Comparison theorems for the minimum eigenvalue of a random positive-semidefinite matrix
Joel A. Tropp
TL;DR
The paper develops a Gaussian-comparison principle for the minimum eigenvalue of sums of independent random PSD matrices, showing that $\lambda_{\min}$ can be controlled by a Gaussian model with matched first and second moments and a computable weak variance $\sigma_*^2$. Central to the approach are trace mgf comparisons via Stahl's theorem, Bernstein/Stein-type tools, and the weak-variance framework, which yield dimension-free tails in many models. The authors provide concise, conceptual proofs for classical and new results in high-dimensional statistics—such as bounds for sample covariance matrices and sparse covariance structures—and resolve an open question on injectivity for very sparse random maps through recursive Gaussian-comparison arguments. The methodology unifies several PSD-sum settings (random weighting and iid sums) and yields applications to projective designs, randomized subspace injections, and matrix-concentration problems, with notable gains over previous moment-based approaches. Overall, the work offers a powerful, general framework for invertibility-like statistics of random PSD sums with broad implications in high-dimensional statistics and numerical linear algebra.
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.