  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.
