Improved analysis of randomized SVD for top-eigenvector approximation
This work addresses a fundamental problem in fields requiring eigenvector computation, offering improved theoretical guarantees for a widely used algorithm, though it is incremental as it builds on existing randomized SVD methods.
The paper tackles the problem of approximating the top eigenvector of a symmetric matrix, focusing on the Rayleigh quotient accuracy rather than reconstruction error, and presents a novel analysis of randomized SVD that provides tight bounds for the first time with any number of iterations, supported by experimental confirmation of efficiency and accuracy.
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 $λ_1$, our goal is to find a vector \hu that approximates the leading eigenvector $\mathbf{u}_1$ with high accuracy, as measured by the ratio $R(\hat{\mathbf{u}})=λ_1^{-1}{\hat{\mathbf{u}}^T\mathbf{A}\hat{\mathbf{u}}}/{\hat{\mathbf{u}}^T\hat{\mathbf{u}}}$. 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.