On Some Extensions of Bernstein's Inequality for Self-adjoint Operators

TL;DR

This paper extends Bernstein's inequality for self-adjoint operators to infinite-dimensional spaces, replacing dimension dependence with an effective rank, thus broadening its applicability in statistics and signal processing.

Contribution

It introduces dimension-free concentration bounds for random matrices based on effective rank, improving upon previous results and enabling applications in infinite-dimensional settings.

Findings

Dimension-free Bernstein inequalities for self-adjoint operators.

Effective rank replaces ambient dimension in bounds.

Extensions applicable to infinite-dimensional spaces.

Abstract

We present some extensions of Bernstein's concentration inequality for random matrices. This inequality has become a useful and powerful tool for many problems in statistics, signal processing and theoretical computer science. The main feature of our bounds is that, unlike the majority of previous related results, they do not depend on the dimension $d$ of the ambient space. Instead, the dimension factor is replaced by the "effective rank" associated with the underlying distribution that is bounded from above by $d$. In particular, this makes an extension to the infinite-dimensional setting possible. Our inequalities refine earlier results in this direction obtained by D. Hsu, S. M. Kakade and T. Zhang.