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Gaps between Singular Values of Sample Covariance Matrices

Nicholas Christoffersen, Kyle Luh, Sean O'Rourke, Calum Shearer

TL;DR

The paper analyzes gaps between consecutive singular values of Σ^{1/2}M with M an n×p random matrix and Σ an n×n deterministic PD matrix, proving that small gaps are unlikely and that the spectrum is simple with high probability. The core methodology blends anti-concentration bounds, regularized LCD techniques, and ε-net arguments to control both compressible and incompressible singular vectors, enabling lower bounds on the squared-singular-value spacings that depend on the location in the spectrum. A principal contribution is resolving Vu's conjecture on simple singular value spectra for random matrices and deriving quantitative gap bounds that reflect edge vs. bulk behavior, consistent with Marchenko–Pastur-type limits. The results have practical implications for numerical linear algebra and graph isomorphism-related problems, including polynomial-time solvability of GI on most bipartite graphs via the non-repetition of singular values.

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 $Σ$ 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 $Σ^{1/2} M$. 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.

Gaps between Singular Values of Sample Covariance Matrices

TL;DR

The paper analyzes gaps between consecutive singular values of Σ^{1/2}M with M an n×p random matrix and Σ an n×n deterministic PD matrix, proving that small gaps are unlikely and that the spectrum is simple with high probability. The core methodology blends anti-concentration bounds, regularized LCD techniques, and ε-net arguments to control both compressible and incompressible singular vectors, enabling lower bounds on the squared-singular-value spacings that depend on the location in the spectrum. A principal contribution is resolving Vu's conjecture on simple singular value spectra for random matrices and deriving quantitative gap bounds that reflect edge vs. bulk behavior, consistent with Marchenko–Pastur-type limits. The results have practical implications for numerical linear algebra and graph isomorphism-related problems, including polynomial-time solvability of GI on most bipartite graphs via the non-repetition of singular values.

Abstract

We study the gaps between consecutive singular values of random rectangular matrices. Specifically, if is an random matrix with independent and identically distributed entries and is a deterministic positive definite matrix, then under some technical assumptions we give lower bounds for the gaps between consecutive singular values of . 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.

Paper Structure

This paper contains 13 sections, 25 theorems, 200 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. The model and assumptions
  4. Main results
  5. Proof Strategy
  6. Applications
  7. Outline
  8. Anti-concentration tools
  9. Compressible singular vectors
  10. Incompressible singular vectors and regularized LCD
  11. Small ball probabilities via LCD
  12. Epsilon nets for regularized LCD level sets and the union bound argument
  13. Proofs of Main Results

Key Result

Theorem 1.3

Let $M$ and $\Sigma$ be as in Assumptions matrix_model and opnorm-assumptions, and assume $K \geq 1$. Then there exist constants $c, C_{main1} > 0$ (depending on $c_{matrix_model}, K,L$ and $m_4$) such that, for any $n^{-c} \leq \alpha \leq c$ and $\delta \geq n^{-c/\alpha}$, where $\mathcal{E}_K$ is defined in def:eK.

Figures (1)

  • Figure 1: A numerical simulation of the spacings of squared singular values of a $2500 \times 2500$ random matrix with Rademacher atom variable and $\Sigma=I$.

Theorems & Definitions (48)

  • Theorem 1.3: Main result
  • Corollary 1.4
  • Corollary 1.5
  • Remark 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Corollary 1.9
  • proof
  • Lemma 2.1
  • proof
  • ...and 38 more