Deterministic sketching for Krylov subspace methods

TL;DR

This paper introduces a deterministic sketching approach for Krylov subspace methods, providing guaranteed subspace embeddings and comparable performance to randomized methods in solving matrix problems.

Contribution

It presents a novel deterministic sketching technique for Krylov methods, replacing randomness with row subset selection for reliable subspace embeddings.

Findings

Deterministic sketching yields subspace embeddings with probability 1.

Deterministically sketched Krylov methods perform similarly to randomized ones.

The approach applies to matrix functions, linear systems, and eigenvalue problems.

Abstract

Randomized sketching is currently introduced into every area of numerical linear algebra. In Krylov subspace methods, it allows runtime savings at the cost of small accuracy reductions. This work offers a different view on sketching in Krylov methods by analyzing what subspace embeddings are obtained by arbitrary sketching matrices. The analysis gives rise to a deterministic sketching approach leveraging row subset selection techniques that yield subspace embeddings holding with probability 1. We propose deterministically sketched Krylov methods for matrix functions, linear systems, and eigenvalue problems that show a similar performance to their randomly sketched counterparts.