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

TL;DR

This paper introduces STAT-SVD, a novel sparse tensor SVD method that improves dimension reduction in high-dimensional, high-order data with sparsity, offering theoretical optimality and practical effectiveness.

Contribution

The paper proposes a new double projection  thresholding scheme for sparse tensor SVD, achieving minimax rate-optimal estimation under weaker assumptions.

Findings

STAT-SVD outperforms existing methods in simulations.

Theoretical bounds confirm minimax optimality.

Application to mortality data demonstrates practical utility.

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.