The smallest singular value of random rectangular matrices with no moment assumptions on entries

TL;DR

This paper establishes exponential bounds on the smallest singular value of large rectangular random matrices with independent entries, without requiring any moment conditions on the entries.

Contribution

It proves that the smallest singular value of such matrices is well-behaved with high probability, even without moment assumptions, extending previous results.

Findings

Exponential decay of the probability that the smallest singular value is small.

Results hold for matrices with no moment assumptions on entries.

Applicable to matrices with arbitrary deterministic perturbations.

Abstract

Let $\delta>1$ and $\beta>0$ be some real numbers. We prove that there are positive $u,v,N_0$ depending only on $\beta$ and $\delta$ with the following property: for any $N,n$ such that $N\ge \max(N_0,\delta n)$, any $N\times n$ random matrix $A=(a_{ij})$ with i.i.d. entries satisfying $\sup\limits_{\lambda\in {\mathbb R}}{\mathbb P}\bigl\{|a_{11}-\lambda|\le 1\bigr\}\le 1-\beta$ and any non-random $N\times n$ matrix $B$, the smallest singular value $s_n$ of $A+B$ satisfies ${\mathbb P}\bigl\{s_n(A+B)\le u\sqrt{N}\bigr\}\le \exp(-vN)$. The result holds without any moment assumptions on distribution of the entries of $A$.