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

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 $\underset{\lambda \in \mathbb{R}}{\sup }\mathbb{P}\{|a_{11}-\lambda |\le 1\}\le 1-\beta$ and any non-random $N\times n$ matrix $B$, the smallest singular value $s_n$ of $A+B$ satisfies $\mathbb{P}\{s_n(A+B)\le u\sqrt{N}\}\le \exp (-vN)$. The result holds without any moment assumptions on distribution of the entries of $A$.

References in corpus (1)

• Smoothed analysis of algorithms