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

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 \underline{s}parse \underline{t}ensor \underline{a}lternating \underline{t}hresholding for \underline{s}ingular \underline{v}alue \underline{d}ecomposition (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.