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.

翻译：本文研究稀疏张量奇异值分解方法，旨在对具有特定稀疏结构的高维高阶数据进行降维处理。我们提出了一种名为稀疏张量交替阈值奇异值分解（STAT-SVD）的新方法。该方法的创新之处在于采用了一种新颖的双重投影与阈值化方案，为每次迭代中的阈值选择提供了精确的判定准则。与常规张量SVD模型相比，STAT-SVD在更弱的假设条件下仍能实现更稳健的估计。我们推导了估计精度的上界与下界，并证明该方法在广泛情境下达到极小极大最优速率。模拟研究表明STAT-SVD在多种配置条件下均表现优异。最后，我们通过欧洲国家死亡率纵向张量数据集展示了所提方法的优势。