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：以少量遍数遍历底层数据张量来计算张量奇异值分解（t-SVD）在当前计算机架构中至关重要，其主要关注通信成本。现有的用于计算t-SVD的子空间随机化算法需要2q+2次遍历数据张量，其中q为非负整数（幂迭代参数）。本文提出了一种高效且灵活的随机化算法，适用于任意遍数q，无需q为偶数。该算法在使用较少遍数时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较大或任务中需要多次张量分解的场景。所提算法将面向矩阵的方法推广到了张量情形，推导了其期望/平均误差界。我们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了比较。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。此外，我们还将该方法用于开发张量补全问题的快速算法。