Simple Relative Deviation Bounds for Covariance and Gram Matrices

2024-10-08

arXiv.org

AI Summary

Conventional uniform bounds on the non-asymptotic relative deviation of eigenvalues of empirical covariance and Gram matrices fail for small eigenvalues. Method: We develop a general theorem that automatically upgrades existing uniform bounds to sharp, spectrum-wide relative deviation bounds—including vanishing eigenvalues—via a constructive proof framework integrating matrix perturbation theory, random matrix theory, and concentration inequalities. Contribution/Results: The resulting bounds are distribution-free, apply to arbitrary data distributions, and significantly improve upon classical results in high-dimensional, low-sample regimes. Crucially, they provide the first tight, relative-error control over the entire eigenvalue spectrum—including eigenvalues approaching zero—thereby enabling precise spectral analysis even in ill-conditioned or rank-deficient settings. The framework is concise, broadly applicable, and readily transferable to related problems in statistical learning and high-dimensional inference.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Other Foundations of Game Theory & Economic Paradigms

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• Non-asymptotic bounds for covariance matrix eigenvalues
• Sharper control across eigenvalue spectrum
• Convert uniform bounds to relative bounds

Innovation

• Non-asymptotic relative deviation bounds
• Convert uniform bounds to relative
• Sharper eigenvalue spectrum control