Big Data Matrix Singular Value Decomposition Based on Low-Rank Tensor Train Decomposition

We propose singular value decomposition (SVD) algorithms for very large-scale matrices based on a low-rank tensor decomposition technique called the tensor train (TT) format. By using the proposed algorithms, we can compute several dominant singular values and corresponding singular vectors of large-scale structured matrices given in a low-rank TT format. We propose a large-scale trace optimization problem, and in the proposed methods, the large-scale optimization problem is reduced to sequential small-scale optimization problems. We show that the computational complexity of the proposed algorithms scales logarithmically with the matrix size if the TT-ranks are bounded. Numerical simulations based on very large-scale Hilbert matrix demonstrate the effectiveness of the proposed methods.