Comparison theorems for the minimum eigenvalue of a random positive-semidefinite matrix

TL;DR

This paper introduces a comparison principle linking the minimum eigenvalue of sums of random positive-semidefinite matrices to Gaussian matrices, enabling new proofs and resolving open questions in high-dimensional statistics and linear algebra.

Contribution

It presents a novel comparison theorem for eigenvalues, facilitating analysis of random matrices and providing insights into their spectral properties.

Findings

Established a new comparison principle for eigenvalues.

Provided simplified proofs for existing high-dimensional statistical results.

Resolved an open problem on the injectivity of sparse random matrices.

Abstract

This paper establishes a new comparison principle for the minimum eigenvalue of a sum of independent random positive-semidefinite matrices. The principle states that the minimum eigenvalue of the matrix sum is controlled by the minimum eigenvalue of a Gaussian random matrix that inherits its statistics from the summands. This methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, the paper presents short, conceptual proofs of some old and new results in high-dimensional statistics. It also settles a long-standing open question in computational linear algebra about the injectivity properties of very sparse random matrices.