Improved analysis of randomized SVD for top-eigenvector approximation

TL;DR

This paper provides a new analysis of the randomized SVD algorithm for approximating the top eigenvector of a matrix, offering tight bounds on the Rayleigh quotient and validating results through experiments.

Contribution

It introduces the first non-trivial bounds on the Rayleigh quotient for randomized SVD with any number of iterations, advancing understanding of its accuracy.

Findings

Tight bounds on the Rayleigh quotient for randomized SVD.

Validation of the method's efficiency and accuracy through experiments.

First analysis providing bounds for any number of iterations.

Abstract

Computing the top eigenvectors of a matrix is a problem of fundamental interest to various fields. While the majority of the literature has focused on analyzing the reconstruction error of low-rank matrices associated with the retrieved eigenvectors, in many applications one is interested in finding one vector with high Rayleigh quotient. In this paper we study the problem of approximating the top-eigenvector. Given a symmetric matrix $\mathbf{A}$ with largest eigenvalue $\lambda_1$, our goal is to find a vector \hu that approximates the leading eigenvector $\mathbf{u}_1$ with high accuracy, as measured by the ratio $R(\hat{\mathbf{u}})=\lambda_1^{-1}{\hat{\mathbf{u}}^T\mathbf{A}\hat{\mathbf{u}}}/{\hat{\mathbf{u}}^T\hat{\mathbf{u}}}$. We present a novel analysis of the randomized SVD algorithm of \citet{halko2011finding} and derive tight bounds in many cases of interest. Notably, this is the first work that provides non-trivial bounds of $R(\hat{\mathbf{u}})$ for randomized SVD with any number of iterations. Our theoretical analysis is complemented with a thorough experimental study that confirms the efficiency and accuracy of the method.