Stronger and Faster Approximate Singular Value Decomposition via the Block Lanczos Method
Cameron Musco,Christopher Musco
(Submitted on 21 Apr 2015 (v1), last revised 1 Jul 2015 (this version, v3))
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 variant of the classic Block Lanczos method 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 like Block Lanczos 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 Lanczos 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.
Subjects:Data Structures and Algorithms (cs.DS); Learning (cs.LG); Numerical Analysis (cs.NA)
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