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Simple Relative Deviation Bounds for Covariance and Gram Matrices

Daniel Barzilai, Ohad Shamir

TL;DR

This work tackles non-asymptotic, relative bounds for the eigenvalues of empirical covariance and Gram matrices, addressing deficiencies of traditional uniform bounds which can be loose for smaller eigenvalues. The authors introduce a simple, general reduction to isotropic random vectors via $Z = X\Sigma^{-1/2}$ and prove a single versatile result (Theorem $\text{gen_bound}$) that translates uniform bounds on $Z^T Z$ into relative eigenvalue bounds for $\hat{\Sigma}$. The main results cover both low- and high-dimensional regimes, with variants for sub-Gaussian as well as bounded-norm settings, and include sharp bounds for square and nearly-square matrices. The approach is simple to apply and broadly applicable, offering sharper control across the spectrum and extending to a range of distributions and dimensions with practical impact for high-dimensional statistics and kernel methods.

Abstract

We provide non-asymptotic, relative deviation bounds for the eigenvalues of empirical covariance and Gram matrices in general settings. Unlike typical uniform bounds, which may fail to capture the behavior of smaller eigenvalues, our results provide sharper control across the spectrum. Our analysis is based on a general-purpose theorem that allows one to convert existing uniform bounds into relative ones. The theorems and techniques emphasize simplicity and should be applicable across various settings.

Simple Relative Deviation Bounds for Covariance and Gram Matrices

TL;DR

This work tackles non-asymptotic, relative bounds for the eigenvalues of empirical covariance and Gram matrices, addressing deficiencies of traditional uniform bounds which can be loose for smaller eigenvalues. The authors introduce a simple, general reduction to isotropic random vectors via and prove a single versatile result (Theorem ) that translates uniform bounds on into relative eigenvalue bounds for . The main results cover both low- and high-dimensional regimes, with variants for sub-Gaussian as well as bounded-norm settings, and include sharp bounds for square and nearly-square matrices. The approach is simple to apply and broadly applicable, offering sharper control across the spectrum and extending to a range of distributions and dimensions with practical impact for high-dimensional statistics and kernel methods.

Abstract

We provide non-asymptotic, relative deviation bounds for the eigenvalues of empirical covariance and Gram matrices in general settings. Unlike typical uniform bounds, which may fail to capture the behavior of smaller eigenvalues, our results provide sharper control across the spectrum. Our analysis is based on a general-purpose theorem that allows one to convert existing uniform bounds into relative ones. The theorems and techniques emphasize simplicity and should be applicable across various settings.
Paper Structure (13 sections, 19 theorems, 35 equations)

This paper contains 13 sections, 19 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Reduction to Isotropic Random Vectors
  3. Setting and Preliminaries
  4. Main Results
  5. Low-Dimensional Case $(d \leq n)$
  6. Bounds Without a Dependence on Sub-Gaussian Norm
  7. Variants and Extensions
  8. High-Dimensional Case ($d \geq n$)
  9. Bounds for Square and Nearly Square Matrices $(d\approx n)$
  10. Omitted Proofs
  11. Proof of Proposition \ref{['prop:ostrowski']}
  12. Auxiliary Lemmas for Proposition \ref{['prop:ostrowski']}
  13. Proof of Thm. \ref{['thm:high_dim']}

Key Result

Proposition 3

Let $Z \in\mathbb{C}^{n\times d}$ for some $n,d\in\mathbb{N}$, and $0\preceq \Sigma\in\mathbb{C}^{d\times d}$ be p.s.d. Then for any $1\leq i\leq \min(n,d)$ it holds that

Theorems & Definitions (22)

  • Definition 1
  • Remark 2
  • Definition 3
  • Proposition 3
  • Theorem 4
  • Theorem 5: vershynin2018high Theorem 4.6.1
  • Theorem 6: Low-Dimensional, Sub-Gaussian
  • Theorem 7: Gaussian Entries
  • Theorem 8: Low-Dimensional, Bounded Norm
  • Theorem 9: High-Dimensional, Sub-Gaussian Entries
  • ...and 12 more