Parallel algorithms for computing the tensor-train decomposition

11/19/2021

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by   Tianyi Shi, et al.

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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 ℝ^n×…× 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.