We present a new restricted SVD-based CUR (RSVD-CUR) factorization for matrix triplets $(A, B, G)$ that aims to extract meaningful information by providing a low-rank approximation of the three matrices using a subset of their rows and columns. The proposed method utilizes the discrete empirical interpolation method (DEIM) to select the subset of rows and columns from the orthogonal and nonsingular matrices obtained through a restricted singular value decomposition of the matrix triplet. We explore the relationships between a DEIM type RSVD-CUR factorization, a DEIM type CUR factorization, and a DEIM type generalized CUR decomposition, and provide an error analysis that establishes the accuracy of the RSVD-CUR decomposition within a factor of the approximation error of the restricted singular value decomposition of the given matrices. The RSVD-CUR factorization can be used in applications that require approximating one data matrix relative to two other given matrices. We discuss two such applications, namely multi-view dimension reduction and data perturbation problems where a correlated noise matrix is added to the input data matrix. Our numerical experiments demonstrate the advantages of the proposed method over the standard CUR approximation in these scenarios.

翻译：我们提出了一种新的基于受限奇异值分解的CUR（RSVD-CUR）分解方法，用于处理矩阵三元组$(A, B, G)$。该方法通过从三个矩阵的行和列子集中提取低秩近似来获得有意义的信息。所提方法利用离散经验插值法（DEIM），从矩阵三元组的受限奇异值分解所得的正交且非奇异矩阵中选取行和列子集。我们探讨了DEIM型RSVD-CUR分解、DEIM型CUR分解与DEIM型广义CUR分解之间的关系，并给出了误差分析，证明了RSVD-CUR分解的精度在给定矩阵受限奇异值分解近似误差的某个因子范围内。RSVD-CUR分解可用于需要相对另外两个给定矩阵来近似一个数据矩阵的应用场景。我们讨论了两种这类应用：多视角降维问题和数据扰动问题（其中相关噪声矩阵被添加到输入数据矩阵中）。数值实验表明，在这些场景下，所提方法比标准CUR近似更具优势。