The smallest singular value for rectangular random matrices with Lévy entries

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 $$C_1n^{\frac{1}{\alpha}}(\log n)^\frac{5(\alpha-2)}{2\alpha}\leq \sigma_{min}(X)\leq C_2n^{\frac{1}{\alpha}}(\log n)^\frac{\alpha-2}{2\alpha}$$ for some constants $C_1,C_2>0$. 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 ant-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. Geometrically, the lower bound shows that the random polytope $X^*(B_1^N)$ generated by heavy-tail distributions will with very high probability contain Euclidean balls $B_2^n$ of a much larger radius compared to its Gaussian counterpart.