A Randomized Algorithm for Tensor Singular Value Decomposition Using an Arbitrary Number of Passes

Efficient and fast computation of a tensor singular value decomposition (t-SVD) with a few passes over the underlying data tensor is crucial because of its many potential applications. The current/existing subspace randomized algorithms need (2q + 2) passes over the data tensor to compute a t-SVD, where q is a non-negative integer number (power iteration parameter). In this paper, we propose an efficient and flexible randomized algorithm that can handle any number of passes q, which not necessary need be even. The flexibility of the proposed algorithm in using fewer passes naturally leads to lower computational and communication costs. This advantage makes it particularly appropriate when our task calls for several tensor decompositions or when the data tensors are huge. The proposed algorithm is a generalization of the methods developed for matrices to tensors. The expected/average error bound of the proposed algorithm is derived. Extensive numerical experiments on random and real-world data sets are conducted, and the proposed algorithm is compared with some baseline algorithms. The extensive computer simulation experiments demonstrate that the proposed algorithm is practical, efficient, and in general outperforms the state of the arts algorithms. We also demonstrate how to use the proposed method to develop a fast algorithm for the tensor completion problem.