Smoothed analysis of the condition number under low-rank perturbations

TL;DR

This paper analyzes how low-rank Gaussian perturbations affect the condition number of matrices, showing they are unlikely to become ill-conditioned and offering computational advantages over dense perturbations.

Contribution

It provides a theoretical analysis of the condition number behavior under low-rank Gaussian perturbations, extending smoothed analysis to this setting with practical computational benefits.

Findings

Low-rank perturbations rarely cause large condition numbers.

The approach is computationally more efficient than dense perturbations.

Results extend to larger rank, symmetric, and complex matrices.

Abstract

Let $M$ be an arbitrary $n$ by $n$ matrix of rank $n-k$. We study the condition number of $M$ plus a \emph{low-rank} perturbation $UV^T$ where $U, V$ are $n$ by $k$ random Gaussian matrices. Under some necessary assumptions, it is shown that $M+UV^T$ is unlikely to have a large condition number. The main advantages of this kind of perturbation over the well-studied dense Gaussian perturbation, where every entry is independently perturbed, is the $O(nk)$ cost to store $U,V$ and the $O(nk)$ increase in time complexity for performing the matrix-vector multiplication $(M+UV^T)x$. This improves the $\Omega(n^2)$ space and time complexity increase required by a dense perturbation, which is especially burdensome if $M$ is originally sparse. Our results also extend to the case where $U$ and $V$ have rank larger than $k$ and to symmetric and complex settings. We also give an application to linear systems solving and perform some numerical experiments. Lastly, barriers in applying low-rank noise to other problems studied in the smoothed analysis framework are discussed.