In this work we obtain results related to 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 case 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 $p=\min\{m,n\}$). In order to do this, we develop a novel perspective for the convergence analysis of the classical deterministic block Krylov methods in this context. Indeed, starting with a matrix $ X\in\mathbb K^{n\times r}$ with $r\geq h$ satisfying a compatibility assumption with some $h$-dimensional right dominant subspace, 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 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 $h$ (which is zero in this case) we exploit the nearest existing singular gaps.
翻译:本文討論了當並不存在奇異值在$h$處產生'隙縫',即$\sigma_h=\sigma_{h+1}$ (其中$\sigma_1\geq\ldots\geq\sigma_p\geq0$是矩陣$A\in\mathbb{K}^{m\times n}$ (其中$\mathbb{K}=\mathbb{R}$或$\mathbb{C}$)的奇異值,$p=\min\{m,n\}$)時,$h$維卓越子空間和矩陣的低秩逼近。為此,本文提出了一種新穎的視角,用於分析該上下文中的經典確定性分塊Krylov方法的收斂性。事實上,從一個滿足與某些$h$維右卓越子空間的兼容性假設的矩陣$X\in\mathbb{K}^{n\times r}$ (其中$r\geq h$)開始,我們展示了分塊Krylov方法對上述問題的逼近可以任意好。我們的方法基於Drineas, Ipsen, Kontopoulou和Magdon-Ismail對結構左卓越子空間逼近的最近工作。我們的工作與關於此主題的先前工作的主要區別在於我們利用最近的現有奇異隙縫,而不是利用在這種情況下為零的奇異隙縫。