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Equation 2.7 remain un- changed under multiplication of ma→bby a constant Ca→bfor alla,b∈A such thatb≤a. Therefore, BP is an algorithm that preserves the equivalenc e classes {C·m}, i.e., it is defined by the relation m∼m′whenever there is a collection of scalars ( Ca→b∝\e}atio\slash= 0;a,b∈A,b≤a) such that ma→b=Ca→bm′ ... | https://arxiv.org/abs/2503.15705v1 |
a(xa) for eacha∈A; denoteya∈/tildewideFathe identification of dhFEa∈F∗ ainFa; then the reformulation of Equation 2.14 is the following, ∀xa∈Fa,ya(xa) =Ha(xa)+lnha(xa)+1 The previous equation is equivalent to ∀xa∈Fa,ha(xa) =e−Ha(xa)+ya(xa)−1 Therefore, we define ∀a∈A,∀xa∈Ea, g+ Ha,a(l)(xa) =e−Ha(xa)+la(xa)−1. (2.15) Thefu... | https://arxiv.org/abs/2503.15705v1 |
a,b:b≤aφ∗ bandφ† 1=/circleplustext a,b:b≤aφ† b. Theorem 1. LetF,Gbe two presheaves from a finite poset Ataking values in finite sets FinSet; letφ:F→Gbe a natural transformation. Then, for any collection of Hamiltonians (Ha:Fa→R;a∈A), and any choice of scalar product on /tildewideFaand/tildewideGawitha∈A, ∆MPG,/tildewideH... | https://arxiv.org/abs/2503.15705v1 |
in turn equips F→with a scalar product. The map ∆MP(F) :F→×/producttext a∈ARFa→F→is defined as, ∀l∈F→,H∈/productdisplay a∈ARFa,∆MP(F)(l,H) = ∆MPH(l) To a natural transformation φ:F→Gbetween two presheaves of ˆAf, ∆MP(φ) = (φ,ψ)is a morphism between ∆MP(F)and∆MP(G)in MPA, where ∀a∈A, ya∈Ga, ψ(H)a(ya) =−ln/summationdispla... | https://arxiv.org/abs/2503.15705v1 |
r supporting this research. References [1] Federico Barbero, Cristian Bodnar, Haitz S´ aez de Oc´ ar iz Borde, Michael Bronstein, Petar Veliˇ ckovi´ c, and Pietro Li` o. Sh eaf neural net- works with connection laplacians. In Topological, Algebraic and Geo- metric Learning Workshops 2022 , pages 28–36. PMLR, 2022. [2] ... | https://arxiv.org/abs/2503.15705v1 |
Juan Pablo Vigneaux. Information structures and their cohomology. Theory and Applications of Categories , 35(38):1476–1529, 2020. [28] Martin J Wainwright, Michael I Jordan, et al. Graphical models, expo- nential families, and variational inference. Foundations and Trends ® in Machine Learning , 1(1–2):1–305, 2008. [29... | https://arxiv.org/abs/2503.15705v1 |
denoted lim F, is the set of collections of elements x= (xa∈ Fa|a∈A) such that they are pairwise compatible: for any a,b∈Awith b≤a,Fa b(xa) =xb. The application of presheaves and sheaves in data science ha s gained more importance in recent years; they were introduced in the context of decentralized optimization, for d... | https://arxiv.org/abs/2503.15705v1 |
µF∗dvFE=dF∗(l). (C.2) Proof.One shows that for any v∈/circleplustext a∈A/tildewideFa, dvFBethe/vextendsingle/vextendsingle lim/tildewideF=/bracketleftbig µF∗dvFE/bracketrightbig/vextendsingle/vextendsingle lim/tildewideF, where FEa(ha) =/summationdisplay xaha(xa)Ha(xa)+/summationdisplay xaha(xa)lnha(xa), forh∈REa. Its ... | https://arxiv.org/abs/2503.15705v1 |
lnma→b− ln/summationtext xbelnma→b. Then,Na→b(lnma→b+C) =Na→b(lnma→b). Let us de- noteN(lnm) = (Na→b;a→b). Consider [m]∗a fixed point of BP. Then, for any representative mrsuch that [mr] = [m]∗, we have ∆MP( N(lnmr)) = 0; therefore, N(lnmr) is a fixed point of MP. E Proof of main theorems E.1 Proof of Theorem 1 Now it is... | https://arxiv.org/abs/2503.15705v1 |
arXiv:2503.15723v1 [math.ST] 19 Mar 2025THE FUNDAMENTAL LIMITS OF RECOVERING PLANTED SUBGRAPHS DANIELZ.LEE˝,FRANCISCOPERNICE˝,AMITRAJARAMAN˝,AND ILIASZADIK: A/b.sc/s.sc/t.sc/r.sc/a.sc/c.sc/t.sc. Given anarbitrarysubgraph /u1D43B“/u1D43B/u1D45Band/u1D45D“/u1D45D/u1D45BP p0,1q,theplantedsubgraphmodel isdefinedasfollows. A... | https://arxiv.org/abs/2503.15723v1 |
p0,1q. We assume that the statisticianobservestheunionofa“signal,”whichisa /u1D458-cliquechosenuniformlyatrandomfrom the complete graph /u1D43E/u1D45B, and the “noise,” which is an instance of an Erd ˝os–R´enyi graph/u1D43Ap/u1D45B,/u1D45Dq. /T_he goal of the statistician is to recover the planted clique from the obser... | https://arxiv.org/abs/2503.15723v1 |
of work. A series of conjectures by Kahn and Kalai, later refined by Talagrand, provid e simple formulas that are conjecturedto yieldthe criticalthreshold /u1D45D/u1D450upto amultiplicative /u1D442plog|/u1D43B|qfactor. Inparticular,aconjectureofKahnandKalai,whichremainsunproven, assertsthatthecritical threshold for any ... | https://arxiv.org/abs/2503.15723v1 |
SUBGRAPHS 5 /T_heorem1.1 (Informalversionof Theorem3.1 andTheorem3.6 (c)).Let/u1D700ą0, and/u1D43B“/u1D43B/u1D45Ban arbitrary weakly dense graph. /T_hen, for sufficiently large /u1D45B, there exists an integer 1ď/u1D440ď |/u1D43B| andthresholds 1“/u1D45E/u1D440ą ¨ ¨ ¨ ą/u1D45E1ą/u1D45E0“0,suchthatthefollowingholdsforthep... | https://arxiv.org/abs/2503.15723v1 |
definetheweak recoverythresholdin our context. Definition 1.2. For any/u1D43B“/u1D43B/u1D45Band/u1D700P p0,1qwe define the /u1D700-recovery recovery threshold /u1D45D/u1D700“/u1D45D/u1D700p/u1D43Bqas /u1D45D/u1D700“/u1D45D/u1D700p/u1D43B/u1D45Bq:“supt/u1D45DP r0,1s: MMSE/u1D45Bp/u1D45Dq ď1´/u1D700u. /T_he planted subgraph... | https://arxiv.org/abs/2503.15723v1 |
aknown cardinality. Inanotherrelatedlineofwork, thegoalistoprovepropertiesof t helimitingMMSEcurvefor specificplantedmodels. Notable examples include [ RXZ20] that understood the limiting MMSE 8 D.Z. LEE,F. PERNICE,A.RAJARAMAN,I.ZADIK curveinsparselinearregressionandprovedtheAoNphenomenon,and[ DWXY23 ,GSXY25]that studie... | https://arxiv.org/abs/2503.15723v1 |
in fact crucial in our work in order to obtain our generalconnectionbetweenminimaxratesandvariantsof thefractionale xpectationthresholds inSection 7. 3.M/a.sc/i.sc/n.scR/e.sc/s.sc/u.sc/l.sc/t.sc/s.sc I:P/l.sc/a.sc/n.sc/t.sc/i.sc/n.sc/g.sc /a.scW/e.sc/a.sc/k.sc/l.sc/y.scD/e.sc/n.sc/s.sc/e.sc /u1D43B 3.1.Characterizingth... | https://arxiv.org/abs/2503.15723v1 |
proof of theseparts are basedon a novelminimaxduality principlet hat couldbe of inde- pendentinterest. We defertheproof to Section 5. Remark3.7. (Onthecomputabilityof /u1D711/u1D45E.) Recallthat /u1D711/u1D45Eisdefinedusingtheminimaxoptimiza- tionproblemwhereboththeminimumandthemaximumoperationisdefined overexponentially... | https://arxiv.org/abs/2503.15723v1 |
that AoN happens for the planted subgraph model for some /u1D43B“/u1D43B/u1D45Bif the MMSE /u1D45Bp/u1D45Dqcurve convergesto a step function from 0 to 1 as /u1D45Bgrows to infinity. An immediate consequence of Theorem3.1 is that the AoN phenomenonhappensfor aweakly densegraphif andonly if for every /u1D45E,/u1D45E1P p0,... | https://arxiv.org/abs/2503.15723v1 |
all 0 ď/u1D461ď/u1D45A´1 and/u1D45D/u1D45A“0,/u1D45D´1“0. Hence,we havethatfor any /u1D6FFą0andfor each ´1ď/u1D461ď/u1D45A´1,if p1`/u1D6FFq/u1D452´2{/u1D436/u1D461`2ď/u1D45Dď p1´/u1D6FFq/u1D452´2{/u1D436/u1D461`1, then itholds MMSE/u1D45Bp/u1D45Dq “1´ř /u1D460ď/u1D461`1/u1D4362 /u1D460ř /u1D460ď/u1D45A/u1D4362 /u1D460`... | https://arxiv.org/abs/2503.15723v1 |
frac- tional expectation threshold [ Tal10] of the monotone property with minimal elements /u1D49C, see Definition 2.4 . Inparticular,thevalueofthefractionalexpectationthresho ldhasbeencalculated THE FUNDAMENTAL LIMITS OF RECOVERING PLANTED SUBGRAPHS 17 up to constants for a number of monotone properties, including the ... | https://arxiv.org/abs/2503.15723v1 |
/o.sc/f.sc T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc3.6 In this section, we establish some key results on the onion decompos ition that will be useful in multiple proofsthroughout thepaperandalsoprove Theorem3.6 . We start with recalling the /u1D70Cnotation from Section 2. Notice that our key quantity p/u1D711/u1D45Eqcan be defin... | https://arxiv.org/abs/2503.15723v1 |
/u1D456/u1D457P/u1D43A|/u1D446/u1D465/u1D456/u1D457“/u1D70Cp/u1D43Dq. Moreover,ř /u1D456P/u1D449p/u1D43Aqz/u1D449p/u1D446q/u1D466/u1D456“1, andthe other two constraints aretriviallysatisfied. Next, given a solution to the LP with value /u1D463˚,we show how to extract a choice of /u1D43Dsuch that/u1D70Cp/u1D43Dq ě/u1D463... | https://arxiv.org/abs/2503.15723v1 |
´/u1D45E. However, in many cases this bound will not be tight. To improve it, using th e intuition from the tightness of the expectation thresholds from [ KK07], one can fix a subgraph /u1D43DĎ/u1D43Band instead upper bound for any /u1D45Dthe probability that a copy /u1D43B1of/u1D43Bin/u1D43Aoverlaps/u1D43B˚on/u1D446by ... | https://arxiv.org/abs/2503.15723v1 |
and of optimization), no such result appea rsapplicable in our discrete THE FUNDAMENTAL LIMITS OF RECOVERING PLANTED SUBGRAPHS 25 vertex-symmetric se/t_ting. Yet, a careful combinatorial argument al lows us to indeed exchange theoperators. /T_hisminimaxresultisthecontentofthenextlemma,pro vedinSection6.6 ,andis provedb... | https://arxiv.org/abs/2503.15723v1 |
asubgraphof /u1D43Dwithℓedges. /T_herefore, MMSE/u1D45Bp/u1D45Dq ´ p1´/u1D45E`/u1D6FFq ď/u1D463p/u1D43Bq/u1D442p/u1D463p/u1D43Bqqÿ ℓďp/u1D45E´/u1D6FFq/u1D458¨/u1D45B´/u1D463ℓp/u1D43Dq¨/u1D440/u1D43D/u1D45D/u1D43F´ℓ Nowobserving that /u1D440/u1D43D“ p/u1D45Bq/u1D463p/u1D43Dq{|Autp/u1D43Bq| ď/u1D45B/u1D463p/u1D43Dq, we g... | https://arxiv.org/abs/2503.15723v1 |
( LemmaA.1 )implythe result. Inthecalculationbelow,wealwaysconditionon /u1D446˚.Forconvenience,weintroducesomenew notation. We denote by p/u1D446˚q/u1D450-graph, any graph on edges only in /u1D43E/u1D45Bz/u1D446˚.We define/u1D43A1to be an independentsamplefromtheproductBernoulli p/u1D45Dqmeasureon/u1D43E/u1D45Bz/u1D446˚... | https://arxiv.org/abs/2503.15723v1 |
/u1D43Dp/u1D456q,0ď/u1D456ď/u1D461`1, it holds /u1D70Cp/u1D43Dp/u1D461`1q|/u1D446q “|/u1D43Dp/u1D461`1qz/u1D446| |/u1D449p/u1D43Dp/u1D461`1qq| ´ |/u1D449p/u1D446q| “ř/u1D461 /u1D456“0|p/u1D43Dp/u1D456`1qz/u1D43Dp/u1D456qq X p/u1D43Dp/u1D461`1qz/u1D446q| ř/u1D461 /u1D456“0|/u1D449p/u1D43Dp/u1D456`1qq| ´ |/u1D449p/u1D43D... | https://arxiv.org/abs/2503.15723v1 |
always supported on sets containing /u1D446. Combining these,itholds for all /u1D44CĎ/u1D4B3that ℙ/u1D45Dp/u1D44Cz/u1D446|/u1D434q “/u1D45D|/u1D44C|´/u1D4581p/u1D434Ď/u1D44Cq “/u1D45D|/u1D446|´/u1D4581p/u1D434Ď/u1D44Cqℚ/u1D45Dp/u1D44Cz/u1D446q, andthereforethe likelihoodratio between ℙ/u1D45Dandℚ/u1D45Dtake thefollowin... | https://arxiv.org/abs/2503.15723v1 |
onion decomposition of any subgraph /u1D43Bthe sequence /u1D70Cp/u1D43Dp/u1D461`1q|/u1D43Dp/u1D461qq,/u1D461“0,1,...,/u1D440 ´1isnon-increasing. Proof.For all/u1D436Ě/u1D435,we have /u1D70Cp/u1D435|/u1D434q “|/u1D435| ´ |/u1D434| |/u1D449p/u1D435q| ´ |/u1D449p/u1D434q|ě/u1D70Cp/u1D436|/u1D434q “|/u1D436| ´ |/u1D434| |/... | https://arxiv.org/abs/2503.15723v1 |
Mmse of probabilistic low-rank matrix es- timation: Universality with respect to the output channel. In 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton) , pages680–687.IEEE, 2015. [LM17] Marc Lelarge and L ´eo Miolane. Fundamental limits of symmetric low-rank matrix est imation. ... | https://arxiv.org/abs/2503.15723v1 |
arXiv:2503.15894v2 [math.PR] 21 Mar 2025March 24, 2025 THE GAUSSIAN CENTRAL LIMIT THEOREM FOR A STATIONARY TIME SERIES WITH INFINITE VARIANCE MUNEYA MATSUI, THOMAS MIKOSCH M. Matsui Department of Business Administration, Nanzan University 18 Yamazato-cho Showa-ku Nagoya, 466-8673, Japan T. Mikosch Department of Mathema... | https://arxiv.org/abs/2503.15894v2 |
infinity, and P(|X|> x) =o(x−2K(x)) asx→ ∞. Hence we can choose an=√nℓ(n) :=/radicalbig nK(an) where ℓ(x)→ ∞asx→ ∞is a slowly varying function. Moreover, the central limit th eorem with normal limitξ2remains valid with the unconventional normalizing constan ts (an). As a matter of fact, the necessary and sufficient condit... | https://arxiv.org/abs/2503.15894v2 |
[Peligrad and Sang (2013), Peligrad et al. (2022)]. [Philip ps and Solo (1992)] provided some gen- eral techniques for dealing with asymptotic theory for line ar processes. In our paper we follow a different path of proof. Assumingmixin g conditions, we split the sample X1,...,Xnintokn→ ∞asymptotically independent block... | https://arxiv.org/abs/2503.15894v2 |
the spectral t ail process (Θ t) for anm-dependent process (Xt). This is due to a large deviation result proved in [Mikosch a nd Wintenberger (2016)]. It depends on the crucial condition that P(Θ−m=···= Θ−1= 0)>0. The derivation of the backward spectral tail process is, in general, a hard proble m but it is simple for ... | https://arxiv.org/abs/2503.15894v2 |
to this condition. In the m-dependent case one can choose rn→ ∞ arbitrarily slowly. If ( Xt) is strongly mixing with mixing coefficients ( αh)h≥0then (2.1) is satisfied if an anti-clustering condition holds ((9.1.7) in [Mikosch and Wintenberger (2024)]) and one can find sequences ( rn) and (ℓn) such that ℓn→ ∞,ℓn/rn→0 andk... | https://arxiv.org/abs/2503.15894v2 |
generic element Zis symmetric. (2) Large deviations of the type (2.5) can be found in [Mikosc h and Wintenberger (2013), Mikosch and Wintenberger (2014), Mikosch and Wintenberger (2016)] for general regularly varying stationary sequences, in [Buraczewski et al. (2013 ), Buraczewski et al. (2016)] and [Konstantinides an... | https://arxiv.org/abs/2503.15894v2 |
have the following large deviation resul t from [Mikosch and Wintenberger (2016)]: P(|Srn|>√y)∼σ2rnP(X2> y) uniformly in the y-region [r1+δ n,∞), asn→ ∞, where we choose r1+δ n=o(a2 n) andσ2is defined in (3.1). Then we obtain I2∼σ2n a2n/integraldisplay(εan)2 r1+δ nP(X2>y)dy, n → ∞. By Karamata’s integral theorem (see [B... | https://arxiv.org/abs/2503.15894v2 |
small, some constantc>0, E/bracketleftbig E[|Sn|2−δ/vextendsingle/vextendsingle(σt)/bracketrightbig/bracketrightbig ≤cE[|Z|2−δ]E/bracketleftBign/summationdisplay t=1σ2−δ t/bracketrightBig ≤cn. Then we can apply Lemma 2.4, 2.since (sn) can be chosen arbitrarily large. 3.2.Linear process. In this section we will provide ... | https://arxiv.org/abs/2503.15894v2 |
m are asymptotically equivalent we can choosea(m) n=aZ n/bardblψ/bardbl2both formfinite and infinite (we do not indicate min/bardblψ/bardbl2). In particular, by the first part of the proof we have for m<∞, (a(m) n)−1Snd→N/parenleftBig 0,/parenleftBigm/summationdisplay j=0ψj/parenrightBig2 //bardblψ/bardbl2 2/parenrightBig... | https://arxiv.org/abs/2503.15894v2 |
negligible, we condition on (ηt) and apply ˇCebyshev’s inequality: I2≤cnE[(Z/aZ n)21(|Z| ≤aZ n)]E/bracketleftbig (σ0−σ(m) 0)2/bracketrightbig . The first factor is bounded by definition of ( aZ n) whileE/bracketleftbig (σ0−σ(m) 0)2/bracketrightbig →0 asm→ ∞. Thus we have proved the following result. Proposition 3.6. Cons... | https://arxiv.org/abs/2503.15894v2 |
moments of AandB as well as suitable mixing conditions. The additional momen t conditions are needed in the large deviation results of Lemma 3.13 below. Proposition 3.9. Assume the aforementioned conditions on the stationary sol ution(Xt)to the affine stochastic recurrence equation (3.11). In addition, we require E[Aγ+|B... | https://arxiv.org/abs/2503.15894v2 |
in (3.12). We start by verifying (2.2) in Theorem 2.2. We choose ( rn) such that a2 n=c∞nlogn > rn(logrn)2Mfor someM >2. This is always possible since rn/n=o(1). By (3.13) we have knP/parenleftbig |Srn−drn|>an/parenrightbig ≤cnP(|X|>an)→0, n→ ∞. (3.15) Next we verify (2.4). Since E[Srn−drn] = 0 it suffices to show that t... | https://arxiv.org/abs/2503.15894v2 |
stochastic recurrence equation (3.11)and the Kesten-Goldie conditions for the unique solution to the stochastic recurrence equation Ct=AtCt−1+1,t∈Z: there exists β >2such that E[Aβ] = 1,E[AβlogA]< ∞, and the law of logAconditioned on {A >0}is non-arithmetic. Then the following large deviation relation holds for every δ... | https://arxiv.org/abs/2503.15894v2 |
the condition εan> r(1+δ)/2 nis trivially satisfied and we can apply (3.18): knP/parenleftbig |Srn−drn|>an/parenrightbig ≤cnP(X >an)→0, n→ ∞. (3.21) The convergence to zero follows from the definition of ( an) in the case α= 2; see the comment after (1.4). Thus (2.2) holds. Next we prove (2.4). Since E[Srn−drn] = 0 it su... | https://arxiv.org/abs/2503.15894v2 |
to zand comparing the coefficients of trigonometric functions. In particular, 0 <C2+D2= (a2+b2)/(1+θ2 0/4)≤1. The function N(x) is not slowly varying. Next we verify the slow variation of K(x). Forx>rwe have K(x) = 2E[X21(0≤X≤x)] = 2/parenleftBig/integraldisplayx2 r2P(X >√y)dy−x2P(X >x)/parenrightBig =cr/integraldisplayx... | https://arxiv.org/abs/2503.15894v2 |
131, 151–171. [Kesten (1973)] Kesten, H. (1973) Random difference equations and renewal theory for products of random matri- ces.Acta Math. 131, 207–248. [Kulik and Soulier (2020)] Kulik, R. and Soulier, P. (2020)Heavy-Tailed Time Series. Springer, New York. 22 MATSUI, MIKOSCH [Konstantinides and Mikosch (2005)] Konstan... | https://arxiv.org/abs/2503.15894v2 |
arXiv:2503.15922v2 [math.ST] 31 Mar 2025General reproducing properties in RKHS with application to derivative and integral operators Fatima-Zahrae El-Boukkouri * Josselin Garnier† Olivier Roustant‡ April 1, 2025 Abstract Dans cet article, nous considérons la propriété reproduisa nte dans les espaces de Hilbert à noyaux... | https://arxiv.org/abs/2503.15922v2 |
revisit the reproducing properties in RKHS for derivativ e and integral operators. Firstly, we focus on the derivative operator, and show that t he reproducing property holds if the cross derivative of the kernel exists and is cont inuous on the diagonal of X×X. Thereby we retrieve the result presented in [ 3,4] by a d... | https://arxiv.org/abs/2503.15922v2 |
x−1/integraltextx 0f(t)dt(with L(f)(0) = f(0)) is written as lim n→+∞Lnwhere Ln=∑n i=11 nTvi,n with vi,n:x/ma√sto→i nx. Here/tildewiderLn(x) =∑n i=11 nK(vi,n(x), .). We now recall the Loève criterion for convergence of sequenc es in Hilbert spaces. Proposition 1 (Loève criterion) .LetHbe a Hilbert space with inner prod... | https://arxiv.org/abs/2503.15922v2 |
Finally, by taking the limit in ( 5), we get: /bardbl/tildewideL(x)/bardbl2=LℓLrK(x,x). Indeed, let F=LrKand x,y∈X. By taking the limit in ( 6), we get F(.,y) =/tildewideL(y)∈ H . Then, taking the limit in ( 7), we obtain LℓF(x,y) =/a\}bracketle{tF(.,y),/tildewideL(x)/a\}bracketri}ht, or equivalently, /a\}bracketle{t/t... | https://arxiv.org/abs/2503.15922v2 |
by Lemma 1, ∂K ∂x1(x, .) =/tildewideL(x). As the limit of /tildewiderLn(x)does not depend on the sequence un, we can conclude that K(x+t,.)−K(x,.) tconverges in Hwhen ttends to 0 to∂K ∂x1(x, .). To conclude the proof of (a), let us show that /bardbl∂K ∂x1(x, .)/bardbl2=∂2K ∂x1∂x2(x,x). From Theorem 1, for all f∈ H ,Lnf... | https://arxiv.org/abs/2503.15922v2 |
f′′has no limit at 0, due to the term sin(1 x), which means that f is not C2. Now, consider the rank-one kernel K: K:R×R→R,K(x,y) = f(x)f(y). K is not of class C2since for all y ∈R, the function x /ma√sto→∂2K ∂x2 1(x,y) = f′′(x)f(y)is not continuous at x =0. Nevertheless, the cross derivative∂2K ∂x1∂x2(x,y) = f′(x)f′(y... | https://arxiv.org/abs/2503.15922v2 |
that Xcan be unbounded and Kcan be unbounded. Theorem 3. LetHbe a RKHS of real-valued functions defined on X, with reproducing kernel K. Assume that K and p are continuous almost everywhere and lo cally bounded on X×Xand Xrespectively, i.e. K (resp. p) is bounded on all bounded subs ets of X×X(resp. X). Assume that the ... | https://arxiv.org/abs/2503.15922v2 |
. Let s,t∈N, as x,y/ma√sto→ | K(x,y)|p(x)p(y)is integrable onX×Xand using Proposition 2with f(x,y) = K(x,y)p(x)p(y)1X×X\Es×Et(x,y) and g(x,y) =|K(x,y)|p(x)p(y)for all x,y∈X, then/integraltext X×X\Es×EtK(x,y)p(x)p(y)dxdy and/integraltext X×X\Es×Et|K(x,y)|p(x)p(y)dxdy exist. Let us denote I(s,t) =/integraltext Es×EtK(x,y... | https://arxiv.org/abs/2503.15922v2 |
proposition. Note, however, that the (almost sure) co ntinuity of the kernel is also required in Theorem 3, which is a mild assumption. Finally, the following example exhibits a kernel Kfor which/integraltext X/radicalbig K(x,x)p(x)dxis not finite but such that/integraltext/integraltext X×X|K(x,y)|p(x)p(y)dx dy is finite... | https://arxiv.org/abs/2503.15922v2 |
separation and convergence with kernel discrepancies,” Journal of Machine Learning Research , vol. 25, no. 378, pp. 1–50, 2024. [9] C. J. Oates, “Minimum kernel discrepancy estimators,” i nInternational Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Co mputing , pp. 133–161, Springer, 2022. [10] K... | https://arxiv.org/abs/2503.15922v2 |
Sequential Monte Carlo with Gaussian Mixture Approximation for Infinite-Dimensional Statistical Inverse Problems Haoyu Lua, Junxiong Jiaa,∗, Deyu Menga aSchool of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China Abstract By formulating the inverse problem of partial di fferential equa... | https://arxiv.org/abs/2503.16028v2 |
the means to navigate and sample from these intricate posterior distributions. In the infinite- dimensional setting, the preconditioned Crank-Nicolson (pCN) and pCN Langevin transition kernel can be derived from the discretization of the Langevin equation [6, 11, 23]. However, the Bayesian inference for computationally... | https://arxiv.org/abs/2503.16028v2 |
a commonly used method for approximation. Gaussian mixture approximations have been applied in SMC filtering. In [35], the authors con- structed the prior samples into a Dirac measure, regarded it as a Gaussian mixture distribution with zero variance, and then linearized the model to update the parameters of the Gaussi... | https://arxiv.org/abs/2503.16028v2 |
convergence theorem for SMC under weaker conditions. Second, we combine the Gaussian mixture with pCN to introduce a new transition kernel, known as the pCN-GM algorithm, which, when utilized as the transition kernel in SMC, forms the SMC-pCN-GM method. Lastly, we present an approximation to SMC- pCN-GM, the SMC-GM alg... | https://arxiv.org/abs/2503.16028v2 |
a positive sequence satisfyingPJ j=1hj=1. Now we have a measure sequences {µj}J j=0, and the importance sampling can be used to sample from µj+1based on samples from µj. The entire SMC algorithm consists of three iterative steps, the re-weighting step, the re- sampling step and the mutation step (See Algorithm 1). Firs... | https://arxiv.org/abs/2503.16028v2 |
transition kernel, we summarize them as follows: •SMC-pCN: The SMC method utilizing the pCN method as the transition kernel. •SMC-pCN-GM: The SMC method utilizing the pCN-GM method as the transition kernel. Here pCN-GM method is an improved version of pCN detailed in Section 2.3. •SMC-GM: The SMC method that replaces t... | https://arxiv.org/abs/2503.16028v2 |
the measure of the next layer in SMC can be expressed in terms of the measure of the previous layer as follows: µN j+1=Lj+1SNQjµN j,0≤j≤J−1, µN 0=µ0. (2.11) To measure the error between µN Jdetermined by (2.11) and the posterior in (2.9), we need to consider the error introduced by these operators, respectively. To det... | https://arxiv.org/abs/2503.16028v2 |
to select the proposal kernel Q(u,dv) and demonstrate the equivalence between µd(du)Q(u,dv) and µd(dv)Q(v,du). It should be noted that only carefully designed Metropolis-Hastings methods have interpreta- tions in infinite dimensions. In fact, most Metropolis-Hastings methods defined in finite dimen- sions will not make... | https://arxiv.org/abs/2503.16028v2 |
of their components are equivalent. Furthermore, µd(du)P(u,dv) andµd(dv)P(v,du) are equivalent. Remark 2.6. The equivalence between the two measures in this theorem is crucial, directly affecting whether the algorithm possesses dimension independence. A simple example is the comparison between the pCN and the random wa... | https://arxiv.org/abs/2503.16028v2 |
[13], each utilizing information of Φ,DΦ, and D2Φrespectively. We need to solve one, two, and four PDEs, respectively, to compute them with the adjoint method [21], which is computationally intensive. To utilize likelihood information while avoiding the computation of the likelihood function, we use a Gaussian mixture ... | https://arxiv.org/abs/2503.16028v2 |
Now Q(u,dv) is independent with respect to u, thus we can drop uand rewrite it as Q(dv). The accept rate function is a(u,v)=min1,e−Φ(v)µ0(dv)Q(du) e−Φ(u)µ0(du)Q(dv). (2.35) 13 Then with the help of Theorem 2.5 we have the equivalence between Q(du) andµ0(du), therefore a(u,v) has a simpler form: a(u,v)=min... | https://arxiv.org/abs/2503.16028v2 |
follows: f1=cos(πx),f2=−cos(πx),f3=cos(2πx),f4=cos(3πx). (3.2) It can be verified that the likelihood defined in (3.1) satisfies the Assumptions 6.1 in [11]. It is easy to see that the posterior distribution should have four modes, i.e., {fi}4 i=1. We drew 2×104samples from the posterior of uusing SMC-GM method, and SM... | https://arxiv.org/abs/2503.16028v2 |
truth (b) Mean of SMC-GM (c) Mean of SMC-pCN Figure 2: (a): The background truth of u. (b): The posterior mean estimated using SMC-GM. (c): The posterior mean estimated using SMC-pCN. (a) Density of u1. (b) Density of u4. (c) Density of u7. (d) Density of u10. (e) Density of u13. (f) Density of u16. Figure 3: Posterior... | https://arxiv.org/abs/2503.16028v2 |
the marginal density estimation by SMC-GM is small in terms of the average total variation distance. Mesh independence. Finally, we show the mesh independence of the SMC-GM method, a key property for a well-defined function space method. Figure 4(a) shows temperature plots for discrete dimensions N={400,1600,3600,6400,... | https://arxiv.org/abs/2503.16028v2 |
4. Discussion In this paper, we propose a novel transition kernel, pCN-GM, which is derived from the Crank-Nicolson discretization of the Langevin system and is proven to be well-defined in infinite- dimensional spaces. As a theoretical foundation for its application in SMC, we prove an SMC convergence theorem under we... | https://arxiv.org/abs/2503.16028v2 |
duM ds=−uM+p 2CMdb ds. Letwjdenote mass, and we will examine the evolution of the average velocity u=PM j=1wjuj PM j=1wj. By multiplying each equation by its respective mass and summing, we obtain MX j=1wjduj ds=−MX j=1wjuj+MX j=1wjq 2Cjdb ds. Discretize the equation using the Crank-Nicolson method, it follows that MX ... | https://arxiv.org/abs/2503.16028v2 |
(5.3) Now we prove that d(µ,ν) is a random version of total variation distance defined by dTV(µ,ν)=Z dµ dη−dν dη dη, whereηis a measure satisfying µ≪ηandν≪η. Ifµandνare not random, it follows from formula (5.3) that d(µ,ν)=sup |f|∞≤1 Z f dµ−Z f dν . Letη=µ+ν 2. Then,µ≪ηandν≪η. Thus d(µ,ν)=sup |f|∞≤1 Z fdµ dηdη−Z fdν dη... | https://arxiv.org/abs/2503.16028v2 |
2(η,β2Cjη)+1 2(ξ+γη,C(ξ+γη)) thatVjis positive. In order to show that Vjis a trace-class operator, we need to compute the eigen pairs ofVj. Note thatCjshares the same eigenfunctions, we claim that Vjhas eigenfunc- tions of the form ( ϕi,tϕi), i.e.,"CγC γCβ2Cj+γ2C#"ϕi tϕi# ="(1+γt)λiϕi (γλi+β2λjit+γ2λit)ϕi# ="(1+γt)λiϕi... | https://arxiv.org/abs/2503.16028v2 |
solutions. Recall that lj1i,l′ j2i∈ℓ2for 1≤j1,j2≤M, thus lj1i,l′ j2i→0,i→∞ . •The sum is ηi++ηi−=β2lj1il′ j2i+(lj1i+l′ j2i)+2 lj1i+1→2,i→∞. Denote l3i:=ηi++ηi−−2=β2lj1il′ j2i+(l′ j2i−lj1i) lj1i+1, then we have l3i∈ℓ2. •The product is ηi+ηi−=l′ j2i+1 lj1i+1→1. Denote l4i:=ηi+ηi−−1=l′ j2i−lj1i lj1i+1, then we have l2i∈ℓ2... | https://arxiv.org/abs/2503.16028v2 |
|ZN−Z|= Z e−Φ(uN)dµN 0(uN)−Z e−Φ(u)dµ0(u) ≤ Z e−Φ(uN)dµN 0(uN)−Z e−Φ(uN)dµ0(u) +Z |e−Φ(uN)−e−Φ(u)|dµ0(u) =Z |e−Φ(uN)−e−Φ(u)|dµ0(u)→0,N→∞. Here we need Φto be continuous. Then we can measure the distance between µpostand the pos- terior derived from the finite-dimensional likelihood. Using Dominated Convergence Theorem,... | https://arxiv.org/abs/2503.16028v2 |
Alexandros Beskos, Ajay Jasra, Kody Law, Youssef Marzouk, and Yan Zhou. Multilevel sequential monte carlo with dimension-independent likelihood-informed proposals. SIAM /ASA Journal on Uncertainty Quantification , 06(2):762–786, 2018. [4] Alexandros Beskos, Ajay Jasra, Kody Law, Raul Tempone, and Yan Zhou. Multilevel s... | https://arxiv.org/abs/2503.16028v2 |
sampling part ii: The nonlinear case. The Annals of Applied Probability , 17(5-6):1657–1706, 2007. [24] Jari Kaipio and Erkki Somersalo. Statistical and Computational Inverse Problems , volume 160 of Applied Mathe- matical Sciences . Springer-Verlag, New York, 2005. [25] Nikolas Kantas, Alexandros Beskos, and Ajay Jasr... | https://arxiv.org/abs/2503.16028v2 |
arXiv:2503.16154v1 [math.ST] 20 Mar 2025STATISTICAL ACCURACY OF THE ENSEMBLE KALMAN FILTER IN THE NEAR-LINEAR SETTING E. Calvello1, J. A. Carrillo2, F. Hoffmann3, P. Monmarch ´e4, A. M. Stuart5and U. Vaes6 Abstract. Estimating the state of adynamical system from partial and n oisy observations is aubiqui- tous problem ... | https://arxiv.org/abs/2503.16154v1 |
evolution: vn+1= Ψ(vn)+ξn, Data aquisition: yn+1=h(vn+1)+ηn+1. This holds for n∈N={0,1,2,···}. We assume that the initial state of the system is a Gaussian random variable v0∼ N(m0,C0) with known mean and covariance. In addition, we assume that the n oise entering in the state evolution is distributed according to the ... | https://arxiv.org/abs/2503.16154v1 |
connection between the finite particle ens emble Kalman filter and the true filter [3]. Before presenting the particle formulation of the ensemble Kalman fi lter, we introduce useful notation: for a probability measure π∈ P(Rd×RK), we write the mean under πasM(π) and the covariance under πas C(π) =/parenleftbiggCvv(π)Cvy(π... | https://arxiv.org/abs/2503.16154v1 |
between µEK nandµn, we write the two evolutions in parallel: µEK n+1=T(QPµEK n;y† n+1), µEK 0=µ0 (5a) µn+1=B(QPµn;y† n+1). (5b) Thus, in order to control the distance between µEK n+1andµn+1given a control at iteration n, it is essential to understand when T≈B.In fact, on the set of Gaussian measures G(Rd×RK), operator ... | https://arxiv.org/abs/2503.16154v1 |
dynamical systems (5a)and(5b), respectively. LetΨ0,h0be affine functions satisfying /bardblΨ0/bardblL∞w,/bardblh0/bardblL∞w≤κand|h0|C0,1≤ℓfor some κ,ℓ >0, and suppose that the noise covariances Σ,Γare positive definite. Then there exists a constant C=C(Σ,Γ,κ,ℓ,N)>0such that for all ε∈[0,1]and(Ψ,h)∈BL∞((Ψ0,h0),ε)satisfying... | https://arxiv.org/abs/2503.16154v1 |
Vaes. Stat istical accuracy of approximate filtering methods. arXiv preprint , 2402.01593, 2024. [5] P. Del Moral and J. Tugaut. On the stability and the unifor m propagation of chaos properties of ensemble Kalman-Bucy fi lters. Ann. Appl. Probab. , 28(2):790–850, 2018. [6] G. Evensen. Sequential data assimilation with a... | https://arxiv.org/abs/2503.16154v1 |
A Statistical Analysis for Per-Instance Evaluation of Stochastic Optimizers: How Many Repeats Are Enough? Moslem Noori, Elisabetta Valiante, and Ignacio Rozada∗ 1QB Information Technologies (1QBit), Vancouver, British Columbia, Canada Thomas Van Vaerenbergh and Masoud Mohseni Hewlett Packard Labs, Hewlett Packard Enter... | https://arxiv.org/abs/2503.16589v1 |
which are essentially special cases of Koza’s computational effort metric. All of these metrics depend on an estimate of the prob- ability of a stochastic optimizer successfully solving a problem. Estimating this probability requires solving the problem with several repeats. Solving the problem with a large number of r... | https://arxiv.org/abs/2503.16589v1 |
tee achieving a given accuracy of estimate for the metrics. Using this bound, we propose an algorithm to adaptively adjust the number of repeats in an experiment to ac- curately estimate the metrics. Our experimental results illustrate the utility of our analysis and proposed algo- rithm to determine the confidence lev... | https://arxiv.org/abs/2503.16589v1 |
discussion that follows can be generalized to using time or the number of operations as a measure of the computational resources. We denote the optimizer’s suc- cess probability, after running i for iiterations, by ps(i). This means that, after iiterations, the optimizer solves the problem or achieves the target value ... | https://arxiv.org/abs/2503.16589v1 |
to calculate CETS c(i). Estimating ps(i) of a solver is indeed a binomial propor- tion estimation (BPE) problem. From this point forward,we use ˆ ps(i) to refer to the point estimate of ps(i). Simi- larly, the sampled estimator of CETSopt cis defined as ˆCETSopt c= min i,ˆCETS c(i) (6) whereˆCETS c(i) =eitriln (1−c) ln... | https://arxiv.org/abs/2503.16589v1 |
and Jeffreys methods, we use the Agresti–Coull method in the remainder of this paper. III. CONFIDENCE INTERVAL OF THE METRICS In this section, we first discuss the effect of the number of repeats on the confidence interval of ps(i). We then extend the analysis to CETS c(i) and CETSopt c. Using statistical analysis, we ... | https://arxiv.org/abs/2503.16589v1 |
use Eq. (14). However, ˆ ps(i)[1−ˆps(i)] is maximized when ˆps(i) = 0 .5 and we can use this as the worst-case scenario. Considering this worst-case scenario and that ˆn=n+z2 α, it is guaranteed that there will be an error margin of at most ϵp(i) if n≥&zα 2ϵp(i)2 −z2 α' . (15) For the 95% confidence interval and assu... | https://arxiv.org/abs/2503.16589v1 |
estimating Rc(i), we study the relationship between the number of repeats and ˆRc(i), as well as its estimate error. We begin by finding the number of repeats needed to guarantee a maximum error of ϵp(i) = 0 .01 and ϵp(i) = 0 .03. This is needed when designing an experiment in order to evaluate the performance of the o... | https://arxiv.org/abs/2503.16589v1 |
replace ˆ ps(i) by ns(i)/(n+z2 α), where ns(i) is the number of success- ful repeats after iiterations, and replaced ϵp(i) from Eq. (18) to express the inequality as a function of n. Then, a numerical root-finding algorithm can be applied to find the smallest value of nthat satisfies the inequal- ity. As a low-complexi... | https://arxiv.org/abs/2503.16589v1 |
2. In reality, these probabilities are unknown to us; hence, to compare the solvers, we solve the problem using nrepeats, estimate p1andp2using these repeats, and then calculate the metric for the op- timizers using the estimated values ˆ p1and ˆp2. In what follows, we use the R99metric (i.e., c= 0.99) to evaluate the ... | https://arxiv.org/abs/2503.16589v1 |
more likely to be an over- lap between the CI αof the R99values than for there to be no overlap. By increasing the number of repeats, it becomes more likely that ˆR1 99<ˆR2 99. For n= 10,000, the better solver can always be identified with a very high level of confidence. We now evaluate the utility of Algorithm 1 in c... | https://arxiv.org/abs/2503.16589v1 |
50 variables and 499 clauses. To solve this problem, we use WalkSAT-SKC [36], which is a stochas- tic local search method for solving SAT problems. First, we solve the problem with the default WalkSAT- SKC walk probability, denoted by w, of 0 .5 and 5000 it- erations using three repeats n∈ {100,1000,10,000 }. For the s... | https://arxiv.org/abs/2503.16589v1 |
(8811.97, 428,066.65) 93 1000 32,015.60 (28,272.11, 36,716.09) 1799 10,000 33,072.04 (32,134.81, 34046.64) 4648 TABLE IV: Effect of non CETSopt cand the optimal number of iterations i∗. Two important observations can be gleaned from this table. First, running the experiment with n= 100 re- peats results in a significan... | https://arxiv.org/abs/2503.16589v1 |
M. Suzuki, Y. Sakai, T. Kanao, Y. Hamakawa, R. Hidaka, M. Yamasaki, and K. Tat- sumura, High-performance combinatorial optimization based on classical mechanics, Science Advances 7, eabe7953 (2021).[4] S. Tsukamoto, M. Takatsu, S. Matsubara, and H. Tamura, An accelerator architecture for combinatorial optimization prob... | https://arxiv.org/abs/2503.16589v1 |
(1996) pp. 21–29. [19] S. Christensen and F. Oppacher, An analysis of Koza’s computational effort statistic for genetic programming, inProceedings of 5th European Conference on Genetic Programming (EuroGP) (Springer, 2002) pp. 182–191. [20] M. Keijzer, V. Babovic, C. Ryan, M. O’Neill, and M. Cat- tolico, Adaptive logic... | https://arxiv.org/abs/2503.16589v1 |
Uniformly consistent proportion estimation for composite hypotheses via integral equations: “the case of location-shift families” Xiongzhi Chen∗ Abstract We consider estimating the proportion of random variables for two types of composite null hypotheses: (i) the means or medians of the random variables belonging to a ... | https://arxiv.org/abs/2503.16590v1 |
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