Small-Deviation Inequalities for Sums of Random Matrices

TL;DR

This paper develops small-deviation inequalities for the largest eigenvalues of sums of random matrices, providing dimension-independent bounds applicable to high- and infinite-dimensional settings.

Contribution

It introduces novel small-deviation inequalities for eigenvalues of sums of random matrices, extending the understanding beyond large-deviation regimes.

Findings

Inequalities are independent of matrix dimension.

Applicable to high-dimensional and infinite-dimensional matrices.

Enhances understanding of eigenvalue behavior in small-deviation regimes.

Abstract

Random matrices have played an important role in many fields including machine learning, quantum information theory and optimization. One of the main research focuses is on the deviation inequalities for eigenvalues of random matrices. Although there are intensive studies on the large-deviation inequalities for random matrices, only a few of works discuss the small-deviation behavior of random matrices. In this paper, we present the small-deviation inequalities for the largest eigenvalues of sums of random matrices. Since the resulting inequalities are independent of the matrix dimension, they are applicable to the high-dimensional and even the infinite-dimensional cases.