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

Computation of a tensor singular value decomposition (t-SVD) with a few passes over the underlying data tensor is crucial in using modern computer architectures, where the main concern is communication cost. The current subspace randomized algorithms for computation of the t-SVD, need 2q + 2 passes over the data tensor where q is a non-negative integer number (power iteration parameter). In this paper, we propose an efficient and flexible randomized algorithm that works for any number of passes q, not necessarily being an even number. The flexibility of the proposed algorithm in using fewer passes naturally leads to lower computational and communication costs. This benefit makes it applicable especially when the data tensors are large or multiple tensor decompositions are required in our task. 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. Several numerical experiments on random and real-time datasets are conducted and the proposed algorithm is compared with some baseline algorithms. The results confirmed that the proposed algorithm is efficient, applicable, and can provide better performance than the existing algorithms. We also use our proposed method to develop a fast algorithm for the tensor completion problem.