  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.
