Information Theory and Reliable Communication

Real and Complex Analysis

Recent years have witnessed an explosion in the volume and variety of data collected in all scientific disciplines and industrial settings. Such massive data sets present a number of challenges to researchers in statistics and machine learning. This book provides a self-contained introduction to the area of high-dimensional statistics, aimed at the first-year graduate level. It includes chapters that are focused on core methodology and theory - including tail bounds, concentration inequalities, uniform laws and empirical process, and random matrices - as well as chapters devoted to in-depth exploration of particular model classes - including sparse linear models, matrix models with rank constraints, graphical models, and various types of non-parametric models. With hundreds of worked examples and exercises, this text is intended both for courses and for self-study by graduate students and researchers in statistics, machine learning, and related fields who must understand, apply, and adapt modern statistical methods suited to large-scale data.

High-Dimensional Statistics: A Non-Asymptotic Viewpoint

Random matrices now play a role in many areas of theoretical, applied,and computational mathematics. Therefore, it is desirable to have toolsfor studying random matrices that are flexible, easy to use, and powerful.Over the last fifteen years, researchers have developed a remarkablefamily of results, called matrix concentration inequalities, that achieveall of these goals.This monograph offers an invitation to the field of matrix concentrationinequalities. It begins with some history of random matrix theory;it describes a flexible model for random matrices that is suitablefor many problems; and it discusses the most important matrix concentrationresults. To demonstrate the value of these techniques, thepresentation includes examples drawn from statistics, machine learning,optimization, combinatorics, algorithms, scientific computing, andbeyond.

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An Introduction to Matrix Concentration Inequalities

Lectures on the Combinatorics of Free Probability: The free commutator

This paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix. The analysis requires a matrix extension of the scalar concentration theory developed by Sourav Chatterjee using Stein’s method of exchangeable pairs. When applied to a sum of independent random matrices, this approach yields matrix generalizations of the classical inequalities due to Hoeffding, Bernstein, Khintchine and Rosenthal. The same technique delivers bounds for sums of dependent random matrices and more general matrix-valued functions of dependent random variables.

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Matrix concentration inequalities via the method of exchangeable pairs

Asymptotically liberating sequences of random unitary matrices

We determine the limiting empirical singular value distribution for discrete Fourier transform (DFT) matrices when a random set of columns and rows is removed.

Limiting Empirical Singular Value Distribution of Restrictions of Discrete Fourier Transform Matrices

Let $\mathbb{H}$ be the general, reduced Heisenberg group. Our main result
 establishes the inverse-closedness of a class of integral operators acting on
 $L^{p}(\mathbb{H})$, given by the off-diagonal decay of the kernel. As a consequence of
 this result, we show that if $\alpha_{1}I+S_{f}$, where $S_{f}$ is the operator given by
 convolution with $f$, $f\in L^{1}_{v}(\mathbb{H})$, is invertible in
 $\B(L^{p}(\mathbb{H}))$, then (\alpha_{1}I+S_{f})^{-1}=\alpha_{2}I+S_{g}$, and $g\in
 L^{1}_{v}(\mathbb{H})$. We prove analogous results for twisted convolution operators and
 apply the latter results to a class of Weyl pseudodifferential operators. We briefly
 discuss relevance to mobile communications.

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Inverse-Closedness of a Banach Algebra of Integral Operators on the Heisenberg Group

We study invertibility of matrices of the form D + R, where D is an arbitrary symmetric deterministic matrix and is a symmetric random matrix whose independent entries have continuous distributions with bounded densities. We show that ||(D + R)^(-1)|| = O(n^2) with high probability. The bound is completely independent of D. No moment assumptions are placed on R; in particular the entries of R can be arbitrarily heavy-tailed.

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Brendan Farrell

Papers

Matrix concentration inequalities via the method of exchangeable pairs

Asymptotically liberating sequences of random unitary matrices

Limiting Empirical Singular Value Distribution of Restrictions of Discrete Fourier Transform Matrices

Inverse-Closedness of a Banach Algebra of Integral Operators on the Heisenberg Group

Smoothed analysis of symmetric random matrices with continuous distributions