A Sparse Singular Value Decomposition Method for High-Dimensional Data

Abstract

We present a new computational approach to approximating a large, noisy data table by a low-rank matrix with sparse singular vectors. The approximation is obtained from thresholded subspace iterations that produce the singular vectors simultaneously, rather than successively as in competing proposals. We introduce novel ways to estimate thresholding parameters, which obviate the need for computationally expensive cross-validation. We also introduce a way to sparsely initialize the algorithm for computational savings that allow our algorithm to outperform the vanilla singular value decomposition (SVD) on the full data table when the signal is sparse. A comparison with two existing sparse SVD methods suggests that our algorithm is computationally always faster and statistically always at least comparable to the better of the two competing algorithms. Supplementary materials for the article are available in an online appendix. An R package ssvd implementing the algorithms introduced in this article is available on CRAN.

Key Words:

• Cross-validation
• Denoising
• Low-rank matrix approximation
• Penalization
• Power iterations
• Principal component analysis
• Thresholding

ACKNOWLEDGMENTS

Andreas Buja was partially supported by NSF grants DSM-1007689 and DSM-1007657. Dan Yang was partially supported by NSF grant DMS-1127914 to the Statistical and Applied Mathematical Sciences Institute.