Gaps between Singular Values of Sample Covariance Matrices
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916988987637760 |
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| author | Christoffersen, Nicholas Luh, Kyle O'Rourke, Sean Shearer, Calum |
| author_facet | Christoffersen, Nicholas Luh, Kyle O'Rourke, Sean Shearer, Calum |
| contents | We study the gaps between consecutive singular values of random rectangular matrices. Specifically, if $M$ is an $n \times p$ random matrix with independent and identically distributed entries and $Σ$ is a $n \times n$ deterministic positive definite matrix, then under some technical assumptions we give lower bounds for the gaps between consecutive singular values of $Σ^{1/2} M$. As a consequence, we show that sample covariance matrices have simple spectrum with high probability. Our results resolve a conjecture of Vu [{\em Probab. Surv.}, 18:179--200, 2021]. We also discuss some applications, including a bound on the spacings of eigenvalues of the adjacency matrix of random bipartite graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15002 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gaps between Singular Values of Sample Covariance Matrices Christoffersen, Nicholas Luh, Kyle O'Rourke, Sean Shearer, Calum Probability 60B20 We study the gaps between consecutive singular values of random rectangular matrices. Specifically, if $M$ is an $n \times p$ random matrix with independent and identically distributed entries and $Σ$ is a $n \times n$ deterministic positive definite matrix, then under some technical assumptions we give lower bounds for the gaps between consecutive singular values of $Σ^{1/2} M$. As a consequence, we show that sample covariance matrices have simple spectrum with high probability. Our results resolve a conjecture of Vu [{\em Probab. Surv.}, 18:179--200, 2021]. We also discuss some applications, including a bound on the spacings of eigenvalues of the adjacency matrix of random bipartite graphs. |
| title | Gaps between Singular Values of Sample Covariance Matrices |
| topic | Probability 60B20 |
| url | https://arxiv.org/abs/2502.15002 |