Optimal Sparse Singular Value Decomposition for High-dimensional High-order Data

Abstract

In this article, we consider the sparse tensor singular value decomposition, which aims for dimension reduction on high-dimensional high-order data with certain sparsity structure. A method named Sparse Tensor Alternating Thresholding for Singular Value Decomposition (STAT-SVD) is proposed. The proposed procedure features a novel double projection \& thresholding scheme, which provides a sharp criterion for thresholding in each iteration. Compared with regular tensor SVD model, STAT-SVD permits more robust estimation under weaker assumptions. Both the upper and lower bounds for estimation accuracy are developed. The proposed procedure is shown to be minimax rate-optimal in a general class of situations. Simulation studies show that STAT-SVD performs well under a variety of configurations. We also illustrate the merits of the proposed procedure on a longitudinal tensor dataset on European country mortality rates.

arXiv comment73 pages; typo fixed

References in corpus (10)

• Sparse PCA: Optimal rates and adaptive estimation
• Guaranteed Non-Orthogonal Tensor Decomposition via Alternating Rank-$1$ Updates
• A statistical model for tensor PCA
• Optimal Estimation of Low Rank Density Matrices
• Rate Optimal Denoising of Simultaneously Sparse and Low Rank Matrices
• Regularized Tensor Factorizations and Higher-Order Principal Components Analysis
• Characterizing Spatiotemporal Transcriptome of Human Brain via Low Rank Tensor Decomposition
• Tensor Decompositions via Two-Mode Higher-Order SVD (HOSVD)
• Homotopy Analysis for Tensor PCA
• Dynamic Tensor Clustering