Computation of a tensor singular value decomposition (t-SVD) with a few passes over the underlying data tensor is crucial in using modern computer architectures, where the main concern is communication cost. The current subspace randomized algorithms for computation of the t-SVD, need 2q + 2 passes over the data tensor where q is a non-negative integer number (power iteration parameter). In this paper, we propose an efficient and flexible randomized algorithm that works for any number of passes q, not necessarily being an even number. The flexibility of the proposed algorithm in using fewer passes naturally leads to lower computational and communication costs. This benefit makes it applicable especially when the data tensors are large or multiple tensor decompositions are required in our task. The proposed algorithm is a generalization of the methods developed for matrices to tensors. The expected/average error bound of the proposed algorithm is derived. Several numerical experiments on random and real-time datasets are conducted and the proposed algorithm is compared with some baseline algorithms. The results confirmed that the proposed algorithm is efficient, applicable, and can provide better performance than the existing algorithms. We also use our proposed method to develop a fast algorithm for the tensor completion problem.

翻译：在使用现代计算机架构时，以较少轮次对底层数据张量进行张量奇异值分解(t-SVD)的计算至关重要，因为其主要关注通信成本。当前用于计算t-SVD的子空间随机算法需要对数据张量进行2q+2次传递，其中q为非负整数（幂迭代参数）。本文提出一种高效灵活的随机算法，适用于任意轮数q，不必要求为偶数。该算法在使用较少传递轮数方面的灵活性自然降低了计算和通信成本。这一优势使其特别适用于数据张量庞大或任务中需要多次张量分解的场景。所提算法是矩阵方法向张量的泛化扩展。推导了该算法的期望/平均误差界。在随机数据集和实时数据集上进行了多项数值实验，并将所提算法与若干基线算法进行比较。结果证实所提算法高效、适用，且能提供优于现有算法的性能。我们还将所提方法用于开发张量补全问题的快速算法。