Randomized block-Krylov subspace methods for low-rank approximation of matrix functions

TL;DR

This paper introduces a theoretically justified randomized block-Krylov subspace method for low-rank approximation of matrix functions, outperforming previous approaches by providing strong error bounds.

Contribution

It provides the first rigorous theoretical analysis and error bounds for a new randomized block-Krylov method applied to low-rank matrix function approximation.

Findings

The method achieves higher accuracy with fewer matvecs.

Error bounds validate the method's efficiency and reliability.

Outperforms naive Krylov-based approaches in experiments.

Abstract

The randomized SVD is a method to compute an inexpensive, yet accurate, low-rank approximation of a matrix. The algorithm assumes access to the matrix through matrix-vector products (matvecs). Therefore, when we would like to apply the randomized SVD to a matrix function, $f(A)$, one needs to approximate matvecs with $f(A)$ using some other algorithm, which is typically treated as a black-box. Chen and Hallman (SIMAX 2023) argued that, in the common setting where matvecs with $f(A)$ are approximated using Krylov subspace methods (KSMs), a more efficient low-rank approximation is possible if we open this black-box. They present an alternative approach that significantly outperforms the naive combination of KSMs with the randomized SVD, although the method lacked theoretical justification. In this work, we take a closer look at the method, and provide strong and intuitive error bounds that justify its excellent performance for low-rank approximation of matrix functions.