The low-rank approximation of big data matrices and tensors plays a pivotal role in many modern applications. Although, a truncated version of the singular value decomposition (SVD) furnishes the best approximation, its computation is challenging on modern, multicore architectures. Recently, the randomized subspace iteration has shown to be a powerful tool in approximating large-scale matrices. In this paper we present a two-sided variant of the randomized subspace iteration. Novel in our work is the exploitation of the unpivoted QR factorization, rather than the SVD, for factorizing the compressed matrix. Hence our algorithm is a randomized rank-revealing URV decomposition. We prove the rank-revealingness of our algorithm by establishing bounds for the singular values as well as the other blocks of the compressed matrix. We further provide bounds on the error of the low-rank approximations of the proposed algorithm, in both 2- and Frobenius norm. In addition, we employ the proposed algorithm to efficiently compute low rank tensor decompositions: we present two randomized algorithms, one for the truncated higher-order SVD, and the other for the tensor SVD. We conduct numerical tests on (i) various classes of matrices, and (ii) synthetic tensors and real datasets to demonstrate the efficacy of the proposed algorithms.

中文翻译：

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低秩矩阵和张量分解的随机两侧子空间迭代

大数据矩阵和张量的低秩近似在许多现代应用中发挥着关键作用。尽管奇异值分解 (SVD) 的截断版本提供了最佳近似值，但其计算在现代多核架构上具有挑战性。最近，随机子空间迭代已被证明是逼近大规模矩阵的强大工具。在本文中，我们提出了随机子空间迭代的两侧变体。我们工作的新颖之处在于利用非旋转 QR 分解（而不是 SVD）来分解压缩矩阵。因此，我们的算法是随机排名揭示 URV 分解。我们通过建立奇异值以及压缩矩阵的其他块的界限来证明我们的算法的等级揭示性。我们进一步提供了所提出算法的低秩近似在 2- 和 Frobenius 范数中的误差界限。此外，我们采用所提出的算法来有效计算低秩张量分解：我们提出了两种随机算法，一种用于截断的高阶 SVD，另一种用于张量 SVD。我们对（i）各类矩阵和（ii）合成张量和真实数据集进行数值测试，以证明所提出算法的有效性。