Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2107.01990 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910571456102400 |
|---|---|
| author | Massey, Pedro |
| author_facet | Massey, Pedro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_01990 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Dominant subspace and low-rank approximations from block Krylov subspaces without a prescribed gap Massey, Pedro Numerical Analysis Functional Analysis 15A18, 65F30 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. |
| title | Dominant subspace and low-rank approximations from block Krylov subspaces without a prescribed gap |
| topic | Numerical Analysis Functional Analysis 15A18, 65F30 |
| url | https://arxiv.org/abs/2107.01990 |