Parallel Algorithms for Computing the Tensor-Train Decomposition

Abstract. The tensor-train (TT) decomposition expresses a tensor in a data-sparse format used in molecular simulations, high-order correlation functions, and optimization. In this paper, we propose four parallelizable algorithms that compute the TT format from various tensor inputs: (1) Parallel-TTSVD for traditional format, (2) PSTT and its variants for streaming data, (3) Tucker2TT for Tucker format, and (4) TT-fADI for solutions of Sylvester tensor equations. We provide theoretical guarantees of accuracy, parallelization methods, scaling analysis, and numerical results. For example, for a [Formula: see text]-dimension tensor in [Formula: see text], a two-sided sketching algorithm PSTT2 is shown to have a memory complexity of [Formula: see text], improving upon [Formula: see text] from previous algorithms.