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

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