Improved analysis of randomized SVD for top-eigenvector approximation

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 A with largest eigenvalue lambda(1), our goal is to find a vector (u) over cap that approximates the leading eigenvector u(1) with high accuracy, as measured by the ratio R((u) over cap) = lambda(-1)(1) (u) over cap (T)A (u) over cap/(u) over cap (T)(u) over cap. We present a novel analysis of the randomized SVD algorithm of Halko et al. (2011b) and derive tight bounds in many cases of interest. Notably, this is the first work that provides non-trivial bounds for approximating the ratio R((u) over cap) using 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. ; QC 20220905