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

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":"55 1","pages":""},"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":"113 1","pages":""},"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":"58 1","pages":""},"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":"160 1","pages":""},"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

Anomalous diffusion limit for a kinetic equation with a thermostatted interface 带有恒温界面的动力学方程的反常扩散极限

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2023-12-29 DOI: 10.1007/s00440-023-01251-3

引用次数: 0

Sums of GUE matrices and concentration of hives from correlation decay of eigengaps 从 eigengaps 的相关衰减中得出的 GUE 矩阵总和与蜂巢浓度

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2023-12-28 DOI: 10.1007/s00440-023-01250-4

Hariharan Narayanan, Scott Sheffield, Terence Tao

引用次数: 0