By exploiting the random sampling techniques, this paper derives an efficient randomized algorithm for computing a generalized CUR decomposition, which provides low-rank approximations of both matrices simultaneously in terms of some of their rows and columns. For large-scale data sets that are expensive to store and manipulate, a new variant of the discrete empirical interpolation method known as L-DEIM, which needs much lower cost and provides a significant acceleration in practice, is also combined with the random sampling approach to further improve the efficiency of our algorithm. Moreover, adopting the randomized algorithm to implement the truncation process of restricted singular value decomposition (RSVD), combined with the L-DEIM procedure, we propose a fast algorithm for computing an RSVD based CUR decomposition, which provides a coordinated low-rank approximation of the three matrices in a CUR-type format simultaneously and provides advantages over the standard CUR approximation for some applications. We establish detailed probabilistic error analysis for the algorithms and provide numerical results that show the promise of our approaches.

翻译：通过利用随机采样技术，本文提出了一种高效的计算广义CUR分解的随机化算法，该算法能同时基于矩阵的部分行与列生成两者的低秩近似。针对大规模数据集存储与计算代价高昂的问题，本文还将一种新型离散经验插值方法——即L-DEIM（其成本显著降低且在实际应用中可显著加速）与随机采样相结合，以进一步提升算法效率。此外，采用随机化算法实现限制性奇异值分解（RSVD）的截断过程，并结合L-DEIM流程，我们提出了一种基于RSVD的CUR分解快速算法。该算法能以CUR型格式同时协调三个矩阵的低秩近似，并在某些应用中展现出优于标准CUR近似的优势。我们为所提算法建立了详细的概率误差分析，并通过数值结果验证了方法的有效性。