Singular vectors under random perturbation

TL;DR

This paper investigates how random noise affects the singular vectors of low-rank matrices, providing improved estimates over classical worst-case bounds by leveraging high-dimensional geometric techniques.

Contribution

It introduces a novel approach using high-dimensional geometry to derive better perturbation bounds for singular vectors under random noise, surpassing classical worst-case results.

Findings

Improved bounds for singular vector perturbations with random noise

Demonstrates advantages of geometric methods over traditional techniques

Applicable to large, low-rank matrices in statistical and numerical contexts

Abstract

Computing the first few singular vectors of a large matrix is a problem that frequently comes up in statistics and numerical analysis. Given the presence of noise, exact calculation is hard to achieve, and the following problem is of importance: \vskip2mm \centerline {\it How much a small perturbation to the matrix changes the singular vectors ?} \vskip2mm Answering this question, classical theorems, such as those of Davis-Kahan and Wedin, give tight estimates for the worst-case scenario. In this paper, we show that if the perturbation (noise) is random and our matrix has low rank, then better estimates can be obtained. Our method relies on high dimensional geometry and is different from those used an earlier papers.