Probability Theory and Related Fields最新文献_第6页

Strong posterior contraction rates via Wasserstein dynamics 通过瓦瑟斯坦动力学实现强后收缩率

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-02-23 DOI: 10.1007/s00440-024-01260-w

Emanuele Dolera, Stefano Favaro, Edoardo Mainini

{"title":"Strong posterior contraction rates via Wasserstein dynamics","authors":"Emanuele Dolera, Stefano Favaro, Edoardo Mainini","doi":"10.1007/s00440-024-01260-w","DOIUrl":"https://doi.org/10.1007/s00440-024-01260-w","url":null,"abstract":"In Bayesian statistics, posterior contraction rates (PCRs) quantify the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of a true model, in a suitable way, as the sample size goes to infinity. In this paper, we develop a new approach to PCRs, with respect to strong norm distances on parameter spaces of functions. Critical to our approach is the combination of a local Lipschitz-continuity for the posterior distribution with a dynamic formulation of the Wasserstein distance, which allows to set forth an interesting connection between PCRs and some classical problems arising in mathematical analysis, probability and statistics, e.g., Laplace methods for approximating integrals, Sanov’s large deviation principles in the Wasserstein distance, rates of convergence of mean Glivenko–Cantelli theorems, and estimates of weighted Poincaré–Wirtinger constants. We first present a theorem on PCRs for a model in the regular infinite-dimensional exponential family, which exploits sufficient statistics of the model, and then extend such a theorem to a general dominated model. These results rely on the development of novel techniques to evaluate Laplace integrals and weighted Poincaré–Wirtinger constants in infinite-dimension, which are of independent interest. The proposed approach is applied to the regular parametric model, the multinomial model, the finite-dimensional and the infinite-dimensional logistic-Gaussian model and the infinite-dimensional linear regression. In general, our approach leads to optimal PCRs in finite-dimensional models, whereas for infinite-dimensional models it is shown explicitly how the prior distribution affect PCRs.\u0000","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139952388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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

Geometry of the minimal spanning tree in the heavy-tailed regime: new universality classes 重尾机制中最小生成树的几何：新的普遍性类别

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-02-17 DOI: 10.1007/s00440-024-01259-3

Shankar Bhamidi, Sanchayan Sen

{"title":"Geometry of the minimal spanning tree in the heavy-tailed regime: new universality classes","authors":"Shankar Bhamidi, Sanchayan Sen","doi":"10.1007/s00440-024-01259-3","DOIUrl":"https://doi.org/10.1007/s00440-024-01259-3","url":null,"abstract":"A well-known open problem on the behavior of optimal paths in random graphs in the strong disorder regime, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade (Braunstein et al. in Phys Rev Lett 91(16):168701, 2003; Braunstein et al. in Int J Bifurc Chaos 17(07):2215–2255, 2007; Chen et al. in Phys Rev Lett 96(6):068702, 2006; Wu et al. in Phys Rev Lett 96(14):148702, 2006) is as follows: for a large class of random graph models with degree exponent (tau in (3,4)), distances in the minimal spanning tree (MST) on the giant component in the supercritical regime scale like (n^{(tau -3)/(tau -1)}). The aim of this paper is to make progress towards a proof of this conjecture. We consider a supercritical inhomogeneous random graph model with degree exponent (tau in (3, 4)) that is closely related to Aldous’s multiplicative coalescent, and show that the MST constructed by assigning i.i.d. continuous weights to the edges in its giant component, endowed with the tree distance scaled by (n^{-(tau -3)/(tau -1)}), converges in distribution with respect to the Gromov–Hausdorff topology to a random compact real tree. Further, almost surely, every point in this limiting space either has degree one (leaf), or two, or infinity (hub), both the set of leaves and the set of hubs are dense in this space, and the Minkowski dimension of this space equals ((tau -1)/(tau -3)). The multiplicative coalescent, in an asymptotic sense, describes the evolution of the component sizes of various near-critical random graph processes. We expect the limiting spaces in this paper to be the candidates for the scaling limit of the MST constructed for a wide array of other heavy-tailed random graph models.\u0000","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139903742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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

The critical variational setting for stochastic evolution equations 随机演化方程的临界变分设置

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-02-02 DOI: 10.1007/s00440-023-01249-x

Antonio Agresti, Mark Veraar

引用次数: 0

Geometric bounds on the fastest mixing Markov chain 最快混合马尔可夫链的几何边界

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-01-30 DOI: 10.1007/s00440-023-01257-x

Sam Olesker-Taylor, Luca Zanetti

{"title":"Geometric bounds on the fastest mixing Markov chain","authors":"Sam Olesker-Taylor, Luca Zanetti","doi":"10.1007/s00440-023-01257-x","DOIUrl":"https://doi.org/10.1007/s00440-023-01257-x","url":null,"abstract":"In the Fastest Mixing Markov Chain problem, we are given a graph (G = (V, E)) and desire the discrete-time Markov chain with smallest mixing time (tau ) subject to having equilibrium distribution uniform on V and non-zero transition probabilities only across edges of the graph. It is well-known that the mixing time (tau _textsf {RW}) of the lazy random walk on G is characterised by the edge conductance (Phi ) of G via Cheeger’s inequality: (Phi ^{-1} lesssim tau _textsf {RW} lesssim Phi ^{-2} log |V|). Analogously, we characterise the fastest mixing time (tau ^star ) via a Cheeger-type inequality but for a different geometric quantity, namely the vertex conductance (Psi ) of G: (Psi ^{-1} lesssim tau ^star lesssim Psi ^{-2} (log |V|)^2). This characterisation forbids fast mixing for graphs with small vertex conductance. To bypass this fundamental barrier, we consider Markov chains on G with equilibrium distribution which need not be uniform, but rather only (varepsilon )-close to uniform in total variation. We show that it is always possible to construct such a chain with mixing time (tau lesssim varepsilon ^{-1} ({text {diam}} G)^2 log |V|). Finally, we discuss analogous questions for continuous-time and time-inhomogeneous chains.","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139648394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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

Tractability from overparametrization: the example of the negative perceptron 过度参数化的可操作性：以负感知器为例

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-01-22 DOI: 10.1007/s00440-023-01248-y

Andrea Montanari, Yiqiao Zhong, Kangjie Zhou

{"title":"Tractability from overparametrization: the example of the negative perceptron","authors":"Andrea Montanari, Yiqiao Zhong, Kangjie Zhou","doi":"10.1007/s00440-023-01248-y","DOIUrl":"https://doi.org/10.1007/s00440-023-01248-y","url":null,"abstract":"In the negative perceptron problem we are given n data points ((varvec{x}_i,y_i)), where (varvec{x}_i) is a d-dimensional vector and (y_iin {+1,-1}) is a binary label. The data are not linearly separable and hence we content ourselves to find a linear classifier with the largest possible negative margin. In other words, we want to find a unit norm vector (varvec{theta }) that maximizes (min _{ile n}y_ilangle varvec{theta },varvec{x}_irangle ). This is a non-convex optimization problem (it is equivalent to finding a maximum norm vector in a polytope), and we study its typical properties under two random models for the data. We consider the proportional asymptotics in which (n,drightarrow infty ) with (n/drightarrow delta ), and prove upper and lower bounds on the maximum margin (kappa _{{textrm{s}}}(delta )) or—equivalently—on its inverse function (delta _{{textrm{s}}}(kappa )). In other words, (delta _{{textrm{s}}}(kappa )) is the overparametrization threshold: for (n/dle delta _{{textrm{s}}}(kappa )-{varepsilon }) a classifier achieving vanishing training error exists with high probability, while for (n/dge delta _{{textrm{s}}}(kappa )+{varepsilon }) it does not. Our bounds on (delta _{{textrm{s}}}(kappa )) match to the leading order as (kappa rightarrow -infty ). We then analyze a linear programming algorithm to find a solution, and characterize the corresponding threshold (delta _{textrm{lin}}(kappa )). We observe a gap between the interpolation threshold (delta _{{textrm{s}}}(kappa )) and the linear programming threshold (delta _{textrm{lin}}(kappa )), raising the question of the behavior of other algorithms.\u0000","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139558969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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

W-entropy and Langevin deformation on Wasserstein space over Riemannian manifolds 黎曼流形上瓦塞尔斯坦空间的 W-熵和朗格文变形

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-01-16 DOI: 10.1007/s00440-023-01256-y

Songzi Li, Xiang-Dong Li

{"title":"W-entropy and Langevin deformation on Wasserstein space over Riemannian manifolds","authors":"Songzi Li, Xiang-Dong Li","doi":"10.1007/s00440-023-01256-y","DOIUrl":"https://doi.org/10.1007/s00440-023-01256-y","url":null,"abstract":"We prove the Perelman type W-entropy formula for the geodesic flow on the (L^2)-Wasserstein space over a complete Riemannian manifold equipped with Otto’s infinite dimensional Riemannian metric. To better understand the similarity between the W-entropy formula for the geodesic flow on the Wasserstein space and the W-entropy formula for the heat flow of the Witten Laplacian on the underlying manifold, we introduce the Langevin deformation of flows on the Wasserstein space over a Riemannian manifold, which interpolates the gradient flow and the geodesic flow on the Wasserstein space over a Riemannian manifold, and can be regarded as the potential flow of the compressible Euler equation with damping on a Riemannian manifold. We prove the existence, uniqueness and regularity of the Langevin deformation on the Wasserstein space over the Euclidean space and a compact Riemannian manifold, and prove the convergence of the Langevin deformation for (crightarrow 0) and (crightarrow infty ) respectively. Moreover, we prove the W-entropy-information formula along the Langevin deformation on the Wasserstein space on Riemannian manifolds. The rigidity theorems are proved for the W-entropy for the geodesic flow and the Langevin deformation on the Wasserstein space over complete Riemannian manifolds with the CD(0, m)-condition. Our results are new even in the case of Euclidean spaces and complete Riemannian manifolds with non-negative Ricci curvature.","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139500645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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

Eve, Adam and the preferential attachment tree 夏娃、亚当和依恋树

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-01-12 DOI: 10.1007/s00440-023-01253-1

Alice Contat, Nicolas Curien, Perrine Lacroix, Etienne Lasalle, Vincent Rivoirard

引用次数: 0

Superconcentration for minimal surfaces in first passage percolation and disordered Ising ferromagnets 第一通道渗流和无序伊辛铁磁体中最小表面的超聚合

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-01-05 DOI: 10.1007/s00440-023-01252-2

Barbara Dembin, Christophe Garban

{"title":"Superconcentration for minimal surfaces in first passage percolation and disordered Ising ferromagnets","authors":"Barbara Dembin, Christophe Garban","doi":"10.1007/s00440-023-01252-2","DOIUrl":"https://doi.org/10.1007/s00440-023-01252-2","url":null,"abstract":"We consider the standard first passage percolation model on ({mathbb {Z}}^ d) with a distribution G taking two values (0<a<b). We study the maximal flow through the cylinder ([0,n]^ {d-1}times [0,hn]) between its top and bottom as well as its associated minimal surface(s). We prove that the variance of the maximal flow is superconcentrated, i.e. in (O(frac{n^{d-1}}{log n})), for (hge h_0) (for a large enough constant (h_0=h_0(a,b))). Equivalently, we obtain that the ground state energy of a disordered Ising ferromagnet in a cylinder ([0,n]^ {d-1}times [0,hn]) is superconcentrated when opposite boundary conditions are applied at the top and bottom faces and for a large enough constant (hge h_0) (which depends on the law of the coupling constants). Our proof is inspired by the proof of Benjamini–Kalai–Schramm (Ann Probab 31:1970–1978, 2003). Yet, one major difficulty in this setting is to control the influence of the edges since the averaging trick used in Benjamini et al. (Ann Probab 31:1970–1978, 2003) fails for surfaces. Of independent interest, we prove that minimal surfaces (in the present discrete setting) cannot have long thin chimneys.","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139103070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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

On the Wiener chaos expansion of the signature of a Gaussian process 论高斯过程特征的维纳混沌扩展

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-01-04 DOI: 10.1007/s00440-023-01255-z

Thomas Cass, Emilio Ferrucci

引用次数: 0

Annealed quantitative estimates for the quadratic 2D-discrete random matching problem 二次方二维离散随机匹配问题的退火定量估计

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-01-04 DOI: 10.1007/s00440-023-01254-0

Nicolas Clozeau, Francesco Mattesini

引用次数: 0