[Paper Review] Note on sampling without replacing from a finite collection of matrices

David Groß, Vincent Nesme|arXiv (Cornell University)|Jan 15, 2010

Mathematical Analysis and Transform Methods13 references64 citations

TL;DR

This paper establishes that operator Chernoff-type bounds for sums of random matrices—previously derived for sampling with replacement—also hold when sampling without replacement from a finite collection of Hermitian matrices. Using a classical argument by Hoeffding, the authors show that the moment generating function of the sum under without-replacement sampling is stochastically dominated by that under with-replacement sampling, preserving concentration inequalities crucial for low-rank matrix recovery algorithms.

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.

Motivation & Objective

• To resolve a technical gap in matrix recovery theory where sampling without replacement introduces dependence, complicating analysis.
• To demonstrate that existing concentration bounds for independent (with-replacement) matrix sampling remain valid under without-replacement sampling.
• To provide a rigorous justification for using with-replacement bounds in the without-replacement setting, simplifying algorithmic analysis.
• To support the validity of matrix recovery algorithms relying on operator norm concentration, particularly in the context of quantum state tomography and low-rank matrix reconstruction.

Proposed method

• Applies Hoeffding's classical argument on stochastic dominance of sampling without replacement over with replacement.
• Constructs a random function Z that generates with-replacement samples from a without-replacement sample via a two-stage randomization process.
• Uses the identity that X and Z(Y) are identically distributed to equate expectations of functions of the sum.
• Applies Jensen’s inequality to convex functions of the matrix sum, particularly the trace of the matrix exponential.
• Leverages the convexity of c ↦ tr(exp(λc)) for Hermitian matrices to extend scalar concentration results to the matrix setting.
• Derives moment generating function domination: M_Y(λ) ≤ M_X(λ), implying that tail bounds for with-replacement sampling apply to without-replacement sampling.

Experimental results

Research questions

• RQ1 Can the operator Chernoff bound for sums of independent random matrices be extended to the case of sampling without replacement?
• RQ2 Does the stochastic dominance of with-replacement sampling over without-replacement sampling preserve concentration inequalities for matrix-valued sums?
• RQ3 Is it valid to use with-replacement bounds in the analysis of matrix recovery algorithms when sampling is actually done without replacement?
• RQ4 What is the relationship between the moment generating functions of sums under with-replacement and without-replacement sampling?
• RQ5 Can classical probabilistic inequalities like Hoeffding’s be adapted to matrix-valued random variables to preserve concentration under sampling without replacement?

Key findings

• The moment generating function of the sum of matrices sampled without replacement is stochastically dominated by that of the with-replacement case.
• As a consequence, the operator Chernoff bound from Ahlswede and Winter remains valid for sampling without replacement.
• The operator norm concentration bound Pr[‖S‖ > t] ≤ 2n exp(−t²/(4V)) for t ≤ 2V/c and Pr[‖S‖ > t] ≤ 2n exp(−t/(2c)) for larger t holds for both sampling schemes.
• The bound on the operator norm of the sampling operator remains effective in the without-replacement setting, avoiding the need for worst-case estimates.
• The analysis of matrix recovery algorithms based on random sampling can safely use with-replacement bounds even when sampling is done without replacement.
• The result justifies the use of simplified analysis frameworks in matrix recovery, such as those in Gross (2009), without loss of generality in the without-replacement case.