Dominant subspace and low-rank approximations from block Krylov subspaces without a prescribed gap
Pedro Massey
TL;DR
This work addresses the problem of obtaining reliable dominant-subspace and low-rank approximations for $A\in\mathbb{K}^{m\times n}$ when there is no singular gap at index $h$ (i.e., $\sigma_h=\sigma_{h+1}$). It extends the deterministic block Krylov analysis by adapting DIKM-I techniques to the gapless setting and showing that, from a fixed starting guess $X$ compatible with some $h$-dimensional right dominant subspace, the block Krylov subspace $\mathcal{K}_\ell$ contains an $h$-dimensional left dominant subspace that is arbitrarily close in principal angles. The paper also provides explicit bounds for the low-rank approximation $\hat{A}_h=\hat{U}_h\hat{U}_h^*A$, including both spectral and Frobenius errors, and shows convergence to the optimal $A_h$ as the power parameter grows, with speeds governed by the gaps $\gamma_j$ and $\gamma_k$ via Chebyshev-filtered iterations. Overall, the results extend gap-based convergence analyses to the gapless case, offering a deterministic path to near-optimal dominant subspaces and low-rank approximations under mild compatibility assumptions.
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 $σ_h=σ_{h+1}$ (where $σ_1\geq \ldots\geq σ_p\geq 0$ 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.