A Generalized CUR Decomposition for Matrix Pairs

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