Smooth analysis of the condition number and the least singular value

TL;DR

This paper extends the analysis of the condition number and least singular value of random matrices by including a fixed matrix component, revealing the influence of the fixed matrix on tail bounds and providing nearly optimal estimates.

Contribution

It generalizes previous results by incorporating the fixed matrix's norm into the tail bounds for the least singular value, especially when the fixed matrix has a small norm.

Findings

The fixed matrix M affects tail bounds for the least singular value.

The estimates involve the norm of M, unlike in Gaussian cases.

Results are nearly optimal when the norm of M is small.

Abstract

Let $\a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $\a$ and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M + N_{n}$, generalizing an earlier result of Spielman and Teng for the case when $\a$ is gaussian. Our investigation reveals an interesting fact that the "core" matrix $M$ does play a role on tail bounds for the least singular value of $M+N_{n} $. This does not occur in Spielman-Teng studies when $\a$ is gaussian. Consequently, our general estimate involves the norm $\|M\|$. In the special case when $\|M\|$ is relatively small, this estimate is nearly optimal and extends or refines existing results.