Gaps between Singular Values of Sample Covariance Matrices

TL;DR

This paper investigates the gaps between singular values of sample covariance matrices, establishing lower bounds and showing that they typically have simple spectra, thereby resolving a conjecture and exploring applications to random graph adjacency matrices.

Contribution

It provides new bounds on singular value gaps of sample covariance matrices and proves they usually have simple spectra, confirming a conjecture by Vu.

Findings

Sample covariance matrices have simple spectra with high probability.

Lower bounds are established for gaps between consecutive singular values.

Applications include bounds on eigenvalue spacings in random bipartite graph adjacency matrices.

Abstract

We study the gaps between consecutive singular values of random rectangular matrices. Specifically, if $M$ is an $n \times p$ random matrix with independent and identically distributed entries and $\Sigma$ is a $n \times n$ deterministic positive definite matrix, then under some technical assumptions we give lower bounds for the gaps between consecutive singular values of $\Sigma^{1/2} M$. As a consequence, we show that sample covariance matrices have simple spectrum with high probability. Our results resolve a conjecture of Vu [{\em Probab. Surv.}, 18:179--200, 2021]. We also discuss some applications, including a bound on the spacings of eigenvalues of the adjacency matrix of random bipartite graphs.