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The TAP free energy for high-dimensional linear regression 高维线性回归的TAP自由能
IF 1.8 2区 数学
Annals of Applied Probability Pub Date : 2022-03-14 DOI: 10.1214/22-aap1874
Jia Qiu, Subhabrata Sen
{"title":"The TAP free energy for high-dimensional linear regression","authors":"Jia Qiu, Subhabrata Sen","doi":"10.1214/22-aap1874","DOIUrl":"https://doi.org/10.1214/22-aap1874","url":null,"abstract":"We derive a variational representation for the log-normalizing constant of the posterior distribution in Bayesian linear regression with a uniform spherical prior and an i.i.d. Gaussian design. We work under the\"proportional\"asymptotic regime, where the number of observations and the number of features grow at a proportional rate. This rigorously establishes the Thouless-Anderson-Palmer (TAP) approximation arising from spin glass theory, and proves a conjecture of Krzakala et. al. (2014) in the special case of the spherical prior.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44919691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Non-universal fluctuations of the empirical measure for isotropic stationary fields on S2×R S2×R上各向同性平稳场经验测度的非普适性涨落
IF 1.8 2区 数学
Annals of Applied Probability Pub Date : 2021-10-01 DOI: 10.1214/20-aap1648
D. Marinucci, Maurizia Rossi, Anna Vidotto
{"title":"Non-universal fluctuations of the empirical measure for isotropic stationary fields on S2×R","authors":"D. Marinucci, Maurizia Rossi, Anna Vidotto","doi":"10.1214/20-aap1648","DOIUrl":"https://doi.org/10.1214/20-aap1648","url":null,"abstract":"","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45159704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
Mixing of Hamiltonian Monte Carlo on strongly log-concave distributions: Continuous dynamics 哈密顿蒙特卡罗在强对数凹分布上的混合：连续动力学
IF 1.8 2区 数学
Annals of Applied Probability Pub Date : 2021-10-01 DOI: 10.1214/20-aap1640
Oren Mangoubi, Aaron Smith
{"title":"Mixing of Hamiltonian Monte Carlo on strongly log-concave distributions: Continuous dynamics","authors":"Oren Mangoubi, Aaron Smith","doi":"10.1214/20-aap1640","DOIUrl":"https://doi.org/10.1214/20-aap1640","url":null,"abstract":"We obtain several quantitative bounds on the mixing properties of an “ideal” Hamiltonian Monte Carlo (HMC) Markov chain for a strongly log-concave target distribution π on R. Our main result says that the HMC Markov chain generates a sample with Wasserstein error in roughly O(κ log(1/ )) steps, where the condition number κ = M2 m2 is the ratio of the maximum M2 and minimum m2 eigenvalues of the Hessian of − log(π). In particular, this mixing bound does not depend explicitly on the dimension d. These results significantly extend and improve previous quantitative bounds on the mixing of ideal HMC, and can be used to analyze more realistic HMC algorithms. The main ingredient of our argument is a proof that initially “parallel” Hamiltonian trajectories contract over much longer steps than would be predicted by previous heuristics based on the Jacobi manifold.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48793816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 17
Stochastic approximation with discontinuous dynamics, differential inclusions, and applications 随机逼近与不连续动力学，微分夹杂，和应用
IF 1.8 2区 数学
Annals of Applied Probability Pub Date : 2021-08-28 DOI: 10.1214/22-aap1829
N. Nguyen, G. Yin
{"title":"Stochastic approximation with discontinuous dynamics, differential inclusions, and applications","authors":"N. Nguyen, G. Yin","doi":"10.1214/22-aap1829","DOIUrl":"https://doi.org/10.1214/22-aap1829","url":null,"abstract":"This work develops new results for stochastic approximation algorithms. The emphases are on treating algorithms and limits with discontinuities. The main ingredients include the use of differential inclusions, set-valued analysis, and non-smooth analysis, and stochastic differential inclusions. Under broad conditions, it is shown that a suitably scaled sequence of the iterates has a differential inclusion limit. In addition, it is shown for the first time that a centered and scaled sequence of the iterates converges weakly to a stochastic differential inclusion limit. The results are then used to treat several application examples including Markov decision process, Lasso algorithms, Pegasos algorithms, support vector machine classification, and learning. Some numerical demonstrations are also provided.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47173572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Cutoff for rewiring dynamics on perfect matchings 在完美匹配的情况下重新布线动态的截止
IF 1.8 2区 数学
Annals of Applied Probability Pub Date : 2021-08-26 DOI: 10.1214/22-aap1825
Sam Olesker-Taylor
{"title":"Cutoff for rewiring dynamics on perfect matchings","authors":"Sam Olesker-Taylor","doi":"10.1214/22-aap1825","DOIUrl":"https://doi.org/10.1214/22-aap1825","url":null,"abstract":"We establish cutoff for a natural random walk (RW) on the set of perfect matchings (PMs). An $n$-PM is a pairing of $2n$ objects. The $k$-PM RW selects $k$ pairs uniformly at random, disassociates the corresponding $2k$ objects, then chooses a new pairing on these $2k$ objects uniformly at random. The equilibrium distribution is uniform over the set of all $n$-PM. We establish cutoff for the $k$-PM RW whenever $2 le k ll n$. If $k gg 1$, then the mixing time is $tfrac nk log n$ to leading order. The case $k = 2$ was established by Diaconis and Holmes (2002) by relating the $2$-PM RW to the random transpositions card shuffle and also by Ceccherini-Silberstein, Scarabotti and Tolli (2007, 2008) using representation theory. We are the first to handle $k>2$. Our argument builds on previous work of Berestycki, Schramm, c{S}eng\"ul and Zeitouni (2005, 2011, 2019) regarding conjugacy-invariant RWs on the permutation group.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41395173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Coexistence in competing first passage percolation with conversion 竞争第一通道渗流与转换的共存性
IF 1.8 2区 数学
Annals of Applied Probability Pub Date : 2021-08-24 DOI: 10.1214/22-aap1792
T. Finn, Alexandre O. Stauffer
{"title":"Coexistence in competing first passage percolation with conversion","authors":"T. Finn, Alexandre O. Stauffer","doi":"10.1214/22-aap1792","DOIUrl":"https://doi.org/10.1214/22-aap1792","url":null,"abstract":"We introduce a two-type first passage percolation competition model on infinite connected graphs as follows. Type 1 spreads through the edges of the graph at rate 1 from a single distinguished site, while all other sites are initially vacant. Once a site is occupied by type 1, it converts to type 2 at rate $rho>0$. Sites occupied by type 2 then spread at rate $lambda>0$ through vacant sites emph{and} sites occupied by type 1, whereas type 1 can only spread through vacant sites. If the set of sites occupied by type 1 is non-empty at all times, we say type 1 emph{survives}. In the case of a regular $d$-ary tree for $dgeq 3$, we show type 1 can survive when it is slower than type 2, provided $rho$ is small enough. This is in contrast to when the underlying graph is $mathbb{Z}^d$, where for any $rho>0$, type 1 dies out almost surely if $lambda>1$.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49356799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Global-in-time mean-field convergence for singular Riesz-type diffusive flows 奇异riesz型扩散流的全局实时平均场收敛性
IF 1.8 2区 数学
Annals of Applied Probability Pub Date : 2021-08-23 DOI: 10.1214/22-aap1833
M. Rosenzweig, S. Serfaty
{"title":"Global-in-time mean-field convergence for singular Riesz-type diffusive flows","authors":"M. Rosenzweig, S. Serfaty","doi":"10.1214/22-aap1833","DOIUrl":"https://doi.org/10.1214/22-aap1833","url":null,"abstract":"We consider the mean-field limit of systems of particles with singular interactions of the type $-log|x|$ or $|x|^{-s}$, with $0<s<d-2$, and with an additive noise in dimensions $d geq 3$. We use a modulated-energy approach to prove a quantitative convergence rate to the solution of the corresponding limiting PDE. When $s>0$, the convergence is global in time, and it is the first such result valid for both conservative and gradient flows in a singular setting on $mathbb{R}^d$. The proof relies on an adaptation of an argument of Carlen-Loss to show a decay rate of the solution to the limiting equation, and on an improvement of the modulated-energy method developed in arXiv:1508.03377, arXiv:1803.08345, arXiv:2107.02592 making it so that all prefactors in the time derivative of the modulated energy are controlled by a decaying bound on the limiting solution.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44266923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 22
Cyclic cellular automata and Greenberg–Hastings models on regular trees 循环元胞自动机和规则树上的Greenberg-Hastings模型
IF 1.8 2区 数学
Annals of Applied Probability Pub Date : 2021-08-13 DOI: 10.1214/22-aap1885
J. Bello, David J Sivakoff
{"title":"Cyclic cellular automata and Greenberg–Hastings models on regular trees","authors":"J. Bello, David J Sivakoff","doi":"10.1214/22-aap1885","DOIUrl":"https://doi.org/10.1214/22-aap1885","url":null,"abstract":"We study the cyclic cellular automaton (CCA) and the Greenberg-Hastings model (GHM) with $kappage 3$ colors and contact threshold $thetage 2$ on the infinite $(d+1)$-regular tree, $T_d$. When the initial state has the uniform product distribution, we show that these dynamical systems exhibit at least two distinct phases. For sufficiently large $d$, we show that if $kappa(theta-1) le d - O(sqrt{dkappa ln(d)})$, then every vertex almost surely changes its color infinitely often, while if $kappatheta ge d + O(kappasqrt{dln(d)})$, then every vertex almost surely changes its color only finitely many times. Roughly, this implies that as $dto infty$, there is a phase transition where $kappatheta/d = 1$. For the GHM dynamics, in the scenario where every vertex changes color finitely many times, we moreover give an exponential tail bound for the distribution of the time of the last color change at a given vertex.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46716962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A general conditional McKean–Vlasov stochastic differential equation 一般条件McKean-Vlasov随机微分方程
IF 1.8 2区 数学
Annals of Applied Probability Pub Date : 2021-08-07 DOI: 10.1214/22-aap1858
R. Buckdahn, Juan Li, Jin Ma
{"title":"A general conditional McKean–Vlasov stochastic differential equation","authors":"R. Buckdahn, Juan Li, Jin Ma","doi":"10.1214/22-aap1858","DOIUrl":"https://doi.org/10.1214/22-aap1858","url":null,"abstract":"In this paper we consider a class of {it conditional McKean-Vlasov SDEs} (CMVSDE for short). Such an SDE can be considered as an extended version of McKean-Vlasov SDEs with common noises, as well as the general version of the so-called {it conditional mean-field SDEs} (CMFSDE) studied previously by the authors [1, 14], but with some fundamental differences. In particular, due to the lack of compactness of the iterated conditional laws, the existing arguments of Schauder's fixed point theorem do not seem to apply in this situation, and the heavy nonlinearity on the conditional laws caused by change of probability measure adds more technical subtleties. Under some structure assumptions on the coefficients of the observation equation, we prove the well-posedness of solution in the weak sense along a more direct approach. Our result is the first that deals with McKean-Vlasov type SDEs involving state-dependent conditional laws.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44679489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices 样本协方差矩阵最大特征值的tracy - wisdom律收敛速度
IF 1.8 2区 数学
Annals of Applied Probability Pub Date : 2021-08-05 DOI: 10.1214/22-aap1826
Kevin Schnelli, Yuanyuan Xu
{"title":"Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices","authors":"Kevin Schnelli, Yuanyuan Xu","doi":"10.1214/22-aap1826","DOIUrl":"https://doi.org/10.1214/22-aap1826","url":null,"abstract":"We establish a quantitative version of the Tracy--Widom law for the largest eigenvalue of high dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix $X^*X$ converge to its Tracy--Widom limit at a rate nearly $N^{-1/3}$, where $X$ is an $M times N$ random matrix whose entries are independent real or complex random variables, assuming that both $M$ and $N$ tend to infinity at a constant rate. This result improves the previous estimate $N^{-2/9}$ obtained by Wang [73]. Our proof relies on a Green function comparison method [27] using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44875604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
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