Comparison theorems for the minimum eigenvalue of a random positive-semidefinite matrix
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915125823275008 |
|---|---|
| author | Tropp, Joel A. |
| author_facet | Tropp, Joel A. |
| contents | This paper establishes a new comparison principle for the minimum eigenvalue of a sum of independent random positive-semidefinite matrices. The principle states that the minimum eigenvalue of the matrix sum is controlled by the minimum eigenvalue of a Gaussian random matrix that inherits its statistics from the summands. This methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, the paper presents short, conceptual proofs of some old and new results in high-dimensional statistics. It also settles a long-standing open question in computational linear algebra about the injectivity properties of very sparse random matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16578 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Comparison theorems for the minimum eigenvalue of a random positive-semidefinite matrix Tropp, Joel A. Probability Numerical Analysis Statistics Theory 15-B52, 60-B20 This paper establishes a new comparison principle for the minimum eigenvalue of a sum of independent random positive-semidefinite matrices. The principle states that the minimum eigenvalue of the matrix sum is controlled by the minimum eigenvalue of a Gaussian random matrix that inherits its statistics from the summands. This methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, the paper presents short, conceptual proofs of some old and new results in high-dimensional statistics. It also settles a long-standing open question in computational linear algebra about the injectivity properties of very sparse random matrices. |
| title | Comparison theorems for the minimum eigenvalue of a random positive-semidefinite matrix |
| topic | Probability Numerical Analysis Statistics Theory 15-B52, 60-B20 |
| url | https://arxiv.org/abs/2501.16578 |