Note on sampling without replacing from a finite collection of matrices

TL;DR

This note confirms that operator Chernoff bounds remain valid for sampling matrices without replacement, extending previous results and impacting matrix recovery methods.

Contribution

It provides a positive answer to whether operator Chernoff bounds hold without replacement, using classical Hoeffding arguments.

Findings

Operator Chernoff bounds apply without replacement.

Sampling without replacement does not invalidate existing bounds.

Implications for matrix recovery problems are discussed.

Abstract

This technical note supplies an affirmative answer to a question raised in a recent pre-print [arXiv:0910.1879] in the context of a "matrix recovery" problem. Assume one samples m Hermitian matrices X_1, ..., X_m with replacement from a finite collection. The deviation of the sum X_1+...+X_m from its expected value in terms of the operator norm can be estimated by an "operator Chernoff-bound" due to Ahlswede and Winter. The question arose whether the bounds obtained this way continue to hold if the matrices are sampled without replacement. We remark that a positive answer is implied by a classical argument by Hoeffding. Some consequences for the matrix recovery problem are sketched.