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Computing the top eigenvectors of a matrix is a problem of fundamentalinterest to various fields. While the majority of the literature has focused onanalyzing the reconstruction error of low-rank matrices associated with theretrieved eigenvectors, in many applications one is interested in finding onevector with high Rayleigh quotient. In this paper we study the problem ofapproximating the top-eigenvector. Given a symmetric matrix $\mathbf{A}$ withlargest eigenvalue $\lambda_1$, our goal is to find a vector \hu thatapproximates the leading eigenvector $\mathbf{u}_1$ with high accuracy, asmeasured 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. Ourtheoretical analysis is complemented with a thorough experimental study thatconfirms the efficiency and accuracy of the method.

Improved analysis of randomized SVD for top-eigenvector approximation