Concentration and moment inequalities for sums of independent heavy-tailed random matrices

TL;DR

This paper develops new concentration and moment inequalities for sums of independent heavy-tailed random matrices, emphasizing intrinsic dimensions like effective rank, with applications to covariance matrices and empirical processes.

Contribution

It introduces Fuk-Nagaev and Rosenthal-type inequalities tailored for heavy-tailed matrices, focusing on low-order moments and intrinsic dimensions, advancing the theoretical understanding of such matrices.

Findings

New moment inequalities for sample covariance matrices with heavy tails

Sharper bounds for empirical process moments

Effective dimension-based bounds outperform ambient dimension estimates

Abstract

We prove Fuk-Nagaev and Rosenthal-type inequalities for sums of independent random matrices, focusing on the situation when the norms of the matrices possess finite moments of only low orders. Our bounds depend on the ``intrinsic'' dimensional characteristics such as the effective rank, as opposed to the dimension of the ambient space. We illustrate the advantages of such results through several applications, including new moment inequalities for sample covariance matrices and their eigenvectors when the underlying distribution is heavy-tailed. Moreover, we demonstrate that our techniques yield sharpened versions of moment inequalities for empirical processes.