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：本文提出一种新的基于受限SVD的CUR分解方法（RSVD-CUR），用于矩阵三元组$(A, B, G)$，通过选取其行和列的子集对三个矩阵进行低秩近似，从而提取有意义的信息。该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获得的正交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分解以及DEIM型广义CUR分解之间的关系，并给出误差分析，证明了在给定矩阵的受限奇异值分解近似误差的因子范围内，RSVD-CUR分解的精度。RSVD-CUR分解可应用于需要近似一个数据矩阵相对于另外两个给定矩阵的场景。我们讨论了两种此类应用：多视角降维和在输入数据矩阵中添加相关噪声矩阵的数据扰动问题。数值实验表明，在这些场景下，所提方法优于标准CUR近似。