Organizers: Tatiana Brailovskaya, Nicholas Cook
Tuesdays 11:45AM–1:15PM, 270 Gross Hall
To receive email updates please get in touch with the organizers.
This is a weekly participating lunch seminar. The subject matter can basically be characterized as "things that are of interest to the organizers", which tend to involve the quantitative study of high-dimensional random (often discrete) structures, such as random matrices and random graphs, but also related topics in harmonic analysis, additive combinatorics, graph limit theory, large deviations, statistical physics, spectral theory, quantum chaos, ...
Suggested papers
Fall 2024 schedule:
Sep 03: Nick Cook, Matrix concentration inequalities
Abstract
Abstract: I'll discuss the Matrix Bernstein Inequality and one of its proofs, and some applications to community detection and/or covariance estimation.Sep 10: Nick Cook, Green's function and eigenvalue rigidity for Wigner matrices
Abstract
The aim of this talk is to give some background on objects and methods in Theo McKenzie's probability seminar on Sept 12. I'll introduce the use of Green's functions (a.k.a. resolvents) to study the spectrum and eigenvectors of random matrices, illustrated in the relatively simple context of Wigner matrices, and say a bit about the spectrum of random regular graphs.Sep 17: No seminar
Sep 24: Tanya Brailovskaya, What do the n-dimensional sphere and the hypercube have in common? Part I
Abstract
I will start by recalling some classical results regarding concentration of measure on the n-dimensional sphere and the hypercube. Even though these two spaces seem wildly different, they can both be viewed as positively curved in the sense of Ollivier–Ricci curvature, which generalizes the notion of Ricci curvature from Riemannian geometry to arbitrary metric spaces. It was shown by Ollivier that positive curvature implies Gaussian concentration (with some additional minor assumptions that are satisfied by a wide range of discrete and continuous examples). This was a remarkable achievement as previously there was no such unified framework for proving concentration for both the sphere and the hypercube. Our goal in Part 1 is to define Ollivier—Ricci curvature and show how to compute it for the hypercube and the sphere and then state the Gaussian concentration result that follows in these spaces from Ollivier's findings.Oct 01: Tanya Brailovskaya, What do the n-dimensional sphere and the hypercube have in common? Part II
Abstract
In Part 2, we will focus on proving the general Gaussian concentration result for spaces with positive Ollivier—Ricci curvature (Theorem 32 in the paper).Oct 08: Víctor Amaya Carvajal, Stein's method for high-dimensional normal approximation (after E. Meckes)
Abstract
In this talk, we will introduce the one-dimensional Stein's method and its extension to the $d \geq 2$ dimensional case, following Meckes' work. Using Stein's method, we will derive explicit error bounds for the normal approximation in multivariate settings, highlighting the impact of dimensionality on convergence rates.Oct 22: Jack McErlean, An Equivalence Principle for the Spectrum of Random Inner-Product Kernel Matrices with Polynomial Scalings Part 1, (after Y. Lu & H-T Yau)
Abstract
In this first talk of a two-part (edit: three-part!) series, we will introduce random inner-product kernel matrices, whose spectral properties are of practical interest in fields such as ML and statistics. We will give an overview of the characterization of the spectra of such matrices as in the work of Y. Lu & H-T Yau. The goal of the talk will be to introduce relevant and basic notions from random matrix theory in order to state the main result of Lu & Yau, and then give a heuristic for the steps in this analysis of RIPK matrix spectra. We will also work through some example computations for the nearest-neighbor graph matrices. Following this set-up and introduction, Sofia Poinelli will provide a more rigorous treatment of the proof of the main result next week.Oct 29: Jack McErlean, An Equivalence Principle for the Spectrum of Random Inner-Product Kernel Matrices with Polynomial Scalings Part 2, (after Y. Lu & H-T Yau)
Nov 05: No seminar
Nov 12: No seminar
Nov 19: Sofia Poinelli, An Equivalence Principle for the Spectrum of Random Inner-Product Kernel Matrices with Polynomial Scalings Part 3, (after Y. Lu & H-T Yau)
Abstract
In this third talk of a three-part series, we will give an overview of the proof of the main result (Theorem 1) of the Y. Lu & H-T Yau paper. We will establish asymptotic equivalence of Random Inner Product Kernel (RIPK) matrices of a certain form. We will establish the self-consistent equation of the Stieltjes transform of these RIPK matrices, which will tell us about the limiting density of the eigenvalues.Nov 26: Bryan Castillo and Scott McIntyre, Log-Sobolev inequalities under curvature-dimension conditions
Dec 03: Bryan Castillo and Scott McIntyre, Log-Sobolev inequalities under curvature-dimension conditions