Randomized block Krylov methods for approximating extreme eigenvalues

TL;DR

This paper develops new theoretical bounds for randomized block Krylov methods, showing their efficiency in approximating extreme eigenvalues and singular values, especially for matrices with polynomial spectral decay.

Contribution

It provides the first theoretical analysis of the performance of randomized block Krylov methods, highlighting the impact of block size and spectral decay on accuracy.

Findings

Efficient spectral norm estimation with few steps for polynomial spectral decay matrices

Algorithm performance depends delicately on block size

Numerical experiments confirm theoretical predictions

Abstract

Randomized block Krylov subspace methods form a powerful class of algorithms for computing the extreme eigenvalues of a symmetric matrix or the extreme singular values of a general matrix. The purpose of this paper is to develop new theoretical bounds on the performance of randomized block Krylov subspace methods for these problems. For matrices with polynomial spectral decay, the randomized block Krylov method can obtain an accurate spectral norm estimate using only a constant number of steps (that depends on the decay rate and the accuracy). Furthermore, the analysis reveals that the behavior of the algorithm depends in a delicate way on the block size. Numerical evidence confirms these predictions.