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

By Cameron Musco, Christopher Musco. Neural Information Processing Systems, 2015

307 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/39c5029a1da27d449bc78440642755a3b912e14e

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Citations in the corpus

Randomized numerical linear algebra: Foundations and algorithms (2020, background): “See (Musco and Musco 2015, Yuan et al. 2018, Tropp 2018) for some results.”