Improved bounds for randomized Schatten norm estimation of numerically low-rank matrices

TL;DR

This paper improves the theoretical bounds on the variance of a stochastic estimator for Schatten norms, showing tighter estimates for matrices that are numerically low-rank, supported by numerical experiments.

Contribution

It provides new, tighter variance bounds for a stochastic Schatten norm estimator, enhancing understanding of its accuracy for low-rank matrices.

Findings

New upper bounds for estimator variance

Tighter bounds compared to previous work

Numerical experiments confirm theoretical improvements

Abstract

In this work, we analyze the variance of a stochastic estimator for computing Schatten norms of matrices. The estimator extracts information from a single sketch of the matrix, that is, the product of the matrix with a few standard Gaussian random vectors. While this estimator has been proposed and used in the literature before, the existing variance bounds are often pessimistic. Our work provides a new upper bound and estimates of the variance of this estimator. These theoretical findings are supported by numerical experiments, demonstrating that the new bounds are significantly tighter than the existing ones in the case of numerically low-rank matrices.