Sensitivity of low-rank matrix recovery
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911936789086208 |
|---|---|
| author | Breiding, Paul Vannieuwenhoven, Nick |
| author_facet | Breiding, Paul Vannieuwenhoven, Nick |
| contents | We characterize the first-order sensitivity of approximately recovering a low-rank matrix from linear measurements, a standard problem in compressed sensing. A special case covered by our analysis is approximating an incomplete matrix by a low-rank matrix. We give an algorithm for computing the associated condition number and demonstrate experimentally how the number of linear measurements affects it.
In addition, we study the condition number of the rank-r matrix approximation problem. It measures in the Frobenius norm by how much an infinitesimal perturbation to an arbitrary input matrix is amplified in the movement of its best rank-r approximation. We give an explicit formula for the condition number, which shows that it does depend on the relative singular value gap between the rth and (r+1)th singular values of the input matrix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_00531 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Sensitivity of low-rank matrix recovery Breiding, Paul Vannieuwenhoven, Nick Numerical Analysis 65J99, 65F22, 65G99, 65K99, 53Z99 We characterize the first-order sensitivity of approximately recovering a low-rank matrix from linear measurements, a standard problem in compressed sensing. A special case covered by our analysis is approximating an incomplete matrix by a low-rank matrix. We give an algorithm for computing the associated condition number and demonstrate experimentally how the number of linear measurements affects it. In addition, we study the condition number of the rank-r matrix approximation problem. It measures in the Frobenius norm by how much an infinitesimal perturbation to an arbitrary input matrix is amplified in the movement of its best rank-r approximation. We give an explicit formula for the condition number, which shows that it does depend on the relative singular value gap between the rth and (r+1)th singular values of the input matrix. |
| title | Sensitivity of low-rank matrix recovery |
| topic | Numerical Analysis 65J99, 65F22, 65G99, 65K99, 53Z99 |
| url | https://arxiv.org/abs/2103.00531 |