Concentration Inequalities for Sample Cross-Covariances

TL;DR

This paper derives sharp, dimension-free concentration bounds for sample cross-covariance matrices, applicable to sub-Gaussian and Gaussian vectors, highlighting their deviation behavior in high-dimensional settings.

Contribution

It provides the first sharp, dimension-free concentration and expectation bounds for sample cross-covariance matrices, including a matching lower bound in the Gaussian case.

Findings

High-probability operator-norm bounds depend on effective ranks.

Matching expectation lower bounds are established for Gaussian vectors.

Results are dimension-free and applicable to high-dimensional data.

Abstract

This paper establishes sharp dimension-free concentration and expectation bounds for the deviation of a sample cross-covariance matrix from its mean. For sub-Gaussian random vectors, we prove a high-probability operator-norm bound governed by the effective ranks of the two marginal covariance matrices. In the Gaussian case, we prove a matching expectation lower bound, allowing arbitrary correlation between the two random vectors.