trace smoothed marginal meaningful even e spectral deviation frequency marginal establish result turn set could guarantee
random hypothesis bayes unknown section present alternative particular say define introduce explain function surrogate satisfy pa aa ig posterior truth surrogate proportion equation even stand come requirement predictor asymptotically vanish support entail yield n equation generalize argument technical definition
fix task accord regular configuration worker second probability r iteration prove k r main theorem sensitive size leave run decay predict even requirement worker range rate suggest error depend effective right crowd affect efficient run comparable majority voting operation ensure run increase majority voting eigenvector suggest top sparse next make remark guarantee assumption distinguish task worker worker performance guarantee second generate need limitation additional cost per step allocation compare estimate agree amongst repeat stop even knowledge affect previous expectation maximization sensitive converge unique computing particular observe transition show significantly small majority surprisingly empirically exhibit fundamentally cf figure get regime get phase transition universal discussion limitation task master core concern reliability number want use degree factor necessary budget allow prove minimax optimal constant restriction assign worker simplify necessary adaptive iterative provide match minimax necessary adaptive bad case worker scenario approach minimax ask pay surprisingly condition algorithm adaptive query constant adaptive optimal large depend application regime restriction allow worker degree establishe achieve adaptive assignment scheme introduce probability show error quality worker need achieve practice assign worker comparison probability different regular run spectral use lead left approach detail voting performance value hand worker observe iterative inference start vote
theorem select loss bound select width histogram easily obtain approximation guarantee optima problem know obtain contain query contain tb column column number b subsection fix extremely width heavy dimension histogram minimum lead practice formulate use sparse transform wavelet haar wavelet orthonormal cardinality haar transform piecewise wavelet wavelet recovery compress describe estimate number zero compress setting obtain unfortunately random range query formal guarantee leave future particular omp omp greedy technique start empty omp large value omp eq note dynamic produce density quadratic attribute frequency modify provide code aa set residual ta rt obtain histogram subsection approach section next sparse general range q I note rectangular restrict histogram set analogously optimal incur compare histogram w bb bl rl detail tighter
analytic form result random mean figure histogram volume deviation histogram recover sampling notation show computed eqs strictly speak likelihood across quite obvious live find mode basically find unimodal live one probability live mode live exceed volume span small mode must actually analysis relate evidence function step evidence sense result note
equally target bias towards well target metric target heterogeneous general max parameter demand edge sample hold grid uniform become demand object exploit selection greedy particular demand proof appeal topology classic membership term adaptive oracle learn search take theorem object normalization bound sequence receive index containing mean rewrite q reach message occur phase link object since therefore dt dt v r tv phase stochastically dominate lie property message close edge object triangle object phase thus linearity thus
nu f satisfy bernstein cb moment gaussian inequality var fx I fx f ef ef ef cb rearrange e expectation v respect hold tv h corollary conjecture invoke imply smooth support respect motivate
purpose appropriately chain reversible exhibit gradient reversible metropolis hasting differential drive wiener correlation primary insight monte hilbert must approximate dimensional dimension order work herein mcmc explore step langevin motivation optimization minima gradient flow computation expensive stochastic partial carefully specify idea know dimensional context gradient flow develop noise go name simulate review reference noise construction method simulated anneal idea directly emphasize apply technique lead degenerate adjoint space gradient flow complex purpose describe walk combine metropolis hasting accept mechanism well idea anneal review reference paper develop dimensional adopt optimize viewpoint chapter dimensional technique diffusion construct basic building evaluate combine proportional finite algorithm respect quadratic form via lebesgue flow limit motivate chain observe
half external benchmark capability new scene semantic labeling tree associate year semantic labeling challenge broad community obvious etc early approach recently problem e learn patch specify field demonstrate success real one reason make difficult notion region demonstrate object inter understand challenge locality brief little part among contrast part label scene pixel pixel broad notable object recent year suit object abstract require part make call limitation image understanding propose global part object tree road tie pixel parameterization rich plausible graph learn complete relation paper manually jointly
hinge reward restrictive many maker allocation reward identically distribute reward arm arm formulation recent reference therein assumption value variant greedy horizon infinity whether notion case type address regret armed bandit underlie functional response mild smoothness encode function describe arm response arm prove first limited upper improve idea elimination static policy deal covariate explore build arm remain horizon tune regret investigate bound upon canonical paper elimination principle solve group similar covariate look bin bin view indexing aforementioned horizon restrict class difficult remain easy adaptively successive elimination severe limitation naive globally
subgraph order subgraph place around set triple disjoint x disjoint contraction implication weak say reverse implication property notice undirecte trivially satisfy hence compositional direct model separation probability variable space disjoint subset conditionally independent measurable almost independence cp independence probabilistic semi converse strictly regular compositional fact entry hence compositional multivariate say pt edge arcs edge line type mixed loop arc em em ki ki follow text let mix three em em ji em em inner
shall dimension subject recent year cluster problem paper point subspace subspace dimension consist partition accounting outlier unless special unit sphere orientation identify assign hide model always subspace affine goal leave introduce treat observe n n quite attention literature ssc algorithm expansion select vector motivate whenever belong subspace detection nonzero entry column subspace detection support subspace might construct vertex construct normalize gap eigenvalue estimate subspace cluster step subspace cluster matrix represent affinity exist involve restriction minimum purpose broad shall ssc correctly intersect principal vanishe prove ssc subspace grow almost linearly ambient sure allow grow root ambient modification ssc data corrupt exceed clean well provably capable handling improvement approach analyze cluster combine tool probability theory functional clear insight ssc successful viewpoint prove propose introduce section ssc give
easy argue enumeration convex feasible approximate correct satisfy hypothesis v ta tv ta f bounding conclude nonnegative hardness result arguably model nmf compute explain explanation hide take occurrence document nmf compute document express topic assertion focus question design algorithm resolve assume nmf whether polynomial geometry empty dominate distinct occur play determine algebraic depend quantity additionally compute successive set time naive constant would run heart algorithm theorem variable define semi algebraic factorization tool geometry obtain exponential time search heuristic maximization run time natural independent nmf express small hidden linearly lemma translate alternatively combination set section associate factorization sf exponential previous exponential system solve check complement use inspire exact nmf hold sf exact sf nmf problem turn nonnegative factorization separability segmentation identity matrix name easy search find row provably work
subject reflect development undirected graph generally region topological measure currently consensus compare mathematical essentially advanced suggest topological respect graph integration literature address integral monte method number estimate topological metric researcher tend involve short path include propose edge propose short
tune dt regret may grow match straightforward forecaster forecaster ds logarithmic satisfy intersection optimization quite small scenario ball reference article convex square aforementioned small proved intersection ball explain gradient satisfie regret differ infimum jx jx tc regret jx jx bound prove latter norm divergence regret look prove aforementioned specific ball contrary see prediction open sometimes essential derive sparsity pac review pac literature refer reader thorough introduction pac inequality recently sequence loss rely online true loss exp inequality automatically apply automatically tune thank regret imply hold solve besides priori latter since unknown adapt regret forecaster fully automatic setting model contain game
truncate expansion give reference side length less achieve obtain value determine evaluate reference develop hierarchical translation operator gauss reference consideration field compute far center first bottom reference expansion translation field shift q moment centroid center moment third entry wise field moment field moment iterate act clean routine approximate query must perform traversal shift centroid child local q center summation center centroid center expansion center node accumulation child wise scalar multi query reference approximation exhaustive computation high approximation exhaustive query reference encounter whether query node field moment accumulation constitute moment evaluate determine asymptotic description fast gauss key even none cost exhaustive method exhaustive query fine approximation p fc dc te reference query direct accumulation right query centroid main overview appendix kde routine maintain store pre
bound write ss lk rt cs il rt expectation arise thank tree calculation exercise prove calculation support fellowship award nsf dms ise introduce toy fact covariance symmetry correlation edge graph odd node adjacent number edge correlation matter explain node fix q instead case computing form edge belong figure eq quickly pp g g seen connect e make success care ss component q hold applicable name also proof proposition assume assumption proximity mean mind straightforward application inequality occur event q small enough verify upper bind ab bc e negative v cauchy v x k small k theorem easy probability bind outline note represent replica lemma j g gx application
flexible module feature implicit building feature datum ill define input important bring interesting idea self context put road formation careful feature drive complex principle genetic programming principle formation configuration approach decide promise hierarchy formation experience combine promise translate
change alarm anomaly frequency detection detection anomaly method frequency job link point st anomaly st st detection relate recent use alarm propose analysis aggregate seem fail detect change detection alarm point among user conference discovery link anomaly figure alarm change detection conference frequency table among twitter
monte former measurement discriminate source truly qualitative processor generators generator architecture hardware programming communication gpu execution code cpu gpu consist compatible device globally memory processor mp finally processor unit perform software provide language device share memory memory barrier barrier processor implicitly schedule problem execute gpu device
selection proposal example example scale mode show correlation obvious tail highly might acceptance ever know front though percentage amount draw demonstrate large class token describe model topic additional include discrete mixed bayesian difficulty lie large discrete difficult estimate discrete popular model close direct tractable advance parallelization use processing might integral practical al case log posterior smooth alternative augmentation kind mcmc approach suffer introduce augmentation weakly normally interested miss small could treat point require require hessian posterior restrict size dataset
cm stanford department science berkeley department berkeley author equally modern essential inspire ease scalable divide thorough characterize divide magnitude enjoy also present collaborative filter background demonstrate collaborative divide distribute randomized video surveillance pose science often entirely infeasible become challenge focus attention massive may satisfactory statistical strength research develop inferential task need write singular decomposition orthogonal onto frobenius matrix framework scalable interest matrix entry sparse outlier dense noise represent onto goal focus problem location zero save entry uniformly replacement combine expensive subproblem subproblem combine framework algorithm differ strategy submatrix submatrix parallel submatrix perform mf retain factor project onto subproblem form rank generalize input input step generate rank approximation use replacement column practice reconstruct rather maintain factor e
correlation cca direction common illustrate tendency cca score loading joint score closely good job distinguish panel signal reflect loading panel reflect loading group loading effect individual loading application object subset measure motivate fit sparse version exploratory pls cca penalty induce variable sparsity accomplish loading induce sparsity loading analogous loading small contribution penalty thresholde penalization loading component color panel color display nonzero loading pair color loading blue otherwise panel linkage correlation sparsity accomplish analogous minimize
trajectory event ms agent c ahead none ahead none avg life lose avg score life make lose ms look ahead ms pac increase considerably power dot deep ahead interesting ahead apart look ahead base could would thus markovian power decrease deep look ahead architecture relate rl algorithms evolutionary emphasize easy learn game branching factored may considerably exponent rl hard machine density correct decision dot dot low argue td decision reflect density parameter scenario however general unclear module module entity consider play module activity ball running involve body pac simple view ball challenge application body module properly ahead factored example rule analog ms illustrate information fig fig ms wrong dot ms one observer simple
anomaly background treat properly possible restriction novel consistent search physics semi supervise detection show successfully search physics systematic error learn extensively datum decade especially multivariate tool ratio search new physics signal background event classification obtain mc physics physics begin certain new event correspond choice generate
comparative competition organize provide blind performance state post new perform conclusion column transpose transpose expect represent calibration prediction provide descriptor make date round involve descriptor remain round ccccc nature round bind affinity affinity complex methodology propagate establish response
absolutely rotation scale family equal range q rotation invariance right side unchanged orthogonal matrix haar orthogonal range statement likewise jacobian claim f nu v dp v f absolute na sx sx transformation random df sx na ia proposition bayesian multivariate model characterize alternative matching interpret offer normal normal center parametrization cluster use cover entire normal model calculate simulation normality nonparametric bayes match integral statistical testing provide simplification nonparametric important parametric scientific hypothesis
optimizer k consider algorithm add remove active find long loss meet terminate backward contribute remove ensure round terminate loss guarantee recall definition restrict convexity specifically rsc restrict rsc parameter function smoothness give similarly loss bound require property loss gradient I capture loss parameter stop theorem
number rank play implementation assume bind meet solution terminology w x tp vb z z u u z w riemannian hessian compute formulae trust numerical complexity handle duality gap completion linear domain candidate final gap candidate trace next random generate accord mean fraction os ratio determine randomly priori mention remove scheme table trust illustration purpose also stop function fall stop duality gap convergence scheme show c incremental right solution right rank characterize look reconstruction reconstruction trace minimization experiment initialize report indeed trace time order compute entire regularization predictor grid illustration geometric fix various demonstrate advantage scheme table warm basis prediction
approximate positive continuous minimum normalizing density family compact exponential estimator take uniformly leibl exponential get small goodness fit fitting separately unconditional compete denote parametric choose goodness ideally goodness evaluated minimize kullback infeasible
guarantee side root pt pt pt pt whole tree select tree implement attain unbiased select true simulate test plan different branching keep paper formalize statistical value decision feasibility obtain approach ideal bind solve stage stage combination simple branch control program poor use valid extreme event scenario supervise section form stochastic scenario tree dataset approximation method statistical describe exploit feasible current describe simulation good statistical notation trivial value run vector single observed stage represent cost uniquely determined domain support
stable vs spatial ideally amongst likelihood criterion logical logical recent cast approximate abc consideration factor realistic set wider less trust leave without model selection might outperform short demonstrate compete exploratory describe statistical limited close log log likelihood yield normal likelihood method widely develop analyze spatial approximate extreme three implementation first third successful benefit show approximate compete composite dependence also bayesian naturally field computationally intensive independent identically distribute univariate q must member location parameter extreme limit fr tail bound extreme hold
obtain illustrate formula validity dt quantity compute expression eq mix asymptotic eq c c discuss mix approximately take representation see ensure desire mixing ap column monte verify approximation kullback optimal mix distribution however mix considerably less go e p accord ccc mix uniform monte side limited practical problem sequential test procedure build combination paper certain implication practical want stop soon either time variable take depend accept say sided threshold show attain alternative
handwritten modelling pose image reconstruction pixel code subset subset assign comparison point measure fraction misclassifie substantial confusion train significant target relative target relative top patch highlight result qualitatively setup color fig target background overlap patch ratio art detection fig tool nonparametric bayesian obtaining start suitably parametric tractable demonstrate argument amenable trick define one predictive evaluation
response enough relate regressor overcome regressor relevance regressor able concern regressor describe use contribution regressor calculate gradually select respective ratio ratio error regressor decrease response often begin gradually decrease remain regressor observe reveal redundancy regressor determine parsimonious suggest help either surface temporal capability analysis flexible build capability regressor principle regressor explain alone often response true local noise naturally cause improve robustness locally use stop regressor analyze converge variable iteration ratio begin uniformly observe examine regressor rich concern regressor dynamic change nonlinearity dimensionality carry depth information may regressor structure training important provide efficiency expansion experimentally order regressor alternatively
abuse terminology equivalence remain show finitely process go class symbol give class symbol stochastic claim ergodic alphabet history triple hmm irreducible hmms hmms definition sake brevity construction example formal definition idea process alphabet machine hmm irreducible symbol generator machine generate cover label right observer nontrivial uniquely even must odd sequence two state finitely exactly positive equivalence x generally generator generator machine finite word depict generate abc choose uniformly inspection generator irreducible shift history machine despite initial law limit limit odd converge initial equivalence state argument like generalize machine noisy depict generating period np alternate symbol state generator sequence probability odd step state even induce machine form grouping
price price storage occur price explain concept convenience yield implicit associate stock hold stock service contract fundamental consider economic relationship imply cost identify factor interest convenience positively stock one need asymmetric basis contract close limited economic effect movement price soon large price correlation near price contract decrease propose general long firstly allow incorporate stochastic discretized volatility option observation methodology advance sequential monte jointly volatility section explain methodology detail methodology rao version markov carlo methodology non typical differential e market price market order risk neutral neutral factor payoff reduce follow motion convenience treat yield fail price secondly extend study model motion convenience model
analysis decay people mutual link mean link seem decay precisely formal statement rise activity social predict decay window decay problem science originally study interval see snapshot longitudinal research link prediction focus evaluate predictive technique order address systematically characterize attribute extent fail attempt importance several identify characteristic evolution email extent structural closure contribute formation new edge people common edge similarly formation edge social thing role formation factor decay neither decay network allow validate social unclear edge effort close persistence mobile phone primary rely set well edge average degree source information record cell phone consistent duration type call text dataset primarily phone dyadic communication exclude exclude auto data phone call week period call information call call decay call phone vertex exist actor actor window criterion
algorithm see vector level initialize index thresholding technical need ax appear rewrite solution exact subtract get rearrange present thresholding algorithm sparse thresholding novel enough also tight thresholding experimental vector million face diverse area finance curse knowledge structure study accordingly huge recent machine signal devote modern processing deal measurement matrix recover see sense setting recover able isolated property restrict isometry rip long measurement rip true vector formulate linear efficiently reader say ensemble provide choose goal weak convexity linear typically suffice rip rip
stream predictor recursively average square summarize detail propose provide letter case letter transpose denote set denote empty denote sub vector form coefficient dimensional comprise observation sequence dimensional standard filter terminology reference filter problem recursively time q factor control
account phenomenon probability distribute response although generalise model amenable simulation likelihood increment increment experimental reaction bottom exclude chance outlier fast describe behaviour pair heterogeneity experimental accumulation rate experimental boundary share therefore prior parameter clear interpretation reaction normally less dominate quantity drift lie outside yield drift rate transform transform attempt abc difficulty choice summary summary experimental summary due inherent curse standard abc take small statistic abc ep notice condition iid employ describe section save go second adopt third stem site case algebra update marginal
quality pmf optimal norm norm time fast pmf implement whereas optimize implementation feature dynamically regression orthogonal consider minima pmf conclude preferable pmf strongly input moderately problem local principled local approximation remark pmf include extra flexibility use capture extension explore could problem improve sense analyze theoretically superiority suggest optimality penalty parallel eqs sequential updating idea gauss update thank code pmf file thank sharp discussion improve manuscript variate see variational feature region unique transition imply quadratic solution twice stroke
requirement estimator influence common influence equal potentially unbounded preferable assess robustness limit easier depend claim evaluate robustness uniformly technique show statistical dirac theory
exactly random effect variance record variance hope estimate bootstrapping effect find approximation around quantity coefficient ideally method expect dominate way resample distinction multinomial apply bootstrap regard factorial bootstrap might naive resample replacement naive tend infinity effect see netflix datum compute variance severe quantity treat bayesian degenerate parameter degenerate thus user need continuously distribute tie bayesian observation get weight bayesian distribution take weight match bag boost integer double nothing independently observation double nothing version remove sum section bootstrap z eq method taylor reliable proxy naive bootstrap bootstrap large datum greatly offer
bit accuracy experiment expand hash accuracy bit hashing per course surprising demonstrate magnitude large random projection issue relatively gb market regression dataset course age growth hashing method test bit solver improvement choose popular familiar validate generating solver without notice code unlike practitioner window experimental agree hashing substantially storage hash platform
learn unsupervised manner prototype dynamical shape field take sign probable automatically illustrate signal modern artificial ai device broad include recognition language medical mention intelligence ai essence able perform optimization make understand merely ai device pre ai device neural
response calculate binary w alternatively allow probit pseudo treat except consider base value consistency vector begin response restriction impose avoid place scenario describe processing section outline conclude predictor influence value matrix denote predictor model link deduce relationship particular wish toward write influence additive interact predictor interact interact predictor practice far simpler distinguished contribute whether demonstrate possibility binary continuous simplest influence category perform alternative consider detect marginal ccccc c ex g main pt useful thought problem pairwise method extension likelihood could often time
step player move player classic play opponent relationship performance drift toy respect drift first fix examine fix row two set depict opponent set affected increase result affect tracking opponent many poor approximation capture opponent decrease opponent depict high result big change small good strategy initial value scenario value whole real choose observe opponent strategy opponent depict red blue fix pre opponent prediction depict perform jump used action opponent action game second action depict opponent depict respectively opponent prediction red occur
marginal call rule bethe algorithm freedom message learn bp initialize run choose initial message stop previous learn approach begin bp bethe place minimum belief marginal possibly point however reason provide bethe bethe attempt learn marginal belief reach vary still reach equilibrium temporal equal marginal bethe equilibrium marginal match belief average bethe equation representation give average reproduce
achieve policy action achieve future sensitive policy linear bottleneck increase contextual expressive overcome sensitive instance policy return accord create weak rule weak learner solve contextual contextual bandit choose exp step set example know probability adversary hope assume context computational scale attempt get perturb computationally special mechanism application policy space oracle aside drawback worse improve adversarial possible adversarial set feedback delay round adversarial setting I previous use epoch regret scale analysis exist reward action lie continuous accord regret round extend vc special case new simultaneously small argument context reward
lebesgue manifold deconvolution receive attention deconvolution raise interesting hausdorff distance author contaminate noise deconvolution interest bound wasserstein distribution homology additive estimating homology principal surface approach manifold establish bound hausdorff noise denote open center set point euclidean supremum fold product eq denote measure fourier denote use dot product nc expression
finally summary quantity use characterize population element mean converse pdf number imply measure rate support support point wise great pdfs pdfs homogeneous pdfs imply homogeneity pdfs large great heterogeneity aggregate exist aggregate normalize general support relative population support quantify raw measurement individual population circumstance collection come identical per denote individual arrive imply element measure insight rate iii rate metric diversity characterize rely force homogeneity recall individual variance support simple instance minima maxima length intersection support differ support maxima length support characterize way maximum band population pdfs infimum pdfs union pdfs collection coincide collection population population substantially near zero supremum supremum near nearly method sensitive heterogeneity rest maximize second method diversity pdfs population quantify support union support pdfs maximize maximally orthogonal calculate l analysis pt quantify average quantify aggregated population quantify aggregated support support population aggregate homogeneity pdf estimator bias variety way bias individual permutation bias preserve relative quantify diversity quantify imply diverse cardinality homogeneity support diversity population diversity p population point individual rough individual interpretation split three broad perform preliminary interpretation iii understanding use pdfs yield understanding proportion algorithmic depicted circumstance useful population calculation variance distribution pdfs interpretation scientific circumstance purely
satisfy kolmogorov appendix basic definition kolmogorov shannon item process markov process class bernoulli variable outcome random class typical outcome kolmogorov form former code alphabet constrain index put last additive item take string string represent source take account encoding outcome outcome joint x x copy stochastic chain dependent whether stochastic
transform base introduce pointwise continuously statistic channel unnormalized haar wavelet remarkable orthogonal transform risk wavelet optimize dependent article organize first unbiased risk chi square arbitrary degree freedom apply pointwise transform threshold give chi unnormalize haar propose inter conduct mr evaluation method international conference independent draw chi degree recall see random characterize data q k order modify function kind chi understand transform first straightforward designing consider deterministic realization distribution requirement
decrease depend x I remain eq term positive plug obtained verify decrease material I lemma I clearly dominant utilize simultaneously lipschitz general present synthetic one highlight fit theoretical finding real world even early two difference iteration monotonic datum adjacent monotonic generally difference theoretical require outcome particular might try computationally quite subsection sometimes fit
light monotone call interval even bad interval interval light conditioning contribution bad note bind monotone interval side suffice sample probability conditioning equation follow pi pi pi pi turn clear contribution partition g p monotone light interval tv sample pi tell either accurate sample hypothesis distribution accurate draw sample pi I argument henceforth hypothesis output probability monotone interval consideration hold claim fact distance I c bound ki statement monotone tv k ii expression maximize final equality fact prof run simple perform increase motivate level dominant let modal time monotone know efficiently monotone learner separate clear hope efficiently mode natural try decompose modal level novel modal sample yes monotone monotone distribution stress modal essential whether monotone monotone care running parameter decompose monotone
offer scope interaction offer way interact average user thus increase hope meaningful projection pursuit exploratory previous narrow argue allow exploratory much insight class make dataset demonstrate task far demonstrate quite subtle spatial explore formal application easy straightforward histogram tool traditionally principal transform exploratory pursuit chance use question necessity exploratory pursuit replacement benchmark motivation mean benchmark notion thus really occur meaningful certain context really context usage year rather poor successive generator
handle aware restrict seem restrictive class action guarantee section restrict base cost meet requirement bound bandit trade reward number key use r dependent propose simply difference bad trading way bandit reasonable reward reward satisfy leave observe term p second consider relate function uncertainty outcome trading cost affect like come observation maker position outcome independent outcome independent bundle share exactly bind much perturbation change need argue
mix allow consistency I dimensionality mix nj ix balance consistency assumption uniformly sufficiently depend q coefficient decrease confirm theorem play fundamental role implement thresholding estimate threshold select minimizer function q observe time consecutive segment remain illustration procedure convergence oracle loss perform oracle justification adapt band choice let x w oracle estimate define empirical theorem satisfied c p generality justify coincide end white satisfying constant cross validation minimize oracle problem high subsection procedure moderate variable explanatory covariance thresholding generality standardize actually pearson correlation distribution population classic person correlation pearson two former
small looking interval cover simple plot long abstract automatic parametric kind important green article quantity svms stochastic error term several establish x iii three directly however time quantity interval special wavelet univariate nonparametric
expert many expert insight study expert rate likelihood mle fast divergence converge produce rate nonparametric exponential model I mixture tool density know covariate space adequate fall generalization classical constant covariate expert widely recognition audio finance distinct include among one conditioning variable parameter glm example binomial fall might lead discussion many simple new consider use interpret gain increase complexity direction
bioinformatic homogeneity performance mmd depend width mmd permutation investigate nan hypothesis hypothesis sample set perform homogeneity summarize show adaptive type situation two generate denominator thus numerator positive experimental summarize show adaptive tend outperform mmd term homogeneity propose outli detection outli another regular tend use wide support regard evaluation sample denominator numerator outlier outli outli outli detection outlier trade adopt auc bold face illustrate behave outli dataset let dimensionality setup regard sample table describe input auc get small outli detection become challenge result input dimensionality tendency sharp point ratio preferable density
rest globally discuss possible interior rest sake monotonically equal temperature straightforward check along possible rest depend ratio depict light grey region correspond ne intuitively correspond ne rest ne exploration vanish correspond anti game three rest light grey additional rest point sufficiently obtain characterization consider obtain quadratic equation reasoning intersection meet fig anti game illustrative original variable recover briefly interior single globally nature multiple rest jacobian eqs symmetric equilibria recall critical
piece foreground put overlap sake result specific lasso strong correlation lar independent simply simple converge apply problem separable scheme pixel resolution rgb channel add result test remove lack structural constraint encode neither group sparsity improve norm correspond reconstruct method foreground mask original detected overlap present foreground color aim linear patch adapt dictionary prove interested less multiply scalar inverse prove practice prevent induce norm element natural image another put encode decomposition relate arrange independent corresponding element model dependency natural like grid call dictionary norm dictionary grid group spatial neighborhood grid grid cyclic norm task consider project one perform orthogonal batch drawing signal mention recently hierarchical dictionary image dictionary group notation subtree dictionary idea propose prune irrelevant use additional decomposition remove w operate instance overlap group define alternate optimize respect protocol element improve denoise sparse hierarchical predefine dimension training dictionary impose handle
apply simple univariate strategy weakly dependent weakly dependent correlate identify calculate correlation pick correlation variable perform leave dependent dependent strongly dependent partition identify amount estimate direct procedure use learning identify dependent play lead role later reduce correlation accord size depict weakly dependent correlated term domain variable reflect value reflect optimize univariate less cost note gaussian base dependent identical weakly need course imagine define strongly optimize separate dependent one interesting consideration typically strong multivariate gaussian obtain project build impractical increase model sm flow variable sample project subspace extent trust reliable divide subspace project consider dependency among belong subspace probably offer feasible way growth grow validate section construct partition user define multivariate htb x cx x randomly subset multivariate gaussian gaussian subspace matrix approximate diagonal fig mean current capability sample user experience rest later sm perform every generation group interaction strategy also variable belong evaluation subspace partitioning simple straightforward sm indeed performance dimensional course need divide cluster coefficient treat however still disadvantage suffer curse size expect cluster comparison partition sm explicitly reduce sm accord population denote realization variable perform sm form univariate gaussian sm algorithm control need individual individual contribute replacement htbp initialize meet individual individual individual
ghz cpu ram discard thin picking obtain posterior chain converge comparison purpose run variational bayes second attain variational plot shape prevent clutter significantly towards accordance gibbs sampler note informed guess treating improve drastically formulation constitute behavior mixture normal connect broader namely arise inferior normal mixture normal particular guarantee result essential unnecessary signal oracle
develop low output scale fine code closely resemble hidden however subtle neural major difference weight learn independently node propagation minimize typical ensemble converge unless stationary difference seven bit file one size figure rectangle seven bit datum order magnitude mechanism input history fourth return th order context recent operation index mod set history also mixture expert mixture expert et backpropagation network interference effect lead learn naturally interference compose plus feedforward switch select expert expert expert development compression divide subset separate expert train expert train expert difference mixture al feedforward learn algorithm specific one area future learn distribution ideally partition however use mechanism expert govern unit recurrent example lstm hide gradient g deterministic mechanism intend prediction vanish computation recommend work rnn
linearize bregman journal science pp bregman minimization application compress sciences invariant rank vision pp mc low mc like know entry stands sample mc low feasible lead objective popular alternative summation rank exactly cardinality portion entry corrupt low portion entry corrupt sparse stand np exploit convex convex summation balance recover assumption exist solve problem large aforementioned approximation involve objective mc use solve contain analysis
design single maximize function appendix lemma relate x x strictly necessarily often convenient support set find interval reduce polynomial k well table notation design
round since loo stable shown result potentially binary potentially randomized uniform loo online exist potentially uniform loo stable c minimal need cover loo e g exist sequence round bound exist sequence om learnable loo particular playing round suppose must round apply setting diameter tt notion I learn adversary pick time always setting supervise input instantaneous regret z h instantaneous interested characterize condition set stability stable asymptotic minimizer output regularize mirror weight interpret thought empirical erm dataset erm stability albeit strong require guarantee type form stability slightly strong weak
solve simplex optimal finite set linear function piecewise uniqueness quantify metric task suppose simplex nr ir ir small object weight whenever later paper weight respective notation move restrict index formulate distance agree favor histogram histogram criterion balance favor metric idea pair formal perspective ordering criterion neighbor weighted neighbor sum set stand necessarily subset kn kk consider positively homogeneous unbounded solve issue restrict search variable upper piecewise consequence admit least cast convex
help approach problem directional sphere ray event observe set begin interested propose framework test directional datum spherical specific show perform situation competitive tend really priori spherical test incoming particle energy simple threshold implement joint directional energy numerous dependent multiscale use cr make public one tune use sophisticated drive method tune high although threshold approach threshold test power however possible give although compactly support window appropriate spatially concentrated function knowledge distribution ray spirit directional specific like fine coverage take benefit full significantly obtained investigate finally addition extension stress two theoretical experimental theory part acknowledge package interface package transform thank concern le fr ed make aspect thank anonymous associate suggestion improve proposition france paris france decade origin ground attempt discriminate try arrival energy suppose toward directional sphere nontrivial make turn detect form explore wavelet
base expensive computation algorithm score trajectory without computation complexity state practice number training example feature quadratic number able feature research sequentially classify step linear computation choose action action already acquire classify score contribution newly order feature equivalent constrain practice choose second couple minute minute hour category indeed name number binary breast cancer binary diabetes binary heart binary binary guide segment multiclass
run time repetition vary various linear weight age seminal predictor first standardized shrinkage try aic bss sub list h lp aic bss shrinkage estimator prediction ten error repeat l bias correct validation small base bss aic instance bss sub clearly shrinkage estimator utility positive practically may well correct shrinkage estimator less sensitive misspecification reduce maintain superiority misspecification behaviour detail monte carlo section call case percent
matrix element diagonal element converse let position let column agree index form generalization let satisfy equal non let element column goal case claim large subtracting expand positive definite together proved need conclusion maximal take denote consist establishe acknowledgement acknowledge arc centre mathematic via team corollary lemma comment mathematics university edu university france fr maximal determinant search design find optimal design
range increasingly complex grow automatic density adaptive nature box exploratory mcmc aim practitioner exploratory stage preliminary exploratory carlo fitting attention grow complexity tool tuned author publish acknowledgement support e support fellowship acknowledge analysis author helpful wang interact parallel chain mechanism comparison flat histogram bin split interact hence former chain state chain chain use present bin split desire new bin reduce desire experiment however iteration xx sequence typically update bias invariant parametrize proposal ad I extend necessary extend split bin bin flat histogram I I tx stepsize monte chapter recent ergodicity fail explain parallel version prof infinity precise statement impact upon require impact add proof consistency flat stepsize flat stepsize stay rely threshold result result flat histogram remain stochastic algorithm adaptation homogeneous hasting
q depend apply practical unknown validate heuristics oracle spectrum rank large corollary finally rise appear estimator slightly heuristic notice index auxiliary distribution yx lead lead conclusion quantile unconditional enyi representation law number e write entail collect conclude denote u k
include median minimum variance dissimilarity center dissimilarity method mention cluster specify dissimilarity agglomerative hierarchical dissimilarity method formula center diameter median centroid j j attribute name coefficient define recurrence formula agglomerative similar cluster center table dissimilarity must equivalence two point formula use q center distance expression readily verify agglomerative centroid variance store dissimilarity store examine dissimilarity two point cluster representative center treat remain object center method consideration since storage object linkage term refer theory single link dissimilarity couple store implementation look reciprocal mutual near use practice agglomerative hierarchical conclude agglomerative
normalize convenience homotopy prove view monotonic know need condition fact correlate natural mean full symmetric ii ii eq jj kronecker move right side dd matrix dd cone principal minor principal point cone absolute nonzero
weak hard extract rescale useful property formalize empty size margin produce loss generality orthogonal margin norm assume pick subsequence limit nan firstly margin extremely example contradiction secondly otherwise arbitrarily example attain arbitrarily contradiction margin everywhere nan space contradiction positive subset margin reduce empty sequence zero restrict inductive zero margin property combination margin within existence example suffice row f kf I kf corollary prove item set loss margin less upper contradiction suppose margin bound element present mx exist whose loss lemma achieve empty subset contain margin convex combination f false margin show arbitrarily large entry rate hypothesis doubly feature
strength part several restrict elastic limited past subject treat processing align e computation study comparison jointly energy careful convex combination f ds ds pp recommend analyze statistical use match disjoint change alignment may additional orientation interval composition parameterization domain space separately jointly solve problematic symmetric one enforce double solution however degenerate ensure symmetry symmetric eqn commonly function keep close suffer step seem problematic empirically conceptually template use different basis turn relate observation nice solving implicitly eqn geometric natural fundamental function shape parametrize train problematic eqn understand issue
trivially hash consumption avoid disk use task many value preprocesse gram response generate disk occur mask hashing note substantially low indeed concern cost accuracy continue increasingly resource partition compact representation sparse binary hashing hash positive definite naturally logistic leading bit variance random projection significantly interestingly bit hashing improvement speed department science ny edu microsoft research storage efficient progress among integrate logistic
row dot product although clearly let onto span matrix span rank extension statistical leverage quantify leverage row square influence square leverage score quantify leverage low quantity widely diagnostic large exploit example degree cluster coherence value customer network dense thousand hundred even traditional qr dna single nucleotide snp increasingly genetic snp possible previously confusion emphasize main contribution important concept statistic several random score crucial require qr partial fast existence fourth implication actual row arbitrary related streaming contribution rigorous understand practical constant realistic size numerical interested consider brief summary bottleneck section similar base approximation also random art hadamard algebra library
definition bregman eq expand gradient inequality consequence convexity give xt xt sum optimality old algebra assume eq desire may integrate take respect shorthand old apply valid control term inequality invoke combine inequality bind provide provide subgradient descent receive converge stationary modify subgradient mirror generalize recent incremental markov subgradient control couple autoregressive sensor wireless sensor attempt sequence statistical come dependent experiment example ram perform stochastic mirror descent step describe paragraph provably convergent procedure govern quantity radius lipschitz function familiar previous stochastic rate converge main mirror make expectation ergodic high show general remainder paper description expand corollary throughout explore provide fy fx x c describe description algorithm familiar literature mirror generalize address geometry prox
single follow object similarly distinct estimate n manner unbiased block derivation generic detail interpretation popular agglomerative mean spectral tp calculate c static cluster dissimilarity return flat evolutionary agglomerative interesting function agglomerative cluster merge agglomerative merge low cost iteration snapshot make merge dissimilarity temporal merge merge dissimilarity give see expand recursive membership begin static initialization employ speed reduce require evolutionary initialize time initialization evolutionary snapshot cost temporal dx modify measure fit past datum compute rather cost function static average spectral clustering perform optimize front snapshot simply term note case modify incorporate note easily accommodate history use exponentially factor factor cluster evolve ratio spectral form laplacian admit interpretation modify operate rather assume scenario object object may scenario proximity combine describe observe remove object handle add perform factor contribute result illustrate cluster
I incomplete run poorly optimum estimation offer advantage na I form respect em generally ascent property ensure reason algorithm accommodate high dimensional space increase many perform covariate example speedup discrete observation parallel speed issue graphic useful computational em algorithm evolution effort often ignore evolutionary relationship incomplete repeat separate categorical nucleotide composition equilibrium intend sophisticated model inform consideration costly maximum various show compete technique accelerate convergence finally synthetic mutation birth death likelihood markov death time negative particle one give birth particle decrease count popular biology characterize epidemic dynamic probabilistic alignment dna sequence traditionally birth death particle birth
whereas company provide day estimation france consumption measure long channel individual equip home able channel around day consumption people send determine advance profile relate economic sample behavior total people sample day aggregate measure observation belong spend representative really center draw great majority simulation minimum error optimum center automatically start draw order
simplicity distribute allocation problem static channel associate game equilibrium show distribute learn fact game converge extension sort achievable efficient constraint e g transmission power user importantly extended strategy constrain long suitably modify value allow recall mind convexity r star strictly star global weakly unique minimum star convex game star convex potential start employ gr evolutionary index name iff fact calculation dt h l qp lyapunov z z definition qp prove need k shorthand direction convex f q follow quadratic far k f z plugging noting near
horizon item find strategy particular ucb present use probabilistic support finite cardinality performance good ucb oracle describe indeed oracle slightly make reason appear expert request notation generic setting lead bind armed bandit time good miss quantify virtual aware time optimally draw assumption sense set infinity maintain precisely find go uniformly infinity present optimal allocation turn
change vast exist segmentation detection correspond continuous signal precisely different notably profile technology collect extent change translate increase empirically aware theoretical set sense correctly change towards center share change position zero th respect center correct infinite jump let characterize first select assume generality point group fuse tend give deduce find unweighted fuse lasso position unweighte fuse tend tend comment position monotonically boundary detect change detect consistently asymptotically profile relative identify close enough correct point find benefit unweighted suffer boundary fact tell middle group fused fuse correctly find position tend independently position increase problem detect allow change model th value identically result extend show discover unweighted condition
previously prove enforce descent challenge address introduce follow splitting model stepsize computational uniformly eq vanish q reasonable still nonconvex split monotonicity analysis ultimately gx inexact point subdifferential exercise equivalently equation splitting define inexact stationarity residual call prescribe satisfie iterate stepsize become correspondingly strictly
network difference particular bic size assess true parameter element probability network estimate independence provide situation black dot represent pearson permutation plot structure size structure learn test predict behaviour alarm interesting improve improvement sample
attractive variation poor censor associate ht contamination performance decrease adequate censor recommendation conduct extensive carlo several divergence theoretically efficiency robustness go beyond scope paper weight devote previously continue proceed technique largely make little nh consequently inequality expansion w probability analogous hand taylor series expansion write straightforward combine infer introduce algebra probability analyze rewrite keep use
strictly concept site positivity everywhere positivity though concept start useful regard consider event belief formally uninformative let
reader paper address discriminative view dimensionality representation two tend goal vice versa important overfitte objective simultaneously seek case name paper inherently include semi case drive learn combination cost dictionary clear dictionary share overfitte learn dictionary predefine fisher despite drawback form reconstruction consequently generalize incorporate classification cost loss cross validation misclassifie sensitive outli optimization
observe synthetic zero draw p k p kp kp marginal stop show curve artificial average consistently outperform accuracy summarize performance algorithm aspect value almost identical outperform thresholding kind nonzero numerical like emphasize dimension limitation expression measurement experimental
method propose specification calculate resample increase highly deviation close estimate evaluate calculate divide ols ol know enter tell traditional prediction prediction summarize rate decrease rapidly relative slowly observe correct set parameter deal regressor variance relative decrease extension adaptive literature method converge oracle enter model candidate enter sample size allow ols integrate another
would type work improper benchmark absence frequentist confidence interval value continuous bayes posterior posterior interesting widely variation confidence simultaneously work complete moderate plausible sufficient meet absence work bayesian posterior expense knowledge base hand allow make less dependent precise extent uncertain third moderate posterior confidence illustrate implication corollary variation px px entail assess strategy function set mass biology department road k
mixture dependent notice stick break straightforwardly rich stick break rest briefly discuss atom analyze feature stick specification stick multivariate atom otherwise measurable correspond case eventually law covariate eq mean mean th component reasonable essentially choice correlation measure discuss definition give subsection stick variable assume correlation simplicity shall every measurable exchangeable array indeed expectation joint array joint permutation observation practical exchangeability represent suitable measure marginally dirichlet process representation consider beta tractable procedure follow independent beta result independent beta product beta give independence follow specification independent beta r ik
top seven log scale capture simulation single show beyond decay reduce well significantly square wave include collect set learn drive uniform illustrative figure empirical involve propose balanced reduction general key knowledge literature reduction dimensionality reduction pca reproduce hilbert space rkh balanced functional analytical understanding ready approach case dynamic simulate trajectory emphasis broadly begin empirical estimate dimensional direction observable behave linearly linearity perform state therefore map feature situation closely separability linearly separable separate map nonlinear feature decision
parameter number find minimizer problem rest want nonzero want avoid minimizer zero key exceed accomplish suitable desirable minimizer empty cardinality satisfy minimizer contradict suppose nonzero contradict solution note know minimizer large minimizer
besides causal ordinary relate conditional obvious direct cause observe quantity intuitive mean direct cause correlation theoretic study causality lead direct extension generalization direct prediction source investigate causality artificial advanced particularly causality functional relationship stable mechanism together thus infer relationship term discussion ideal set observable another mechanism force take result
vector family task spherical candidate freedom also difficult leave investigation aspect idea previous estimate function unit sphere spherical proceed interested would second portion infeasible spherical result accord subsection step feature fix mixture spherical coordinate consider spherical spherical feature formally graph kf estimate equation share choose parameter carry optimize respect kl lagrange multipli minimized sample graph identify associate accord result multinomial distribution metropolis hasting outline
reward know arm reward status static propose logarithmic system status evolve accord show logarithmic policy achieve pre agreement rewrite regret markovian transition algebra reward stop show logarithmic sufficient term third term reason play arm show term logarithmic caused spend arm exploration epoch consequently cause epoch term playing expect arm exploitation
norm crucial heavily influence dictionary rank admit decomposition refer contrary component zero refer structure introduce pca decade alternative potentially notably nmf many support factor seem consider structure analysis essentially retrieve partly face seem meaningful priori induce enforce priori factorization support reflect pixel organize grid support factor localize respect expression pattern microarray involve group pathway neighbor protein base remark early readily explain respect priori relevant hand slight define define appear tool exploratory naturally analyze decomposition learn obviously exercise assessment face consist image pose resolution computational reason show dictionary pca nmf spatially pattern sparse segment meaningful
nx consider expect experiment hope exhibit experiment exhibit though goal mechanism suggest pool section viewpoint web feature relevance query irrelevant perfectly many hence may prediction fact know vector cluster insight datum distinct suppose diverse exhibit obvious unit discuss various mn tm denote least motivate suppose vector index iteratively estimate algorithm require guess initialize vector step eq
description aim build help traditionally could train could interest direct benefit technique visible light measure perform compare search common group together datum currently present object preliminary show concentrate spatially occur concentrate plot highly analyze frequency see range
cdf information focus share deep list c top mkl rank near column refer first among interesting classical start say top correct candidate list gene short list candidate half position mkl soon accept place gene candidate start list wish gene preferable correctly top mkl beneficial typically top set find gene soon interested list hundred candidate disease become share disease disease share ht curve curve mkl across number causal disease gene identify disease disease know gene require know disease one share disease tested context discover disease context contribution dirac vanish know disease disease disease association involve disease associate remove scoring association disease remove rank disease turn method similar result relative performance overall perform list
recently construct shall follow slightly large desirable use rkh discard automatically combine improvement estimate advantage reader strategy sampling applicable space norm different semi
eqn replace appear understand mean failure position algebra overall assume rescale failure hold w r ok feature compute follow row ok value preserve done call multiplication indeed matrix sign reading sign however sign therefore compute correspondence matrix partitioning running follow notice kk ok create projection run low running mean datum result low run recover subspace matrix drop increase frobenius k term eqn eqn statement triangle norm projection drop increase eqn replace appear give indicator statement eqn give failure union approximation project clustering result straightforward reduction distance preserve preserve clustering preserve way suffice preserve preserve
take note winner select testing improve average combination adopt equal mean note decade nn organize computational intelligence method challenge multi ahead miss time competition represent daily amount machine uk competition forecast value day method assess absolute relative denote h five forecasting nn competition know selection want forecasting configuration increase compose preprocesse step step way g two alternative model come configuration detail specificity series mean occur observation value corrupt gap adopt removal gap possess decide role forecasting adopt discuss week month forecast embed equation select central apply state review realization function select relevant past final dimensionality vector series step require estimate describe explore adopt dt
train namely original corresponding significance versus simultaneous cost reject nan size run training model inference training characterize environment gb ram core machine case analyze effect problem discuss section discuss create pick seven computing distance additionally priori cost second cost cost option problem decision equation pick distance solve solve cost box plot simultaneous process problem cost node generally hard computation second nm nm take iteration subproblem second time time among three use scalability initially find hypothesis generalization case think much constrain hypothesis predict probability generalization ml lf I lf capture space f small since failure lagrange multipli explicit constraint
strongly objective fail objective issue boost present meta combine hypothesis complex superior achieve weak marginally boost focused optimize meta perhaps adaboost specifically via together near look adaboost boost trivial recent attempt successful generalize problem boost multiclass analysis restrict multiclass utilize optimize boost analysis learner provide intuitive adaboost boost reformulate modify functional al additionally functional specific pac weak rigorously boost notion performance extend foundation
create thresholding obtaining thresholded covariance interest thresholded broken construction describe component solve exhaustive different scope summary per coordinate roughly problem complexity complexity possibly larger desire computation impractical role play solver thresholded total k op jj difference numerical example large property help fold speed section consider discuss real microarray example problem glasso code
successive cycle run enable concept concept concept equal multiply factor analogy discover average evaluation affect expansion quality keep expansion maximum possible expansion limit expansion knowledge basis mainly mit medium contribution people maintain graph relation use knowledge grouping word thing article useful mit interface database file stage plan add base
seem suit round pair correspond entry select reveal suffer goal minimize w reality pick measure predict collaborative filter remark mention trace learn predictor predictor stochastic mirror descent technique bound matrix regret scale get obtain computationally work predict see convex w expert remark perhaps surprising forecaster advance forecaster key namely necessarily bound forecaster entry forecaster know reveal convex forecaster consist bound
count similarly prior dag compose addition dag neighbor obtain single start add edge mode mass dag recursively lead greatest mode view discrete counterpart identical maximum recursion ascent manner differentiable implement md dags move proposal jump dag define otherwise need node pair common dag update limit work randomly choose modify dag dag reverse retain e bb add analogously lastly check cycle reverse graph parent follow application interested adjacency domain edge identify local mass adjacency node dag million representation truth independently please simulation sampler burn local adjacency adjacency true dag enumeration confirm indeed mode mode maximum graph report table md single mode demonstrate search recall detect burn mode global mode identify dataset observation confirm burn serve number calculate mse ignore probability report ccccc
random course non usual inequality past generalize provide let measurable couple immediate bound iid wherein need iid start functional deriving bad measurable suitable suppose deterministic say long
control fluctuation deterministic state procedure update avoid theorem state dual nonnegative convex respect smooth behave smoothing density affine hull constant respect almost fx f respect measure ensure f gaussians satisfy corollary follow subdifferential sample respect lebesgue subdifferential distribution ball corollary follow ensure use decrease f g note explicitly corollary state require suitable streaming result precede provide convergence bound g si achieve involve error mean reader smooth author corollary condition compact convergence identical decrease discuss corollary appropriate yield corollary optimality guarantee focus concrete algorithm choice corollary begin corollary provide uniform hold lipschitz assume b attain normal lipschitz parameter l remark deep corollary constant essentially whose smoothed version close necessary
high record average content nucleotide loo cv box plot utilize negative centroid plot utilize negative bar end rank name nucleotide pair information content whether rank centroid testing procedure sign centroid centroid correct method measure weight content follow relationship justify similarity centroid low average show test pair ic ic wang propose transform vector shift left alignment scoring distance measure consider st one see distance centroid compare measure propose tf median across general median give low rank tf well produce high tf loo cv present knowledge negative assume bind
conditional univariate take I logistic dimensional triangular logistic conditional logistic univariate logistic regression arbitrarily parametrization observe impact mind regression system typically instance section fit drawback close fitting discuss fitting construct conditional estimate fast work parsimonious weight di mean component precisely correspond logistic firstly really separation might conditional construct parsimonious regression identify component small significant acceptance reduce via iteration update formula iteratively least square quasi work h di maximizer complete complete separation proceed numerical option computationally might variance likelihood target grow beyond threshold logistic approach procedure elaborate add cause extra newton ensure decomposable procedure absence information naturally smooth parameter algorithm regression four move logistic qp justify simple parametric product p denote requirement likelihood hold trivially
empty intersection pairwise choice lie regularize interpolation prove equivalence minimal interpolation regularize fix acceptable minimizer bf b q functional compact attain space belong regularization functional choice vanish recover satisfie also satisfy linear interpolation nk restrict subsequence go shall x evaluation functional continuous arbitrary letting minimal characterization theorem learn minimal interpolation introduction x gx gb interpolation g k x xx prove ready minimal norm kx n add repeat sequence give since converge j j draw conclusion acceptable linear satisfie likewise regularize
increase appear distribution shape error already logarithm median essentially even error eventually
suffice polynomially grow stay almost set neighbor belong subspace rank space complete uniformly constant local neighbor issue form seed incomplete third neighbor us uniformity correct seed sufficient seed observation seed seed column neighborhood sample notation minimum distance support seed entire pattern support seed let seed binomial seed seed eq tt thin variable value pz desire distribution accomplish follow draw subset support remainder seed eliminate seed probability perfectly complete perfectly eq thus chernoff p
argue arbitrarily close exactly bayesian shrinkage frequentist posteriori observe estimate rely simple gibbs model computation excellent performance study double pareto shrinkage broad variety suggest close penalize likelihood prior generalized shrinkage modeling prove note right fu eq kkt optimality proof prove n n p u p u u fu normal prove acknowledgment award es national environmental health sciences content solely environmental sciences national lee national education science robust bayes multivariate normal statistical york heuristic instability estimating journal
however computation predictive propose drive fit scope burden learn knot knot hereafter write mt deal property factorize replace justify st place knot tend eq hence small news play provide probabilistic zero process process appendix constant knot dimensionality smoothness arbitrary entire bind value modification addition give pointwise cost equal apply continue process choose knot remain canonical knot possible adapt modern gaussian independent covariance loop
way regularity condition concern normality tend consistent asymptotically value subject divergence bayesian posteriori integration easier set order monte markov give compute construction confidence interval
toolbox publish correction mean classification instrumental importance double base analysis quantify exist variance remain consume black hybrid strategy introduce smooth counterpart accounting practice failure decide increase rare prove carlo intractable world output expensive black box finite remark frequent exhibit simulation run approach first replace known order reliability
outline model nonlinear might impose distribution scale idea matrix adequate define impose derive process consume hellinger proof adapt metric part summarize lemma cover intersection intersection sub pack information serious drawback depend raise see undesirable consequence optimal prior design want calculate allocate available sequential jeffreys jeffreys crucial surprisingly overcome current parametrization much informative expect intuitively analyse problem lead coverage shape
normality multivariate investigation residual auto residual residual hypothesis series series whether see weak test reject significance level whereas evidence run relationship test perform constant agree impulse response series show week response cause impulse series response response normalise compare impulse causal ordering follow order write orthogonal source variation cause variation alternatively could positively except response strongly much impulse response around slowly response affect affect affect
thank helpful lemma david find blind controller np np place science np nonetheless outline admit keyword partially observable markov bilinear complexity square root fractional decision prove conceptual tool ai include plan briefly
co justification matrix view semidefinite semidefinite engine cut ensemble body work please overview reference ensemble phase ensemble multiple instance aggregation separately instance instance bootstrap sample instance project clustering projection subspace contrast quality individual fashion rf performance essential ambient possible mean main co count time hyper solve cut problem add vertex meet another likelihood aggregation incorporate similarity thresholded improve clustering demonstrate study closely aggregation finding agrees assume ensemble applicable unsupervised forest rf distance metric randomization fundamentally analysis
policy bad would policy unfortunately seem consequence ergodic environment currently process negative computable discount strong asymptotically weak function deterministic computable discount asymptotically policy asymptotically weak policy say strong asymptotically computable discount computable discount policy discount somewhat asymptotic weak asymptotically necessarily real idea computable rule stochastically computable asymptotically discount function discount computable geometric discount analytically easy contradiction basic weak asymptotically asymptotically indistinguishable asymptotically reward omit correspond strongly asymptotically I
non fs fs ss define natural economic lack submodular decrease marginal marginal base fs fs fs fs fs fs necessarily combinatorial represent tree max leave location location set fs w j ji appear tree every permutation build marginal value role sum submodular call stand denote demand weight demand max leaf weight several unit demand tree also weight demand way ks k gs great informally g expensive g formal review far strict hierarchy separate class approximate learn approximate primarily input target hypothesis exist mostly pac model value allow unknown target polynomial number approximate everywhere fs fs definition framework
dictionary use pseudo appendix technique type mainly focus image signature dictionary well dictionary use dictionary classical problem vice descent dictionary prove efficient remain nonconvex method experimentally enough gradient material technique algorithm potentially require sometimes note updating matrix introduce
summarize follow matrix svd compute n operation interest immediate usual characterize bring set identity identity q solution algorithm common problem hundred note achievable reasonably sized dataset implication step order practically require computable eigenvalue singular represent additional storage requirement conclusion prove
rbm sampling perform manual expert short assess bayesian optimization manually tune cube manual intervention rbm adapting advantage approximation discuss apply space complementary technique adopt wang propose randomize mcmc bayesian objective exploration costly discrete densely graphical problematic optimize adaptation mechanism tend optima finite practice parameter bayesian stochastic
completely atom union two must location atom atom location ordinary poisson c also calculate atom give put together j proceed atom far location eq identify weight put together full propose distribution propose counting may integrate marginal prior since calculate calculation also work component atom number ordinary counting process atom atom atom ordinary equality increment poisson useful refer mean notation usual increment measure find q step integrate integration exactly case set intensity main measure tend posterior case merely note imply start define may measure prior may count distribution proof laplacian enough observation finite enough fact combine limit completely finite draw similarly binomial completely measure keep intensity height generate completely consist ordinary component jump less count eq establish henceforth shorthand quantity assumption choose chebyshev inequality write I follow note continuous c n yield every eq desire truncated case turn propose therefore integration choice approach exist atom bind bound next need part atom desire follow transform difference component share notational convenience new ni ni ci two together conditional conditional proportional otherwise univariate hasting slice sampling binomial conjugacy
determine agent action random size current randomly product mh mh three linear programming theoretic reinforcement long trajectory provide since sampling must convergence chain hybrid mh prior reward bernoulli mh sampler conditioning mh form mh suggest verify reinforcement learn actual estimate cumulative discount feature expectation appropriately discrete environment state expectation first transition robust performance
dt give proceed duality ix mx mx inequality present rademacher class convolution operator hilbert highlight generality apply specific case mixed norm recover bound remove see regularizer learning kernel regularizers linear associate exist admit novel countable certain
describe main scale sde dx bx z dt gx fast solution ergodic rapidly unique converge diffusion govern sde construct average define exist probability generate topology convergence begin general describe introduce seek reduce dimension present expansion paper probabilistic dimensional backward doubly transition bx z component slow n kf function assume measurable rapidly measure later assume dx aim calculate eq q brownian f ix z j z b transpose vector tell suitable sde suitably generator estimate slow marginal
mean allow local arm neighborhood exploit notably al process focus parametric curse query typically structural main request convex cost imply convexity natural function make polynomial note focus bandit stochastic vast emphasize imply bind detail jensen optimization converse necessarily optimization candidate suffer iteration setup prefer solution involve distinction free book sufficient descent concrete emphasis work nesterov schemes et evaluation function accelerate analyze set rate degradation regret plan attack query outline first every pass function order interest source regret opt give center device might act unlike discretization device solution clean intuitive develop version zero optimization vanish center device require center device conjunction ellipsoid motivate device slope value distinguish slope
root induce parent may imply combine variable combine child work entire tree decide subtree combine return nan test neighbor leave subroutine along current return variable return evidence subroutine subroutine determine merge parent child check parent child avoid merge recover empirical moment maximize exist return h u u return node leaf set root parent parent u x h h h x x grouping enjoy guarantee model vector use eq eq recursive return tree undirected consistency imply reveal solely spectral applicable setting complete aa award fa copy tail first tail type average independent provided define random copy tight exponential probability simplex copy probability least eq
convex let theorem corollary extension nesterov loss corollary mirror descent stability note eq extend mix corollary follow concrete gradient descent dual strong assume function corollary online svms regularize satisfy sharp guarantee begin martingale concentration martingale define previous random theorem give martingale sequence note bound ff f use deviation sequence define give precede inequality side q note union relate sum definition sum final complete turn mix application markov strongly great use bound
hamiltonian node weak enforce free saddle exactly used generation infer know determine assignment hamiltonian want correctly probable group marginalization maximize label configuration boltzmann correct size correct number ground minimum create group
vb deal section unknown iii describe symbol transmission iv combinatorial bayes symbol transmission numerical receiver current section vi implication remark future action redundancy know code partition message analog map alphabet
update accelerate proximal propose update proximity fast quadratic count value rate scalability large operation involve zero arbitrary tw u version influence nesterov recursion fista proximity define stage possible case wide solve next study estimation aim competitive art employ lasso choice optimization toolbox stop decrease proximity stop relative smaller study
account non calculate path reveal burden shift conditioning phase ode output treat input inefficient disadvantage want retain form stochastic model basis mention knowledge retrieve e
particular pose level al extend recently level mrfs shape g know advance initial pose shape product expert black therein mathematically high auxiliary overlap high potential use crf show gibbs expressive shape obviously complex graphical become learn potential consider application point view instance shape instance composite explore expressive first separately relation segment shape capability shape finally discrimination shape node image define neighbourhood I edge unary graph obviously subset composite
p x yx infimum x nx subset denote coupling independent identically empirical suppose couple attain concave likelihood empirical reduce follow term weak concave deduce another convergence nb hand notation eq even nb nb nb exchangeable deduce theorem almost surely converge compact strong lebesgue converge interior immediately convergence notation suffice continuous fix n definition lemma smoothed concave
word least sentence cover concave discount multiple time analogous submodular example illustrate analogous choice word importance heuristic highly occur sentence document bf x length hard achieve linear monotone empty relative objective relative add sentence terminate rule select I score greedy despite additional provide propose submodular scoring summary similarity allow pairwise feature cosine share whether
huge sometimes prohibitive integrate approximate variational technique kullback kl approximate improved distribution choose unfortunately log configuration lead involve smoothing presence optimal divergence end problem differ approximate derive estimation go back specific unsupervised classification collection family complete simulation study highlight average classification also public surveillance average average account evaluate compute field allow mainly bayesian
pointwise mapping frequent branch unlike branch outperform consider chance pointwise mode practically marginally sometimes demonstrate mode good potentially achieve lie selection mode information mask forward isolated g mask huge regression nonsmooth trajectory unimodal symmetric mode reconstruct trajectory contain branch reason wrong priori suboptimal candidate reconstruction correct appearance trajectory reconstruction constraint despite approach recommend trajectory reconstruct smooth trajectory reduce eliminate part original trajectory reconstruct global particular break ambiguity change candidate reconstruction vary pattern miss type spurious reconstruction shorter jump conditional become unimodal series change dramatically run reconstruct point shorter long appear nonsmooth pattern never vary pattern toy mask problem mask c robot arm mask component one toy toy nonsmooth density arm nearly mode location mixture superposition tendency develop log still represent qualitatively well mixture mode trajectory mode area reconstruction mode close achieve largely insensitive mask crowd reconstruct shorter due appear horizontal nearly vertical extent inside period scatter become trajectory geometry spurious appear distribution remain low usually get conditional spurious mode also full besides hard log small rate trajectory trajectory whose normal trajectory start trajectory particularly bad lack trajectory surprising give mask nonsmooth still trajectory long actual pointwise reconstruction short branch branch really constraint trajectory
alternative seek test statistic split three expand definition adjust reader appendix q n nn k section substitute limit equation integrate apply assumption limit suffice chebyshev last hold term jensen eq step degradation replace bs propose replace bs trivial statistic refined chen show achieve without explicit restriction du sd short demonstrate general satisfying moment allow function scaling paper set project average matrix working framework allow tend constant outperform test bs term efficiency conceptual differ past bs sd estimator analysis limited estimation capture effectively pool testing consider previously cl remainder intuition formally define devoted number performance condition achieve great relative roc curve comparison discrepancy test
van discussion theorem corollary phone phone author propose allow detect shape algorithm linear consistency algorithmic paper new technique quick image theory object noisy image primary diverse automate limitation essentially scope application object picture color colour intensity object thick colour application present paper rescale colour process digital image crucial section natural
identify quality new restaurant low restaurant review restaurant restaurant deal quality price quality restaurant low restaurant deal price customer restaurant restaurant high restaurant restaurant quality restaurant improve make potential true quality try deal price order part game chinese restaurant analyze part illustrate specific application chinese restaurant game cloud social spectrum cloud exact activity service platform deal tend signal affect paper chinese restaurant game social chinese consider chinese restaurant table customer label size restaurant restaurant restaurant customer customer distribution customer learn signal chinese restaurant game sequentially request request receive customer payoff determine rational maximize chinese restaurant game customer customer restaurant grouping
layer flat flat model combination figure recover flat close across model sampling model cccc flat use data set dash blue outer contour represent respectively solid black blue solid dash blue inner outer contour represent product peak figure plot prediction versus value validation sufficient see model number hide physical allow error final significantly specify flat range network large increase log log likelihood
boundedness hierarchical q right evidence exploit distribution evidence estimate independent hyperparameter implement treat variable thus derivation exposition efficiently subsection initialization value accord repeat guarantee subspace idea signal sensitivity measurement incorporate
complete regard complete whose edge expect slightly remove option agreement intuition ready lemma state concept properly slow graph deterministic compute average slow graph carefully need graph random outer eq vertice fast electrical circuit appendix grow low fast upper approach fast probability one ratio bound zero translate mutual subgraph concentrate section exact truncate noticed behavior walk behavior difference slow fast graph confirm experimentally expansion actually separation fast slow subgraph fuse graph experiment eigenvalue fuse envelope fast notice slow connection therefore slow slowly away display graph exhibit graph similarity fast os right appear decay dynamic control mix measure step necessary reduce distance justify fact spectral fast take walk fused graph walk begin fuse step probability find subgraph walk initialize subgraph impose slow subgraph decrease fuse confirm experimental fused exhibit thereby rate fuse graph join histogram experimentally transition fuse affect h slow fuse right subgraph slow subgraph
dimension show separation model particular publication paper substantial follow interactive mechanism query way query ask expense notion differential privacy mechanism extend rather extend arbitrary sensitivity differential privacy de achieve achieve set arrive online analyst pay query ask course query ask necessary interactive multiplicative weight interactive set offline version mechanism agnostic learner et evaluation unify mistake private release reduction achieve improve special query et al give error privacy time database universe variable guarantee laplace requires run proportional mechanism size block time al query prove low et synthetic giving count universe query improve run size database representation possible release answer size synthetic
least posterior probability mass posterior similar variance seem model look derive single posterior might suggest ratio proposal jump approximately mix improve jump determine pilot automatic propose automatic proposal mainly reduce trial identifiability center determine proposal move weak non choice value essentially simple efficient detail implemented know approximate expand taylor call center density order center
uci california repository detection case sequentially update real time uci dataset part training determine use stage combine framework assume arrive sequential uci learning repository free kind structure good response bad response sample element pass perceptron back neural test nn classifier use bad scale range accuracy sub algorithm intersection turn solution include scheme dataset get uci c c data success nn adaptive fusion vision concept framework compose number center represent weight projection onto describe sub apply
sign recover high nesterov quantity lasso x practitioner severe detailed proof rely view result lasso belong uniqueness continuity piecewise affine support sx tx tx tx covariate easily guarantee automatically
exponential theoretical guarantee independence discuss open future advance independence produce test reliable underlie statistical learn present assume learn discard structure error cascade error produce learn polynomial evidence proportional row execution compare without numerical markov strength guarantee complete ht sound use local test sound global reduce useful sound strategy design computing use useful domain comparable dynamic number test domain comparable respect cm quality correct use test design learn exact improve accuracy improve accuracy markov relate improve quality approach direct markov network require improve efficiency exploiting axiom infer unknown observe avoid redundant environment run design improve global efficiently structure improve efficiency subroutine order publish show independence reliable uncertainty outcome show accuracy independence exact present improve cost produce quality improvement cm
motion motion characteristic self diffusion calculate pe examine motion motion start flow profile ns correspond roughly profile flow examine plot velocity without profile presence use number equal contact contact vary intermediate jump good balance fluctuation within detect unfold order separate ensemble unfold ensemble copy classify trajectory unfold ensemble trajectory regardless reach trajectory reach return rather region show see single region entry list divide point list dedicate come come lower help ensemble force velocity store trajectory counter determine streaming unnecessary
unless approximation exploratory probability analysis furthermore exploratory abc software option software abc select genetic scenario shift abc approximation towards whole bayesian select assessment practitioner project grateful look since jacobian hence discrepancy connect lack consider sufficient satisfie illustrate behaviour operate fundamental integrate approximation carlo practical indeed unstable survey wide array application implementation argue abc bayesian discriminate insufficient lead fail entire infeasible due
attempt answer element conjunction provide challenge seek consequence phenomenon try decision uncertainty area science decision likely review statistical understanding conduct decade familiar wide array statistical player mention significantly especially run mean speed player fewer equal combination status able obtain walk evaluation player shorter argue first base type sum triple home double triple triple home triple home run perform analysis remain transformation indicate statistically trend state trend towards imply become recent seem reasonable assumption player try year occurrence dominate mention proportion score increase derive measure argue player already effective seek different lead evaluate markovian recurrence seem logical modification team team performance measure evaluate team team notice production model influence trading team rely production aspect game measure team serve team consequently contain availability
detail true cluster convex show observation model density unique optimum technical require gradient smooth make true construct requirement constant apply refined guarantee via direct definition singular u entry matrix similarly except plant spectral norm infinity easy probability completely determined distribution plant partial first result handle spirit j sign disagreement sign vanish paper optimum universal discussion appendix guarantee condition high existence desire dual lemma probability strong allow
merely form sample representative compute th patch index thresholding feed entry new matrix entry indice iii train classifier scoring cost effort large take hour often performance drop sample aim achieve reasonable accuracy co adopt propose paper extremely idea separate use classifier iteratively classify assign example illustration co natural redundancy relationship training loop however co superior coupling boundary naive margin proxy confidence margin define vote element list algorithmic description respectively label example membership f kf kf kx k co rarely obtain feature possibility constitute natural split subset acceptable independence label fortunately represent naturally property construct among
concave laplace cavity remain may induce numerical moment convergence report fractional improve initialize remain variance cavity problematic multimodal extra care discuss loop rigorous stationary loop parallel early stage approximation marginal improvement parallel update double necessary ep prove adopt propose modification double fast ep parallel double loop optimization moment matching loop marginal inner site update loop distribution message loop consideration direction extend modification check obtain additional evaluation every site integral update site cavity variance choose computationally modification illustration small tolerance modification result double loop optimization initialization iteration achieve distribution loop usually inner loop utilize objective determine respect cubic interpolation derivative site I efficiently comparison sensible hyperparameter achieve parallel double loop update ill many cavity utilize modification cost limit maximum inner loop additional size adjustment site outer loop evaluation rare may cavity though loop optimality
want previously financial index change consequence type drive change specific financial evaluate supervision concluding remark iv asset order volatility deviation specify information majority occur great
subroutine say within therefore meta procedure operate meta simplify explanation ignore constraint result fundamentally focus correctly identical meta recall iff universal uniform meta become case meta improve passive need large round behave follow thus contain vs put would vote infer request sufficiently high least step request x limited precision example necessary target function target z z type classifier one interval nonzero far interval split exclude total large number interval use require classifier represent particular among classifier represent separation vote request large vote favor particular decrease associate reasoning sufficiently satisfy label correctly separate point intuition operation result show return abstract follow include vc strength recall provide meta achieve complexity nontrivial target function know advance exist algorithm mention approach remove merely calculation actually somewhat general result meta achieve target corollary theorem class function passive algorithm achieve imply satisfie observation characterize possess vast array question passive describe question give learning efficiency specifically suppose label find return indicate concept method capability subroutine body store simply try possible additional check evaluation prohibitive easily calculate occurrence q calculate appendix determining total meta remain question polynomial efficiency subroutine actually elegant efficient specifically bind finding need infinite correct difficulty simply search number take restrict unlabele show state probability select modification unlabeled guarantee increase concrete corollary suffice use meta example result efficient running find previous always gap label complexity refine advantage characterize achieve gain address open problem similar explore say label p foundation upon begin gain achievable disagreement active gain serve improve quantify sophisticated disagreement subsection already publish slightly version meta disagreement learn result begin disagreement though refinement meta great aspect two meta request focus disagreement therefore region request request meta budget output request request else l procedure define satisfy inductive well thus infer meta stage version space passive return classifier request add guarantee passive achieve surprisingly meta meta claim universal however specifically improvement achievable step address passive label region small example request union f f consider request example meta positive say restrict small large disagreement reduce fraction request complexity dependent accounting observe positive observe positive request upon reach passive similar case union improvement observe interval negative region separate interval j z ji h difference interval disagreement meta improvement passive toward generalize example concept informally disagreement characterize label complexity disagreement wang disagreement include related variety satisfie condition explain implication go calculation toy take threshold example simplicity distribution uniform section z rr z rr h z consider r b b h p disagreement mean consider interval c set label region arbitrary location point separate gap ci extra remain mention disagreement complexity achievable disagreement reason request version disagreement request meta disagreement coefficient specifically essentially though vc class achieve learning modification instead formal detail implicit theorem essentially identical
measure cube variant endow pf ft uniform orthonormal indeed unique property concentrate conjecture follow endow measure universal confirm numerous implication imply influence tight clique currently conjecture would answer state formula polynomial fourier concentrate
lag matter technique lag lags forecast grouping remark group grouping grouping grouping select optimize forecast variable forecasting average forecasting segment pattern coefficient strong mention lasso detail universal grouping estimate whole combination modeling type underlie month suggest grouping estimate realization mainly grouping estimator depend problem produce appropriate proof closely upon consider ar equation regressor entry lag assume coefficient move fractional base variable say subset variable subset proper j spirit far state tp positive complement form vector w eigenvalue gram see cardinality remove column gram state
I quantitative ergodicity constant hasting knowledge setting kernel proposal metropolis hasting target density density vx f p fx vx readily px q qp vx choose vx pt projection finitely eventually stable ill frequently projection occur expand difference first focus allow random size analysis adaptive monte carlo generic expand set define stand algebra x practical mechanism include expand projection ensure feasible potentially adaptive particularly provide significantly well setting expense require convergence certain criterion step family smooth g include framework dependent particularly former share quantify ergodicity crucially projection present may situation
score see number redundant feature feature indicate score nonetheless give similar evaluation incremental classifier add irrelevant task increasingly reliable goal literature principle fundamental study known redundancy result rely illustrate amount redundancy evaluation reliable principled combination reliable assessment subset work extend rich continuous scoring repeat self def subset attribute motivate certain definition fundamental study control measure compute behaviour solution size relevance reliable relevance presence sophisticated matter study filter case error loose set actually feature benefit less
construction hypothesis reject identified reduce health physical activity week health option health good fair poor include indicator economic status formal education status origin parent product moreover information age gram per yes variable percent overall percent missing miss assess imputation available alternative surrogate imputation vision speak mass weak lack range head back activity getting accounting use major turn range phone call friend phone friend association variable category limitation index restriction activity formal education consumption gram day coefficient rate functional health social support connect five node background health false positive quite bound identical
operation reconstruction perform estimate node internal let fix satisfy leave pair vertex q child repeat role rely topology used subroutine building child think neighboring subtree weight leaf form standard bias technique calculation ready complete correctness fix proposition divide topology correctly reconstruct hide sample everything level four test proposition choice least reconstruct third account reconstruct ensure weight conclude state general outside polynomially q balance root say state mutual go
attractive fundamental terminology fisher mean would prefer statistic variation describe student variable acknowledge abstraction student many abstraction random toward statistical aim statistical article principle model presence inductive statistical analyze likely principle identify variation would probability capital sometimes kind call quantify principle principle evaluate procedure explain vary circumstance link connect specifically somewhat advanced elaborate describe detail train follow reasoning picture indicate
interest tag recommendation allocation hypergraph document pass model goal model topic broad tag characterize tag identify tag publish treat compose word manually label object interest multiple tag tag illustrate document tag author author annotate tag build document fig compose tag image link bipartite graph times corpus times build implicit link encourage topic document like motivate variant topic focus generation influence document document cite document labeling depict co citation document large parameter document distribution topic label generate limit relax relational topic hadamard product value generate citation adapt account use topic markov field multiple type model either proportion cite document regularization regularizer encourage labeling topic among nevertheless pairwise topic expressive insufficient document fig associate tag word uniform tag use
calculate entry code
partially projective algebra product topology sequential automatically either though arise unless countable product adequate projective construct constitute projective turn consist finitely additive set crucial set finitely additive call additive algebra countable additive countable helpful projective mapping define space product necessary algebra word measure projective projective projective limit countable sec consist measure definition facilitate projective specify sec start countable fix topology open ball rational since separable countable set disjoint q element partition algebra partial iff refinement index projective limit partition
exceed ill sample poor scenario variance belief generate model approach generative identical impose additional identity symmetric specify factor efficiently principal pca established involve ms last loading span top estimation illustrate conceptually obtain procedure thing study comparison estimation consider subset validation detail appendix pca unclear impose constraint number could motivate shall experiment constraint constraint efficiently straightforward together motivate follow residual variance pricing closely relate trace constrain problem address solution variance stand exist multiplier context practical long practically contribution
network node field laboratory identify node theoretic decide rest generative network connect network stress could location social web stage already decide joint far give block key mutual essential label correlate explore entire label effect node start explore central boundary alternate call agree agreement rest network network
appendix glm heart datum dimensionality range label class letter mnist letter normal gaussians uci learning repository benchmark repository feature range classification split mnist resolution reduce save prevent overfitte split letter perform split train testing letter lie letter time sensitive class heart mnist letter comparative study nn classifier large glm classifier cf distribution follow glm un consistent metric describe energy metric validation overall include number near neighbor interpolation glm margin classification display set dataset though good strategy local metric display split letter scale appendix mnist
svm indicate statistic dependency generally outperform prediction across indicate label elsewhere literature rapidly train relatively performance notably robust demonstrate dependency lda outperform svms rare dependency svms label except frequent dependency outperform document large dataset face large number extensively literature multi label text majority focus corpora relatively label label annotation label long tail rare number increasingly text attention future combine discriminative type generative use discriminative lda hybrid model etc etc text classification multi computer author anonymous journal helpful suggestion improve material upon work support foundation grant intelligence project via air contract fa office research grant microsoft award reproduce annotation contain herein interpret represent express imply new york annotated corpus available linguistic nearly publish york st manually human new york indexing service correspond subject article table numerous example document document descriptor remove article train remain article assign training result corpus remove word document occur set consistent common literature union dataset document descriptor type restrict dataset split equivalent form split first remove document descriptor appear fewer update corpus dataset classification etc consist assign label category original present set commonly convention restrict evaluation randomly select test score margin problematic manually automatically automate avoid automatically subset hierarchy automatically assign hierarchy expansion although sensible lead per relatively multi unique quite small single automatically although nothing inherently wrong lead able assign label statistic real world assume combination yahoo world power law yahoo split work training document remain label assign single label part collect pre wherein category yahoo keep actual add complete set setting motivate setting optimize result optimize cross svms show hyperparameter training experimental applicable lda incorporate indicate value generally total strength priori document large train
discussion permutation write refer distribution refer underlie collection translate poisson approximate binomial translate poisson binomial distribute translate iff write random fractional poisson translate poisson binomial give translate distribution eq description namely positive rational relatively integer sample provide ingredient use give implicitly explicitly cover cover construct size nk integer binomial tv cover theorem part extension algorithm figure modification give detailed description proper algorithm theorem hypothesis run hypothesis return output subroutine sparse sample access design find accurate unknown inside cover design imply close heavy finally hypothesis choose hypothesis subsection return point return choose description start unimodal inductive proof learn unimodal follow unknown unimodal bit operation follow list mass assign sparse bit output description form contain interval explicitly guarantee form cover constant lie level mass unimodal close unimodal sparse obtain list define return put point otherwise run unimodal return
differentiable every far eq rkh inner product dimension continuously differentiable xx eqs sec slight abuse identify operator take space comparison sec interpret gram xx x j x hadamard method propose apply handle strong output gradient method mean subspace span symmetry find various function avoid less
help eq derive starting introduce standard density lebesgue variation ensure please appendix class triple mle mle convex detector label factor small constant phenomenon detector know label achievable matter minimax extra assumption mainly influence slow label smaller slow statistical intuitively challenge passive sensible know density quantitative
old gp equip rescaling offer true old smooth square regression nonparametric model well add convergence hope reader deep fundamental extension try study gp remark estimation restrict define subset unit origin actual rectangle coordinate argument shift domain technical diagonal define measure identity matrix project axis one
ph university california california la currently electrical computer department endow wireless access interest digital digital communication human interaction endow wireless center wireless communication communication channel reveal advantage common measurement reason practical eeg considerable evidence give eeg frequency source exist direction indicate direction recovery numerous basic form impose algorithm algorithm algorithm mixed reweighte among algorithm much achieve bayesian greatly extend researcher example first later extend always much minima application contain structure exploit structure source focus relationship besides purpose theoretical assume source independent identically contradict rich structure exploit decade besides high source exploit design smoothness sparse temporal source structure learn often sufficient limited notice
difference wavelet replace classical space piecewise affine scale operator play classical wavelet show sec imply affine produce organize adapt expect application radius decomposition open j j kb j decomposition structure j notation dyadic cell generate algebra measurable algebra constant dyadic distribute variation map connect nearest kx ik weight construction partition dyadic tree publication computational cell dyadic regular compact embed manifold set property enough ambient therein h close large appear name manifold recent manifold learning complexity positive local pointwise tool build upon geometric especially harmonic therein estimation short account associate dyadic cell covariance identify prescribe value svd projection onto span column affine parallel pass orthogonal onto plane affine projection onto linear coarse geometric onto hausdorff define fact tangent albeit need go fix decompose
task automatically categorization science biology science engineering classify knowledge categorization complex unsupervised classification name task grouping processing analyze validity hypothesis cluster respective pattern matter recognition image bioinformatics region influence g analysis separate drawback cluster arbitrary division repeat reach good division identify community identification able drawback
posterior sampler sequence scale important evaluate integral I chain ie give suggest particle particle normalize unnormalized collection particle time allow explain hasting seem converge find update walk mix demand complexity calculation update intel approximately hour computation setting unify device parallel smc advantage likelihood particle hardware run around speedup run pick start trivial estimate effective computational smc rapidly density map initial value choose remove possible multimodal independence change effect
treatment model dynamical state dependent version modular decomposition conclude discuss direction connection broader distribute quantify coupling modularity shannon measurement q take entropy outcome logarithm entropy bit uncertainty measure provide random px uncertainty measure interpret knowledge hx yx capture marginal reach mutual case variable random x distinct constraint information often two importantly kl equal amount present formal modularity search
hold imply ia pa r case age show case pyramid duration age duration simple duration disease number depend incidence age
represent local
test condition subset sort size give acyclic undirected graph edge record adjacent cardinality order adjacent less triple adjacent adjacent adjacent direct edge orient complexity pc bound maximal degree vertex test pc algorithm hundred number node large even intractable due author grow shrinkage use phase markov dependent condition two phase sequentially real markov shrinkage condition try identify algorithm direct dependent give set great cycle put neither execute long exist gs markov relatively variable sparse complexity graph gs learn graphical minor modification possible try score enforce sparsity structure edge approach establish principle description refined offer sound criterion network approach intractable bayesian score optimize hard criterion score approach scalable combinatorial far away compare candidate structure bayesian involve compute undirected structure
b sx r yx cs disjoint whose intersect ct sx z z
rv set combination assume state contain signature massive period roughly day outer period rv euler cat rv date combine datum discuss analysis set receive bayesian b consist four signal parameter rv amplitude I date reference describe represent measurement analyse combine receive imply star confidence calculate assess denote receive description good model need combine set lc assume variation bias expand every consist
similarly q obtain integral obvious integral case conclude condition evident variable nd accord hold moreover check hence imply hold apply conclude proof fractional fractional simple immediately pp support modern mathematical shift away handle stochastic process develop dependence stochastic process application economic finance devote involve motion well long study
advantage life dataset systematic datum three computational artificial demonstrate smooth critical analysis deal event indicator event occur censor event end last observation cancer survival hazard cox ph cox ph build survival rate originally intend outcome tend covariate cox exceed observation machine cox one artificial sufficient often lead address chance certain interpretation individual early acceptable practitioner
atom distribute define density zero take hasting metropolis walk detailed sample indicator likelihood account possibility great numerically likelihood q slice sampler use sample discrete gibbs eq gibbs sampling discuss process poisson derive sampling atom observe atom round unobserved atom poisson poisson
confidence make useful large dataset social explanation study identify whether could explain despite test point algebraic specify nf directly problem give x algebraic set onto finite second necessary accord algebraic doubly intractable instrumental algebraic gr equality unfortunately limitation experiment small drawback complexity domain domain equality suffer drawback still insufficient graphical instrumental paper general produce inequality optimize
hasting either sampling call start iterate accept fairly weak stationary chain burn converge stationarity initial portion discard case metropolis hasting metropolis hasting metropolis hasting proposal strongly affect algorithm work fraction candidate reject cover important however prefer approximation proposal often limit walk sensitive target frequently nonetheless well true mcmc often consist highly sample therefore estimator posterior separate chain move rarely take stationarity inaccurate seem independent seed hope chain however run generate correctly chain reach mode depend mode mode concept move easy target sequence distribution multiple isolate together sampling idea independently interest
distributional density pf density choice write pz nz sum lebesgue dirichlet process difficulty employ measure advance fit dirichlet microarray nucleotide reach thousand fairly consume pr methodology enjoy mild misspecification proposal mix mixed resolve difficulty marginal mixing parameter output recursion density produce point labeling integrate distribution correspondence exact filtering additionally identically common minimum kullback leibler range weak limit regularize recursion methodology distribution density give assign typical
sampling calculation denote orthonormal dx lx x since q free exist ix I give analyze side ix second tt moreover evaluate equality last symmetric similarly x I early gauss equation work isometry finish rewrite theorem equivalent together together q translate unit vector relationship holds translate orthonormal basis x x kx ie kx kx express orthonormal come rewrite schmidt invariant since otherwise sum square apply away conclude remark demonstrate manifold focus I theorem imply approximation q identical independent hereafter expect equation deviation bind deviation constant term lx complete away inside boundary similar begin lk may plug note integral vanish apply taylor expansion orthonormal express order term py simplify notation expansion calculation q apply conclusion denominator appear numerator taylor expansion integral domain symmetric define slice basis x x xx reduce q next expansion become kernel numerator similarly denominator expand spectrum laplacian diffusion weight complex low space study graph development mathematical orthogonal ji iw optimally affinity actually digit two discrepancy even optimally perhaps connect edge set small patch e usually
program abstraction abstraction body program abstraction program pair abstraction compress compressed abstraction abstraction abstraction abstraction find abstraction match return call match primitive return define match abstraction unify abstraction body abstraction abstraction abstraction false unify si si abstraction abstraction var var abstraction abstraction abstraction illustrate expression anti create occurrence example replacement program begin define v procedure anti term begin f return argument function expression determine abstraction anti process create abstraction opposite list algorithm size return size expression return false return assignment pass place succeed repeat unified repeat unified pair primitive primitive primitive primitive map si si assignment abstraction assignment neither match assignment expression transformation repeat syntactic generate use incorporation syntactic pattern notion first lambda abstraction lambda calculus abstraction two anti via usually correspond incorporation program anti find possible small color abstraction color look abstraction f f f node f program abstraction f color size color color abstraction correspond even replace color draw multiple
one weight krige meet reliability surrogate reliability span span surrogate reliability reliability determine step size along current accuracy check design obtain converge criterion deterministic performance purpose surrogate refine reliability essence purpose validate algorithm long constant service load material characterize behavior modulus load euler allow involved consist mm deterministic design fashion allow service load satisfie limit reliability minimize area target reliability probabilistic multiplication state surface straightforward turn algebra failure denote random cross numerically refinement limit initialize sequentially accurate reliability depict initial design base satisfied design reach stable exact subsection mm approximation error higher optimal reliability thank krige
average explicitly easy take limit retrieval take temperature dynamic embed thus kind recall process explicitly recall asymmetric let embedded follow solution differential two show amplitude ht evolve ab respectively namely quantum pattern process leave panel find getting collapse
right step mean square steady enough accordingly steady random recursion approximate stability guarantee step steady state determine evaluate depend state size sufficient substitute return evaluate become eq approximate matrix shall notation euclidean kronecker u sufficiently size power eq block furthermore show recursion guarantee size accord become q expression proper weight able steady invertible resort choose proper square estimation computing zero elsewhere vector whose th one oppose explain assumption interested weight hold size enough consistent consider connected simulation sequence spatially consider regularization respectively popular choice non need denote go apply minimize manner
show pac theorem h nh bounding pac fix simultaneously eq tight formula relative subsequence first sequence martingale theorem simultaneously minimize grid geometric almost achieve would hoeffding assume theorem variational bernstein ki simultaneously value random take grid close value side almost weight sampling bind reciprocal round bernstein inequality satisfy inequality compare form inequality average
prior simplify choice summarize formulation provide denote inverse gamma numerical analysis parameter mcmc partially collapse integrate example datum bayesian short upon request estimation method denote frequentist require first resolve sign simulation single error seven quantile mcmc burn diagnosis mix start illustrate burn summarize
matrix equality q obtain stack stacking multiplication operation provide way represent equation wise multiplication generalize multiplication multiplication array r matrix multiplication review ib ib stack j n
augment lagrangian lagrangian converge saddle theorem close proper augment saddle factor marginal return problem q p f p value close solve augment special block furthermore well similarly separable fact enable admm marginal residual residual marginal iteration optimum optimization ise figure show
multiple base start introduce random also goal predictor risk introduction stage good consider separate coordinate align ideal eq underlie center k cx kk alignment kernel center state center center alignment effect alignment see underlie alignment center underlie center c k kk kk maximize label shall combination kernel continuously parametrize base kernel combination alignment target kernel come learn kernel omit yy yy k maximizer ascent additive optimize sequentially add previously direction neighborhood always stop iteration reach happen tb n text datum maximum stepsize k terminate eq material directional derivative maximize obtaining function optimize supplementary material use matlab point option final usual optimizer necessary achieve
let decrease amount establish since inequality prop rearrange map ready rank corollary hadamard submatrix particular thm map negative see may ask extend compression extend exist definite max last equality dimension recall since cover ground riemannian ref ref computation need cholesky suffice kernel many cat cat easy verify result establish rewrite relate quantity helpful tu obtain z briefly mention nonconvex similar obtain input nonnegative problem essentially investigate ignore whereas stationarity neither global strictly
lee way york new york usa department university york new york usa institute studies york york purpose sciences
infinity weak coupling correlation contrast net correlate net surprising experiment superior stand stand lasso correlation elastic net add squared term dedicate prior knowledge trace behave provide seek problem exhibit acknowledgement european technique obtain strictly leave singular vector eigenvalue equal associate closed
numerator denominator since lemma observe confidence sde domain technology gene protein chemical sde learn reaction reconstruct crucial model sde whereby drift many equilibrium special broad drift finite function drift chemical reaction ax
mc mesh topological mesh latter generally gauss complex require increasingly refined mesh interpolation gauss provide mc ensure statistical analysis test topological mix experimental formalism introduce subject specific topological fix subject specific specification eq back evaluate test simpler present mixed growth utilize cost integrate illustrative domain integration mc panel clear global domain integration diameter tend one variability cost testing effect global separately test integration integrate integrate statistic back topological metric impact experimental factor different domain yield systematic bias report factor df df cost influence importantly domain affect result large amount variability characterize large domain report statistic cost find influence separation cost topology cost topological metric illustrate interaction plot cost experimental reporting efficiency metric subset visual reach topology global vary investigate coefficient purpose simple unweighted recommendation topological population summarize main finding fix cutoff strength fix topological difference integrate entire successfully connectivity topology monotonic monotonic association iv topological metric weight cost association comparison analyse ease secondly systematically moreover difference per se informative relevance
detector device desirable deep biological engineering aspect problem observation signal consist least part purpose detector device detector device processor analysis detect signal determine finally arrive eq q detector incoming processor basis detector choose environment problem address processor obviously filter light intensity device biology filter transform incoming detector reality detector light brain come detector device fit biological moreover work bound detector device construction corollary lemma remark phone phone object pixel human stop step view
agnostic rely modeling filter filter transform parsimonious estimating coefficient identification polynomial dimensionality kronecker imply p multiply term multiply permutation retain discard redundant dimension exploit module order outline involve admit representation accomplish via due specification interpretability purpose special employ consist impulse cascade nonlinearity filter set redundant tuple length redundant nonzero location drop second wiener module immediately separable likewise sparse filter even case model high locate case aid sound environment adaptively impulse comprise undesirable cascade short wiener long approximately room usually follow channel represent impulse expansion also bioinformatic filter model ensemble spike record neuron probit selection neuron furthermore genome sparse determine phenotype human trait specie reveal multiplicative gene fact phenotype occurrence disease pose multilinear regression apart
bind together assumption bind exist event event general estimate bind mention every quantity role dimension justify hence conditionally inequality event bind let event trivially cube inequality convenience brevity subsample imply take unconditional true combine discuss sampling subroutine eq view run use notation resolution dyadic estimator always finite dyadic cube enough resolution find piecewise sort
part action take iteratively objective iterative nature well htp important uncertainty quantity bias bias gps straightforward mathematical incorporate aspect characteristic date specifically original discussion gp little relevance uncertainty g determine action objective problem aspect objective already subsection infinitely candidate action infinitely high carlo scheme problem domain computationally likewise increase seem restrictive spirit anneal nonconvex priori due costly method rely abundance arise give satisfactory answer multivariate compact admit random guarantee probability tight important note sampling nonzero measurement gp determinant newly point entropy multivariate entropy point maximized formulated computed matrix scalar summarize
n log markov give distribution innovation assumption asymptotic parameter stationary assumption c far autocorrelation dynamic innovation mention satisfie innovation process matrix symmetric eq could tractable seek n expectation recurrence ahead q one ahead q simply one additional ahead expression get parameter terminology cause cause gaussian var interpretation diagonal I cause cause triangle cause cause conversely causality innovation constrain innovation margin q causality constrain independence triangular causality denote conditional constrain model innovation
I n resample move metropolis hasting provide good instrumental parametric long easy histogram
contain elastic fista without outer augment lagrangian albeit magnitude truncate sparse range yield accuracy effectively cc fista solve linear avoid solve complement instead definite yield background separation marginally elastic net yield separation order fista work fista typical run fista take outer hand take iteration meet stop criterion percent elastic fista interestingly majority inner iteration test synthetic breast cancer mean behave like number inner step require second p fast linearization counterpart fista suggest reason fista whose gradient soft thresholding operation system show return varied setting tune contrast fairly rarely general well fista lasso without augment lagrangian fista take fista p solve believe evidence balancing demonstrate problem build unify
hilbert reproduce hilbert hilbert always functional natural space first thank space vector space understand secondly input many application orient usually provide systematic inclusion rkh concrete appear machine recent study embed rkhs requirement show rkhs small demand rkh embed rkh requirement outline inclusion devote investigation translation invariant schmidt reproduce relation translation relation require set refer input space reproduce hermitian
positive phenotype share target disease phenotype solution identify use phenotype cluster gene association validate finding throughput validate complex gene association help closely gene similarity disease phenotype phenotype future tool infer gene go weight straightforward acknowledgment support seed grant institute university tc cm cm zhang department engineering usa college correspondence gene identify throughput association gene phenotype fail reveal association phenotype exist annotation incomplete discover association set phenotype association rank gene disease phenotype rank phenotype formulate coherence disease gene association efficient coupling ridge propagation evaluate baseline leave task predict discover association experiment demonstrate ranking baseline apply identify disease disease recent dna rank candidate top rank list across author summary determination molecular
vector mn thus direction form j np nevertheless alternate maximization easy maximize fixing implement candidate candidate mention pair large reader correct addition section perform replacement attempt procedure previously utilize procedure use iteratively direction perform plot error rank small test medium start increase beyond avoid trace start overfitte
connect edge add connect vertex black white picture white lattice special lost sense overcome obstacle critical case satisfied get picture white black within predict formation shape relation explain formally split vertex consists belong vertex set call cluster go mention high form relatively black cluster region mathematical
finally multiple show multiple individual six doubly family table notation operational advanced measurement capital estimation become accept distributional lda process ii business line paper derive class doubly ability typical also compound aggregated compound particular develop scenario loss compound poisson arrival increment year loss increment dependent increment capture binomial success change couple comprise per derive analytic operational distributional doubly poisson process ii statistic email service pt operational take prominent occur ii requirement operational risk complex financial product information technology combine
mcmc technique markov diffusion numerical normality markov mcmc become tool ergodic great markov quantitative bound work fairly property satisfy bad iteration mcmc procedure category step early number satisfy performance procedure poor da mixture know performance da procedure
probability subsample dr building reward predict ridge estimation dr error rmse dr dr reduce rmse std confirm dr tend reward doubly always widely doubly give dr improve algorithm develop offset take acknowledgement doubly sensitive multiclass error specify vector policy give adjust aa work policy randomly perturb return learn try
square rsc basically define norm level increase particularly agreement furthermore dedicated discussion matrix complex otherwise brevity depend denote support zero vector denote hermitian matrix project gradient choose kk operator minimization q iteration outline broad sensing except solve impose whose great
volume volume fig analysis confirm web traffic informative trading reverse beyond causality power engine volume test check various hypothesis predict volume trading trading predict trading trading predict query volume trading query volume model hypothesis bootstrap sample bootstrap empirical p sort significance reject suggest reject find support third column cross trading direction compute clean large trading volume volume report small much direct value model regression square compare run clean significance outcome support four trading volume trading volume predict trading trading volume predict query volume volume use query consider compute model residual compute aim assess independent tend nan sample opposite bootstrap clean result testing search engine answer question user even user ask regular basis group user drive science statistical insight social human traffic request www track successful include car
open exact rate game separation er call build game remove split game action action game must armed bandit regret action change subtract column game subtract second one bandit armed bandit action split put half second recall dominate action accord loss split game recursive splitting binary strategy play internal outcome action conversely action optimal boundary share game matter restrict outcome hope know outcome reveal step idea action game stay large previously action start play happen switch fail boundary outcome optimal action action half game enough recover play avoid repeat problematic happen hence minimal generalize mention thus
book explanation black size great lying marked pixel black denote white black useful lattice orientation coordinate inequality limit finite lattice side symmetry arbitrary subset triangular lattice affect event measurable event decrease event ci ci e nn c moreover following let matching definition black rectangle vertex side event white path side black
almost mean unlike marginal beta formal beyond build model measure accord bernoulli place ibp drawing atom beta turn finally observation stick beta process stick length stick length segment break nonparametric mixture call chinese restaurant dirichlet factor latent specify advance allow flexible sequential group relational datum organize detail counterpart mixture latent limitation extension hope flexible study available software analysis grouping identity understand estimate population cognitive contribute produce behavioral generative g cognitive might rise worth fundamentally analysis behavioral attempt explain cause response recover distribution cause finite model observation generate rt rt mixture accommodate mixture condition
rl r dl anchor direction decide threshold nonzero along linear software http edu tw correlation basic continuous experimental study performance page size window kernel estimate equal sample replicate simulation bivariate normal distribution equal let sample variable diagnostic roc degree line cut hypothesis five statistic simulation versus increase true coefficient increase current multi variate deviations standard bootstrap deviation show standard table gold three cauchy three association correlation estimate roc index
issue stochastic bandit bandit usage introduce difficulty sample play weight handle hoeffding variance control variance action improve improvement bernstein evolve way analysis far improvement detail work emphasize although bandit
strength convexity highly dimensionality dimension subspace covariate dimensional gp c prior widely matlab package take run tolerance variance burn model extremely insensitive hyperparameter hyperplane set test dimension important likewise hyperparameter test little relationship proposal hyperparameter poor constraint region subspace general outperform outcome traditional problem range decision convex compute surface create maximizer result function unconstrained function solver noisy square place rather produce estimate posterior demonstrate objective produce stochastic randomly piecewise smooth contour
hence currently investigate employ transform arise challenge general nonparametric noise amount functional set estimator satisfy condition maximum quantile cover selector vast area arise situation neighboring signal gain average neighboring residual subproblem square orthogonal feasible hand demanding application illustrate estimation nonparametric regression image applicability algorithmic novel inherent behaviour consistency inequality extent area attempt value error gaussian discrete observation tend indicate cardinality get likely degenerate unless properly normalize utilize guarantee boundedness bound true regularity eq condition interest expect purpose future imaging application rather algorithmic framework select partition location concentration issue deconvolution step great explore f support support support department institute provide grateful associate anonymous helpful
