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arXiv · 2405.07147

Randomized algorithms for computing the tensor train approximation and their applications

Maolin Che·Yimin Wei·Hong Yan
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Abstract

In this paper, we focus on the fixed TT-rank and precision problems of finding an approximation of the tensor train (TT) decomposition of a tensor. Note that the TT-SVD and TT-cross are two well-known algorithms for these two problems. Firstly, by combining the random projection technique with the power scheme, we obtain two types of randomized algorithms for the fixed TT-rank problem. Secondly, by using the non-asymptotic theory of sub-random Gaussian matrices, we derive the upper bounds of the proposed randomized algorithms. Thirdly, we deduce a new deterministic strategy to estimate the desired TT-rank with a given tolerance and another adaptive randomized algorithm that finds a low TT-rank representation satisfying a given tolerance, and is beneficial when the target TT-rank is not known in advance. We finally illustrate the accuracy of the proposed algorithms via some test tensors from synthetic and real databases. In particular, for the fixed TT-rank problem, the proposed algorithms can be several times faster than the TT-SVD, and the accuracy of the proposed algorithms and the TT-SVD are comparable for several test tensors.

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Maolin Che, Yimin Wei, Hong Yan. 2025-02-10. Randomized algorithms for computing the tensor train approximation and their applications. https://arxiv.org/abs/2405.07147

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