A Generalized CUR Decomposition for Matrix Pairs

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We propose a generalized CUR (GCUR) decomposition for matrix pairs $(A, B)$. Given matrices $A$ and $B$ with the same number of columns, such a decomposition provides low-rank approximations of both matrices simultaneously, in terms of some of their rows and columns. We obtain the indices for selecting the subset of rows and columns of the original matrices using the discrete empirical interpolation method (DEIM) on the generalized singular vectors. When $B$ is square and nonsingular, there are close connections between the GCUR of $(A, B)$ and the DEIM-induced CUR of $AB^{-1}$. When $B$ is the identity, the GCUR decomposition of $A$ coincides with the DEIM-induced CUR decomposition of $A$. We also show a similar connection between the GCUR of $(A, B)$ and the CUR of $AB^+$ for a nonsquare but full-rank matrix $B$, where $B^+$ denotes the Moore--Penrose pseudoinverse of $B$. While a CUR decomposition acts on one data set, a GCUR factorization jointly decomposes two data sets. The algorithm may be suitable for applications where one is interested in extracting the most discriminative features from one data set relative to another data set. In numerical experiments, we demonstrate the advantages of the new method over the standard CUR approximation; for recovering data perturbed with colored noise and subgroup discovery.