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 d-dimension tensor in $\mathbb{R}$ $n\times∙∙∙$$\times$$n$ a two-sided sketching algorithm PSTT2 is shown to have a memory complexity of $O(n^{[d/2]})$, improving upon $O(n^{d—1})$ from previous algorithms.