Sums of random Hermitian matrices and an inequality by Rudelson

TL;DR

This paper presents a new elementary proof of a key inequality used in bounding random sums of rank-one operators, utilizing operator Chernoff bounds, and provides a concentration inequality with explicit constants.

Contribution

It introduces an elementary proof of Rudelson's inequality and derives a concentration inequality for sums of rank-one random matrices with explicit constants.

Findings

Elementary proof of Rudelson's inequality

Concentration inequality for rank-one matrices with explicit constants

Enhanced understanding of random Hermitian matrix sums

Abstract

We give a new, elementary proof of a key inequality used by Rudelson in the derivation of his well-known bound for random sums of rank-one operators. Our approach is based on Ahlswede and Winter's technique for proving operator Chernoff bounds. We also prove a concentration inequality for sums of random matrices of rank one with explicit constants.