Faster Eigenvector Computation via Shift-and-Invert Preconditioning

TL;DR

This paper introduces faster algorithms for estimating the top eigenvector of a matrix, improving runtime and sample complexity by leveraging shift-and-invert preconditioning and stochastic optimization techniques.

Contribution

It presents novel offline and online eigenvector estimation algorithms with improved theoretical runtime and sample complexity bounds using a robust shift-and-invert framework.

Findings

Improved runtime bounds for offline eigenvector estimation.

Enhanced sample complexity for online eigenvector refinement.

Better dependencies on eigengap, stable rank, and approximation accuracy.

Abstract

We give faster algorithms and improved sample complexities for estimating the top eigenvector of a matrix $\Sigma$ -- i.e. computing a unit vector $x$ such that $x^T \Sigma x \ge (1-\epsilon)\lambda_1(\Sigma)$: Offline Eigenvector Estimation: Given an explicit $A \in \mathbb{R}^{n \times d}$ with $\Sigma = A^TA$, we show how to compute an $\epsilon$ approximate top eigenvector in time $\tilde O([nnz(A) + \frac{d*sr(A)}{gap^2} ]* \log 1/\epsilon )$ and $\tilde O([\frac{nnz(A)^{3/4} (d*sr(A))^{1/4}}{\sqrt{gap}} ] * \log 1/\epsilon )$. Here $nnz(A)$ is the number of nonzeros in $A$, $sr(A)$ is the stable rank, $gap$ is the relative eigengap. By separating the $gap$ dependence from the $nnz(A)$ term, our first runtime improves upon the classical power and Lanczos methods. It also improves prior work using fast subspace embeddings [AC09, CW13] and stochastic optimization [Sha15c], giving significantly better dependencies on $sr(A)$ and $\epsilon$. Our second running time improves these further when $nnz(A) \le \frac{d*sr(A)}{gap^2}$. Online Eigenvector Estimation: Given a distribution $D$ with covariance matrix $\Sigma$ and a vector $x_0$ which is an $O(gap)$ approximate top eigenvector for $\Sigma$, we show how to refine to an $\epsilon$ approximation using $ O(\frac{var(D)}{gap*\epsilon})$ samples from $D$. Here $var(D)$ is a natural notion of variance. Combining our algorithm with previous work to initialize $x_0$, we obtain improved sample complexity and runtime results under a variety of assumptions on $D$. We achieve our results using a general framework that we believe is of independent interest. We give a robust analysis of the classic method of shift-and-invert preconditioning to reduce eigenvector computation to approximately solving a sequence of linear systems. We then apply fast stochastic variance reduced gradient (SVRG) based system solvers to achieve our claims.