Seminars and Colloquia by Series

On the geometry of polytopes generated by heavy-tailed random vectors

Series
Stochastics Seminar
Time
Friday, September 1, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Felix KrahmerTechnical University of Munich

In this talk, we present recent results on the geometry of centrally-symmetric random polytopes generated by N independent copies of a random vector X. We show that under minimal assumptions on X, for N>Cn, and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector - namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors we recover the estimates that have been obtained previously, and thanks to the minimal assumptions on X we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when X is q-stable or when X has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing - noise blind sparse recovery. This is joint work with Olivier Guédon (University of Paris-Est Marne La Vallée), Christian Kümmerle (UNC Charlotte), Shahar Mendelson (Sorbonne University Paris), and Holger Rauhut (LMU Munich).

Bio: Felix Krahmer received his PhD in Mathematics in 2009 from New York University under the supervision of Percy Deift and Sinan Güntürk. He was a Hausdorff postdoctoral fellow in the group of Holger Rauhut at the University of Bonn, Germany from 2009-2012. In 2012 he joined the University of Göttingen as an assistant professor for mathematical data analysis, where he has been awarded an Emmy Noether Junior Research Group. From 2015-2021 he was assistant professor for optimization and data analysis in the department of mathematics at the Technical University of Munich, before he was tenured and promoted to associate professor in 2021. His research interests span various areas at the interface of probability, analysis, machine learning, and signal processing including randomized sensing and reconstruction, fast random embeddings, quantization, and the computational sensing paradigm.

(Skew) Gaussian surrogates for high-dimensional posteriors: tighter bounds and tighter approximations

Series
Stochastics Seminar
Time
Thursday, August 31, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anya KatsevichMIT

Computing integrals against a high-dimensional posterior is the major computational bottleneck in Bayesian inference. A popular technique to reduce this computational burden is to use the Laplace approximation, a Gaussian distribution, in place of the true posterior. Despite its widespread use, the Laplace approximation's accuracy in high dimensions is not well understood.  The body of existing results does not form a cohesive theory, leaving open important questions e.g. on the dimension dependence of the approximation rate. We address many of these questions through the unified framework of a new, leading order asymptotic decomposition of high-dimensional Laplace integrals. In particular, we (1) determine the tight dimension dependence of the approximation error, leading to the tightest known Bernstein von Mises result on the asymptotic normality of the posterior, and (2) derive a simple correction to this Gaussian distribution to obtain a higher-order accurate approximation to the posterior.

Quantitative Generalized CLT with Self-Decomposable Limiting Laws by Spectral Methods

Series
Stochastics Seminar
Time
Thursday, May 18, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benjamin ArrasUniversité de Lille

In this talk, I will present new stability results for non-degenerate centered self-decomposable laws with finite second moment and for non-degenerate symmetric alpha-stable laws with alpha in (1,2). These stability results are based on Stein's method and closed forms techniques. As an application, explicit rates of convergence are obtained for several instances of the generalized CLTs. Finally, I will discuss the standard Cauchy case.

Coalescence, geodesic density, and bigeodesics in first-passage percolation

Series
Stochastics Seminar
Time
Thursday, April 20, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jack HansonCity College, CUNY

Several well-known problems in first-passage percolation relate to the behavior of infinite geodesics: whether they coalesce and how rapidly, and whether doubly infinite "bigeodesics'' exist. In the plane, a version of coalescence of "parallel'' geodesics has previously been shown; we will discuss new results that show infinite geodesics from the origin have zero density in the plane. We will describe related forthcoming work showing that geodesics coalesce in dimensions three and higher, under unproven assumptions believed to hold below the model's upper critical dimension. If time permits, we will also discuss results on the bigeodesic question in dimension three and higher.

Random Laplacian Matrices

Series
Stochastics Seminar
Time
Thursday, April 13, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrew CampbellUniversity of Colorado

The Laplacian of a graph is a real symmetric matrix given by $L=D-A$, where $D$ is the degree matrix of the graph and $A$ is the adjacency matrix. The Laplacian is a central object in spectral graph theory, and the spectrum of $L$ contains information on the graph. In the case of a random graph the Laplacian will be a random real symmetric matrix with dependent entries. These random Laplacian matrices can be generalized by taking $A$ to be a random real symmetric matrix and $D$ to be a diagonal matrix with entries equal to the row sums of $A$. We will consider the eigenvalues of general random Laplacian matrices, and the challenges raised by the dependence between $D$ and $A$. After discussing the bulk global eigenvalue behavior of general random Laplacian matrices, we will focus in detail on fluctuations of the largest eigenvalue of $L$ when $A$ is a matrix of independent Gaussian random variables. The asymptotic behavior of these Gaussian Laplacian matrices has a particularly nice free probabilistic interpretation, which can be exploited in the study of their eigenvalues. We will see how this interpretation can locate the largest eigenvalue of $L$ with respect to the largest entry of $D$. This talk is based on joint work with Kyle Luh and Sean O'Rourke.

Stein kernels, functional inequalities and applications in statistics

Series
Stochastics Seminar
Time
Thursday, April 6, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE via Zoom https://gatech.zoom.us/j/94387417679
Speaker
Adrien SaumardENSAI and CREST

Zoom link to the talk: https://gatech.zoom.us/j/94387417679

We will present the notion of Stein kernel, which provides generalizations of the integration by parts, a.k.a. Stein's formula, for the normal distribution (which has a constant Stein kernel, equal to its covariance). We will first focus on dimension one, where under good conditions the Stein kernel has an explicit formula. We will see that the Stein kernel appears naturally as a weighting of a Poincaré type inequality and that it enables precise concentration inequalities, of the Mills' ratio type. In a second part, we will work in higher dimensions, using in particular Max Fathi's construction of a Stein kernel through the so-called "moment maps" transportation. This will allow us to describe the performance of some shrinkage and thresholding estimators, beyond the classical assumption of Gaussian (or spherical) data. This presentation is mostly based on joint works with Max Fathi, Larry Goldstein, Gesine Reinert and Jon Wellner.

The sample complexity of learning transport maps

Series
Stochastics Seminar
Time
Thursday, March 30, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Philippe RigolletMassachusetts Institute of Technology

Optimal transport has recently found applications in a variety of fields ranging from graphics to biology. Underlying these applications is a new statistical paradigm where the goal is to couple multiple data sources. It gives rise to interesting new questions ranging from the design of estimators to minimax rates of convergence. I will review several applications where the central problem consists in estimating transport maps. After studying optimal transport as a potential solution, I will argue that its entropic version is a good alternative model. In particular, it completely escapes the curse of dimensionality that plagues statistical optimal transport.

Implicit estimation of high-dimensional distributions using generative models

Series
Stochastics Seminar
Time
Thursday, March 16, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yun YangUniversity of Illinois Urbana-Champaign

The estimation of distributions of complex objects from high-dimensional data with low-dimensional structures is an important topic in statistics and machine learning. Deep generative models achieve this by encoding and decoding data to generate synthetic realistic images and texts. A key aspect of these models is the extraction of low-dimensional latent features, assuming data lies on a low-dimensional manifold. We study this by developing a minimax framework for distribution estimation on unknown submanifolds with smoothness assumptions on the target distribution and the manifold. The framework highlights how problem characteristics, such as intrinsic dimensionality and smoothness, impact the limits of high-dimensional distribution estimation. Our estimator, which is a mixture of locally fitted generative models, is motivated by differential geometry techniques and covers cases where the data manifold lacks a global parametrization. 

Large-graph approximations for interacting particles on graphs and their applications

Series
Stochastics Seminar
Time
Thursday, March 2, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Wasiur KhudaBukhshUniversity of Nottingham

Zoom link to the talk: https://gatech.zoom.us/j/91558578481

In this talk, we will consider stochastic processes on (random) graphs. They arise naturally in epidemiology, statistical physics, computer science and engineering disciplines. In this set-up, the vertices are endowed with a local state (e.g., immunological status in case of an epidemic process, opinion about a social situation). The local state changes dynamically as the vertex interacts with its neighbours. The interaction rules and the graph structure depend on the application-specific context. We will discuss (non-equilibrium) approximation methods for those systems as the number of vertices grow large. In particular, we will discuss three different approximations in this talk: i) approximate lumpability of Markov processes based on local symmetries (local automorphisms) of the graph, ii) functional laws of large numbers in the form of ordinary and partial differential equations, and iii) functional central limit theorems in the form of Gaussian semi-martingales. We will also briefly discuss how those approximations could be used for practical purposes, such as parameter inference from real epidemic data (e.g., COVID-19 in Ohio), designing efficient simulation algorithms etc.

Covariance Representations, Stein's Kernels and High Dimensional CLTs

Series
Stochastics Seminar
Time
Thursday, February 23, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christian HoudréGeorgia Tech

In this continuing joint work with Benjamin Arras, we explore connections between covariance representations and Stein's method. In particular,  via Stein's kernels we obtain quantitative high-dimensional CLTs in 1-Wasserstein distance when the limiting Gaussian probability measure is anisotropic. The dependency on the parameters is completely explicit and the rates of convergence are sharp.

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