A DEIM Induced CUR Factorization

We derive a CUR approximate matrix factorization based on the discrete empirical interpolation method (DEIM). For a given matrix A, such a factorization provides a low-rank approximate decomposition of the form A C U R, where C and R are subsets of the columns and rows of A, and U is constructed to make C U R a good approximation. Given a low-rank singular value decomposition A V S WT, the DEIM procedure uses V and W to select the columns and rows of A that form C and R. Through an error analysis applicable to a general class of CUR factorizations, we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of V and W. For very large problems, V and W can be approximated well using an incremental QR algorithm that makes only one pass through A. Numerical examples illustrate the favorable performance of the DEIM-CUR method compared to CUR approximations based on leverage scores.