On generic chaining and the smallest singular value of random matrices with heavy tails

TL;DR

This paper introduces a versatile chaining technique to analyze the smallest singular value of heavy-tailed random matrices and provides non-asymptotic bounds for empirical processes in high-dimensional probability.

Contribution

It develops a general chaining method applicable to heavy-tailed data and derives non-asymptotic singular value bounds and empirical process estimates.

Findings

Non-asymptotic Bai-Yin type theorem for heavy-tailed matrices

Sharp empirical process bounds for isotropic log-concave measures

Versatile chaining method for complex empirical processes

Abstract

We present a very general chaining method which allows one to control the supremum of the empirical process $\sup_{h \in H} |N^{-1}\sum_{i=1}^N h^2(X_i)-\E h^2|$ in rather general situations. We use this method to establish two main results. First, a quantitative (non asymptotic) version of the classical Bai-Yin Theorem on the singular values of a random matrix with i.i.d entries that have heavy tails, and second, a sharp estimate on the quadratic empirical process when $H=\{\inr{t,\cdot} : t \in T\}$, $T \subset \R^n$ and $\mu$ is an isotropic, unconditional, log-concave measure.