Randomized algorithms for computing the tensor train approximation and their applications
<|reference_start|>Randomized algorithms for computing the tensor train approximation and their applications: 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.<|reference_end|>
@article{che2024randomized,
    title={Randomized algorithms for computing the tensor train approximation and
  their applications},
    author={Maolin Che and Yimin Wei and Hong Yan},
    journal={arXiv preprint arXiv:2405.07147},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.07147},
    primaryClass={math.NA cs.NA}
}