Weighted sampling of outer products

TL;DR

This paper provides a simplified analysis of a randomized matrix multiplication approximation scheme using weighted sampling of outer products, establishing bounds based on stable rank and spectral norm error.

Contribution

It offers a straightforward proof for the sampling complexity of approximating matrix products using outer products with spectral norm guarantees.

Findings

Sampling complexity depends on stable ranks of matrices.

Spectral norm error bound achieved with high probability.

Analysis leverages matrix Bernstein's inequality.

Abstract

This note gives a simple analysis of the randomized approximation scheme for matrix multiplication of Drineas et al (2006) with a particular sampling distribution over outer products. The result follows from a matrix version of Bernstein's inequality. To approximate the matrix product $AB^\top$ to spectral norm error $\varepsilon\|A\|\|B\|$, it suffices to sample on the order of $(\mathrm{sr}(A) \vee \mathrm{sr}(B)) \log(\mathrm{sr}(A) \wedge \mathrm{sr}(B)) / \varepsilon^2$ outer products, where $\mathrm{sr}(M)$ is the stable rank of a matrix $M$.