Given a matrix $\bf A$ with numerical rank $k$, the two-sided orthogonal decomposition (TSOD) computes a factorization ${\bf A} = {\bf UDV}^T$, where ${\bf U}$ and ${\bf V}$ are orthogonal, and ${\bf D}$ is (upper/lower) triangular. TSOD is rank-revealing as the middle factor ${\bf D}$ reveals the rank of $\bf A$. The computation of TSOD, however, is demanding. In this paper, we present an algorithm called randomized pivoted TSOD (RP-TSOD), where the middle factor is lower triangular. Key in our work is the exploitation of randomization, and RP-TSOD is primarily devised to efficiently construct an approximation to a low-rank matrix. We provide three different types of bounds for RP-TSOD: (i) we furnish upper bounds on the error of the low-rank approximation, (ii) we bound the $k$ approximate principal singular values, and (iii) we derive bounds for the canonical angles between the approximate and the exact singular subspaces. Our bounds describe the characteristics and behavior of our proposed algorithm. Through numerical tests, we show the effectiveness of the devised bounds as well as our proposed algorithm.

中文翻译：

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基于随机旋转分解的矩阵的有效低秩逼近

给定一个矩阵 $\bf A$ 有数字等级 $千$，两侧正交分解 (TSOD) 计算因式分解 ${\bf A} = {\bf UDV}^T$， 在哪里 ${\bf U}$ 和 ${\bf V}$ 是正交的，并且 ${\bf D}$是（上/下）三角形。TSOD 是排名显示的中间因素${\bf D}$ 显示等级 $\bf A$. 然而，TSOD 的计算要求很高。在本文中，我们提出了一种称为随机枢轴 TSOD (RP-TSOD) 的算法，其中中间因子是下三角。我们工作的关键是利用随机化，而 RP-TSOD 主要设计用于有效构建低秩矩阵的近似值。我们为 RP-TSOD 提供了三种不同类型的边界：（i）我们提供了低秩近似误差的上限，（ii）我们限制了$千$近似主奇异值，并且（iii）我们推导出近似和精确奇异子空间之间的规范角度的界限。我们的界限描述了我们提出的算法的特征和行为。通过数值测试，我们展示了所设计的边界以及我们提出的算法的有效性。