论文标题
矩阵三胞胎的受限SVD类型CUR分解
A Restricted SVD type CUR Decomposition for Matrix Triplets
论文作者
论文摘要
我们提出了一个新的基于SVD的CUR（RSVD-CUR）分解，用于矩阵三重性$（A，B，G）$，旨在通过使用其行的子集和列的子集提供三个矩阵来提取有意义的信息，以提取有意义的信息。所提出的方法利用离散的经验插值法（DEIM）从通过矩阵三重态的受限奇异值分解获得的正交和非矩阵中的行的子集和列的子集选择。我们探讨了DEIM型RSVD-cur分解，Deim类型的cur分解和DEIM类型的通用CUR分解之间的关系，并提供了一个误差分析，该分析确定了RSVD-CUR分解的准确性，这是在限制的给定组合限制单数值分解的因素内。 RSVD-cur分解可用于相对于其他两个给定矩阵的应用程序，需要近似一个数据矩阵。我们讨论了两个这样的应用程序，即减少多视图维度和数据扰动问题，其中相关的噪声矩阵添加到输入数据矩阵中。在这些情况下，我们的数值实验证明了所提出的方法比标准Cur近似的优势。
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.
