Intrinsic dimension concentration inequalities for self-adjoint operators

TL;DR

This paper introduces new intrinsic-dimension-based concentration inequalities for self-adjoint operators, offering tighter bounds and simplified proofs that unify and improve upon existing ambient and intrinsic dimension inequalities.

Contribution

The paper presents a general master theorem for operator concentration inequalities that applies to various tail behaviors and dependence structures, advancing the theoretical understanding of operator norm concentration.

Findings

Derived tighter concentration inequalities with intrinsic dimension dependence.

Unified intrinsic and ambient dimension inequalities under independence.

Extended bounds to martingale-dependent operators with improved tightness.

Abstract

We derive novel concentration inequalities for the operator norm of the sum of self-adjoint operators that do not explicitly depend on the underlying dimension of the operator, but rather an intrinsic notion of it. Our analysis leads to tighter results (in terms of constants) and simplified proofs. Our results unify the current intrinsic-dimension and ambient-dimension inequalities under independence, strictly improving both categories of bounds (such as by Tropp and Minsker). We present a general master theorem that we instantiate to obtain specific sub-Gaussian, Hoeffding, Bernstein, Bennett, and sub-exponential type inequalities. We also establish widely applicable concentration bounds under martingale dependence that provide tighter control than existing results.