Randomized Block Krylov Methods for Stronger and Faster Approximate Singular Value Decomposition

TL;DR

This paper introduces a randomized block Krylov method for approximate SVD that converges faster and performs better than existing methods, providing strong theoretical guarantees without relying on singular value gaps.

Contribution

It presents the first provable runtime improvement on Simultaneous Iteration using a block Krylov approach, with nearly optimal PCA guarantees for any matrix.

Findings

The new method converges in  O(1/\u221Aepsilon) iterations, outperforming previous algorithms.

It achieves spectral norm low-rank approximation within (1+epsilon) error without dependence on singular value gaps.

Experimental results show substantial performance improvements over existing randomized SVD methods.

Abstract

Since being analyzed by Rokhlin, Szlam, and Tygert and popularized by Halko, Martinsson, and Tropp, randomized Simultaneous Power Iteration has become the method of choice for approximate singular value decomposition. It is more accurate than simpler sketching algorithms, yet still converges quickly for any matrix, independently of singular value gaps. After $\tilde{O}(1/\epsilon)$ iterations, it gives a low-rank approximation within $(1+\epsilon)$ of optimal for spectral norm error. We give the first provable runtime improvement on Simultaneous Iteration: a simple randomized block Krylov method, closely related to the classic Block Lanczos algorithm, gives the same guarantees in just $\tilde{O}(1/\sqrt{\epsilon})$ iterations and performs substantially better experimentally. Despite their long history, our analysis is the first of a Krylov subspace method that does not depend on singular value gaps, which are unreliable in practice. Furthermore, while it is a simple accuracy benchmark, even $(1+\epsilon)$ error for spectral norm low-rank approximation does not imply that an algorithm returns high quality principal components, a major issue for data applications. We address this problem for the first time by showing that both Block Krylov Iteration and a minor modification of Simultaneous Iteration give nearly optimal PCA for any matrix. This result further justifies their strength over non-iterative sketching methods. Finally, we give insight beyond the worst case, justifying why both algorithms can run much faster in practice than predicted. We clarify how simple techniques can take advantage of common matrix properties to significantly improve runtime.