Dominant subspace and low-rank approximations from block Krylov subspaces without a prescribed 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 $A\in K^{m\times n}$ (where $K=R$ or $C)$ in the case that there is no singular gap at the index h i.e., if ${\sigma }_h={\sigma }_{h+1}$ (where ${\sigma }_1\ge \dots \ge {\sigma }_p\ge 0$ denote the singular values of A, and $p=\min \{m,n\}$). Indeed, starting with a (deterministic) matrix $X\in K^{n\times r}$ with $r\ge h$ 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. We include a section with numerical examples that test the performance of our main results.