On the Smallest Singular Value of Log-Concave Random Matrices
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908500906475520 |
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| author | Fernandez V, Manuel Livshyts, Galyna V. Mui, Stephanie |
| author_facet | Fernandez V, Manuel Livshyts, Galyna V. Mui, Stephanie |
| contents | Let $A$ be an $N\times n$ random matrix whose entries are coordinates of an isotropic log-concave random vector in $\mathbb{R}^{Nn}$. We prove sharp lower tail estimates for the smallest singular value of $A$ in the following cases: (1) when $N=n$ and $A$ is drawn from an unconditional distribution, with no independence assumption; (2) when the columns of $A$ are independent and $N\geq n$; (3) when $A$ is sufficiently tall, that is $N\geq (1+λ)n$ for any positive constant $λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17745 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Smallest Singular Value of Log-Concave Random Matrices Fernandez V, Manuel Livshyts, Galyna V. Mui, Stephanie Probability Combinatorics Functional Analysis Metric Geometry Let $A$ be an $N\times n$ random matrix whose entries are coordinates of an isotropic log-concave random vector in $\mathbb{R}^{Nn}$. We prove sharp lower tail estimates for the smallest singular value of $A$ in the following cases: (1) when $N=n$ and $A$ is drawn from an unconditional distribution, with no independence assumption; (2) when the columns of $A$ are independent and $N\geq n$; (3) when $A$ is sufficiently tall, that is $N\geq (1+λ)n$ for any positive constant $λ$. |
| title | On the Smallest Singular Value of Log-Concave Random Matrices |
| topic | Probability Combinatorics Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2508.17745 |