RSVD-CUR Decomposition for Matrix Triplets
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We propose a restricted SVD based CUR (RSVD-CUR) decomposition for matrix triplets (A, B, G). Given matrices A, B, and G of compatible dimensions, such a decomposition provides a coordinated low-rank approximation of the three matrices using a subset of their rows and columns. We pick the subset of rows and columns of the original matrices by applying either the discrete empirical interpolation method (DEIM) or the L-DEIM scheme on the orthogonal and nonsingular matrices from the restricted singular value decomposition of the matrix triplet. We investigate the connections between a DEIM type RSVD-CUR approximation and a DEIM type CUR factorization, and a DEIM type generalized CUR decomposition. We provide an error analysis that shows that the accuracy of the proposed RSVD-CUR decomposition is within a factor of the approximation error of the restricted singular value decomposition of given matrices. An RSVD-CUR factorization may be suitable for applications where we are interested in approximating one data matrix relative to two other given matrices. Two applications that we discuss include multi-view/label dimension reduction, and data perturbation problems of the form A_E=A + BFG, where BFG is a nonwhite noise matrix. In numerical experiments, we show the advantages of the new method over the standard CUR approximation for these applications.
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