The smallest singular value of inhomogenous random rectangular matrices
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911076442963968 |
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| author | Dabagia, Max Fernandez, Manuel |
| author_facet | Dabagia, Max Fernandez, Manuel |
| contents | Let $A \in \mathbb{R}^{N \times n}$ ($N \geq n$) be a random matrix with with independent entries that have mean 0 variance 1 and bounded $2+β$ moment. We show that the smallest singular value $σ_n(A)$ satisfies
\[
\Pr \left(σ_n(A) \leq \varepsilon(\sqrt{N+1} - \sqrt{n})\right) \leq (C\varepsilon)^{N-n+1} + e^{-cN},
\]
for all $\varepsilon > 0$, where $c,C$ depend only on $β$ and the $2+β$ moment. This extends earlier results of Rudelson and Vershynin, who showed that such lower tail estimates held for rectangular matrices with i.i.d. mean 0 subgaussian entries. When the $2+β$ moment assumption is replaced with a uniform anti-concentration assumption, $\sup_z \Pr\left(|X-z| < a\right) < b$, we show that
\[
\Pr\left(σ_n(A) \leq \varepsilon(\sqrt{N+1} - \sqrt{n})\right) \leq (C\varepsilon\log(1/\varepsilon))^{N-n+1} + e^{-cN},
\]
where $c,C$ now depend only on $a$ and $b$. This extends more recent work of Livshyts, whose showed that such lower tail estimates held for rectrangular matrices with i.i.d. rows. To prove these results we employ a number of new technical ingredients, including a new deviation inequality for the regularized Hilbert-Schmidt norm and a recently proven small ball estimate for the distance between a random vector and a subspace spanned by an inhomogeneous rectangular matrix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14389 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The smallest singular value of inhomogenous random rectangular matrices Dabagia, Max Fernandez, Manuel Probability Metric Geometry Let $A \in \mathbb{R}^{N \times n}$ ($N \geq n$) be a random matrix with with independent entries that have mean 0 variance 1 and bounded $2+β$ moment. We show that the smallest singular value $σ_n(A)$ satisfies \[ \Pr \left(σ_n(A) \leq \varepsilon(\sqrt{N+1} - \sqrt{n})\right) \leq (C\varepsilon)^{N-n+1} + e^{-cN}, \] for all $\varepsilon > 0$, where $c,C$ depend only on $β$ and the $2+β$ moment. This extends earlier results of Rudelson and Vershynin, who showed that such lower tail estimates held for rectangular matrices with i.i.d. mean 0 subgaussian entries. When the $2+β$ moment assumption is replaced with a uniform anti-concentration assumption, $\sup_z \Pr\left(|X-z| < a\right) < b$, we show that \[ \Pr\left(σ_n(A) \leq \varepsilon(\sqrt{N+1} - \sqrt{n})\right) \leq (C\varepsilon\log(1/\varepsilon))^{N-n+1} + e^{-cN}, \] where $c,C$ now depend only on $a$ and $b$. This extends more recent work of Livshyts, whose showed that such lower tail estimates held for rectrangular matrices with i.i.d. rows. To prove these results we employ a number of new technical ingredients, including a new deviation inequality for the regularized Hilbert-Schmidt norm and a recently proven small ball estimate for the distance between a random vector and a subspace spanned by an inhomogeneous rectangular matrix. |
| title | The smallest singular value of inhomogenous random rectangular matrices |
| topic | Probability Metric Geometry |
| url | https://arxiv.org/abs/2408.14389 |