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.

翻译：本文聚焦于固定TT秩和精度问题，旨在寻求张量的张量列（TT）分解近似。注意到TT-SVD和TT-cross是解决这两个问题的两种经典算法。首先，通过将随机投影技术与幂方法相结合，我们针对固定TT秩问题提出了两类随机化算法。其次，利用子随机高斯矩阵的非渐近理论，我们推导了所提随机化算法的上界。第三，我们提出了一种新的确定性策略，用于在给定容差下估计目标TT秩，并设计了一种自适应随机化算法，该算法能够找到满足给定容差的低TT秩表示，尤其适用于目标TT秩未知的情况。最后，通过来自合成数据集和真实数据库的若干测试张量，我们验证了所提算法的准确性。特别地，对于固定TT秩问题，所提算法比TT-SVD快数倍，且在多个测试张量上，所提算法与TT-SVD的精度相当。