Invertibility via distance for non-centered random matrices with continuous distributions

TL;DR

This paper establishes a bound on the probability that the smallest singular value of a non-centered random matrix plus a fixed matrix is small, under certain density conditions on the rows, extending results to non-centered log-concave vectors.

Contribution

It introduces a novel method to bound the smallest singular value of non-centered random matrices without relying on covering arguments or Gaussian-specific techniques.

Findings

Bound on the probability of small singular values for non-centered matrices

Applicable to matrices with independent log-concave rows with identity covariance

Method avoids traditional covering and Gaussian-specific approaches

Abstract

Let $A$ be an $n\times n$ random matrix with independent rows $R_1(A),\dots,R_n(A)$, and assume that for any $i\leq n$ and any three-dimensional linear subspace $F\subset {\mathbb R}^n$ the orthogonal projection of $R_i(A)$ onto $F$ has distribution density $\rho(x):F\to{\mathbb R}_+$ satisfying $\rho(x)\leq C_1/\max(1,\|x\|_2^{2000})$ ($x\in F$) for some constant $C_1>0$. We show that for any fixed $n\times n$ real matrix $M$ we have $${\mathbb P}\{s_{\min}(A+M)\leq t n^{-1/2}\}\leq C'\, t,\quad\quad t>0,$$ where $C'>0$ is a universal constant. In particular, the above result holds if the rows of $A$ are independent centered log-concave random vectors with identity covariance matrices. Our method is free from any use of covering arguments, and is principally different from a standard approach involving a decomposition of the unit sphere and coverings, as well as an approach of Sankar-Spielman-Teng for non-centered Gaussian matrices.