Tensor train decomposition is one of the most powerful approaches for processing high-dimensional data. For low-rank tensor train decomposition of large tensors, the alternating least squares (ALS) algorithm is widely used by updating each core tensor alternatively. However, it may suffer from the curse of dimensionality due to the large scale of subproblems. In this paper, a novel randomized proximal ALS algorithm is proposed for low-rank tensor train decomposition by using TensorSketch, which allows for efficient implementation via fast Fourier transform. The theoretical lower bounds of sketch size are estimated for approximating the optimal value of subproblems. Numerical experiments on synthetic and real-world data also demonstrate the effectiveness and efficiency of the proposed algorithm.
On low-rank tensor train approximability for linear nearest neighbor systems
Zuse Institute Berlin（柏林齐苏研究所）；Theory Department, Fritz Haber Institute of the Max Planck Society（马克斯·普朗克学会弗里茨·哈伯研究所理论部）；Institute of Mathematics, Technical University of Berlin（柏林工业大学数学研究所）；Institute of Mathematics & Centre for Advanced Analytics and Predictive Sciences, University of Augsburg（奥格斯堡大学数学与高级分析与预测科学中心）
Automatic differentiation for Riemannian optimization on low-rank matrix and tensor-train manifolds
Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences（俄罗斯科学院马尔丘克计算数学研究所）；HSE University（高等经济学院）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）
