Dominant subspace and low-rank approximations from block Krylov subspaces without a prescribed gap

TL;DR

This paper introduces a new convergence analysis for block Krylov methods that effectively approximates dominant subspaces and low-rank matrices even when there is no clear singular value gap at the target index.

Contribution

It provides the first convergence guarantees for block Krylov methods without relying on a prescribed singular gap, using the nearest existing singular gaps instead.

Findings

Block Krylov methods can approximate dominant subspaces without a singular gap.

The analysis applies to matrices with equal singular values at the target index.

The approach generalizes previous methods that required a singular gap.

Abstract

We develop a novel convergence analysis of the classical deterministic block Krylov methods for the approximation of $h$-dimensional dominant subspaces and low-rank approximations of matrices $ A\in\mathbb K^{m\times n}$ (where $\mathbb K=\mathbb R$ or $\mathbb C)$ in the case that there is no singular gap at the index $h$ i.e., if $\sigma_h=\sigma_{h+1}$ (where $\sigma_1\geq \ldots\geq \sigma_p\geq 0$ denote the singular values of $ A$, and $p=\min\{m,n\}$). Indeed, starting with a (deterministic) matrix $ X\in\mathbb K^{n\times r}$ with $r\geq h$ satisfying a compatibility assumption with some $h$-dimensional right dominant subspace of $A$, we show that block Krylov methods produce arbitrarily good approximations for both problems mentioned above. Our approach is based on recent work by Drineas, Ipsen, Kontopoulou and Magdon-Ismail on the approximation of structural left dominant subspaces. The main difference between our work and previous work on this topic is that instead of exploiting a singular gap at the prescribed index $h$ (which is zero in this case) we exploit the nearest existing singular gaps.