The smallest singular value for rectangular random matrices with L\'evy entries

TL;DR

This paper investigates the smallest singular value of rectangular random matrices with heavy-tailed entries following a stable law, providing bounds that extend known results to cases with infinite variance.

Contribution

It establishes new bounds for the smallest singular value of matrices with stable law entries, solving a problem about its asymptotic behavior in the infinite variance regime.

Findings

Lower bound for $\sigma_{min}(X)$ with heavy-tailed entries is established.

Upper bound from recent work is confirmed, with the lower bound being new.

Results apply to matrices with diverging aspect ratios and shifted matrices.

Abstract

Let $X=(x_{ij})\in\mathbb{R}^{N\times n}$ be a rectangular random matrix with i.i.d. entries (we assume $N/n\to\mathbf{a}>1$), and denote by $\sigma_{min}(X)$ its smallest singular value. When entries have mean zero and unit second moment, the celebrated work of Bai-Yin and Tikhomirov show that $n^{-\frac{1}{2}}\sigma_{min}(X)$ converges almost surely to $\sqrt{\mathbf{a}}-1.$ However, little is known when the second moment is infinite. In this work we consider symmetric entry distributions satisfying $\mathbb{P}(|x_{ij}|>t)\sim t^{-\alpha}$ for some $\alpha\in(0,2)$, and prove that $\sigma_{min}(X)$ can be determined up to a log factor with high probability: for any $D>0$, with probability at least $1-n^{-D}$ we have $$C_1n^{\frac{1}{\alpha}}(\log n)^\frac{2(\alpha-2)}{\alpha}\leq \sigma_{min}(X)\leq C_2n^{\frac{1}{\alpha}}(\log n)^\frac{\alpha-2}{2\alpha}$$ for some constants $C_1,C_2>0$. The upper bound was derived in a recent work of Bao, Lee and Xu \cite{bao2024phase2} but the lower bound is new and answers a problem posed in that paper in a weaker form. This appears to be the first determination of $\sigma_{min}(X)$ in the $\alpha$-stable case with a correct leading order of $n$, as previous anti-concentration arguments only yield lower bound $n^\frac{1}{2}$. The same lower bound holds for $\sigma_{min}(X+B)$ for any fixed rectangular matrix $B$ with no assumption on its operator norm. The case of diverging aspect ratio is also computed.