The smallest singular value of deformed random rectangular matrices

TL;DR

This paper establishes bounds on the smallest singular value of deformed random rectangular matrices, showing near-optimal estimates under certain conditions, which are crucial for understanding matrix invertibility and stability.

Contribution

It provides new probabilistic bounds on the smallest singular value of deformed random matrices with high probability, extending previous results to more general deformations.

Findings

The smallest singular value exceeds a bound proportional to a0a0a0a0(\

a0a0a0a0(a0a0a0a0(\

a0a0a0a0(a0a0a0a0(\

Abstract

We prove an estimate on the smallest singular value of a multiplicatively and additively deformed random rectangular matrix. Suppose $n\le N \le M \le \Lambda N$ for some constant $\Lambda \ge 1$. Let $X$ be an $M\times n$ random matrix with independent and identically distributed entries, which have zero mean, unit variance and arbitrarily high moments. Let $T$ be an $N\times M$ deterministic matrix with comparable singular values $c\le s_{N}(T) \le s_{1}(T) \le c^{-1}$ for some constant $c>0$. Let $A$ be an $N\times n$ deterministic matrix with $\|A\|=O(\sqrt{N})$. Then we prove that for any $\epsilon>0$, the smallest singular value of $TX-A$ is larger than $N^{-\epsilon}(\sqrt{N}-\sqrt{n-1})$ with high probability. If we assume further the entries of $X$ have subgaussian decay, then the smallest singular value of $TX-A$ is at least of the order $\sqrt{N}-\sqrt{n-1}$ with high probability, which is an essentially optimal estimate.