Comparison theorems for the extreme eigenvalues of a random symmetric matrix

TL;DR

This paper introduces a comparison theorem for the maximum eigenvalue of sums of independent random symmetric matrices, leveraging Gaussian matrix properties to improve bounds in various fields.

Contribution

It presents a new comparison theorem that dominates the eigenvalues of matrix sums with Gaussian matrices, strengthening previous results and enabling new applications.

Findings

Improved eigenvalue bounds for random matrices in spectral graph theory

First complete proof of injectivity for sparse random dimension reduction maps

Enhanced tools for analyzing eigenvalues in high-dimensional settings

Abstract

This paper establishes a comparison theorem for the maximum eigenvalue of a sum of independent random symmetric matrices. The theorem states that the maximum eigenvalue of the matrix sum is dominated by the maximum eigenvalue of a Gaussian random matrix that inherits its statistics from the sum, and it strengthens previous results of this type. Corollaries address the minimum eigenvalue and the spectral norm. The comparison methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, the paper improves on existing eigenvalue bounds for random matrices arising in spectral graph theory, quantum information theory, high-dimensional statistics, and numerical linear algebra. In particular, these techniques deliver the first complete proof that a sparse random dimension reduction map has the injectivity properties conjectured by Nelson & Nguyen in 2013.