A hybrid DEIM and leverage scores based method for CUR index selection

Abstract: The discrete empirical interpolation method (DEIM) may be used as an index selection strategy for formulating a CUR factorization. A notable drawback of the original DEIM algorithm is that the number of column or row indices that can be selected is limited to the number of input singular vectors. We propose a new variant of DEIM, which we call L-DEIM, a combination of the strength of deterministic leverage scores and DEIM. This method allows for the selection of a number of indices greater than the number of available singular vectors. Since DEIM requires singular vectors as input matrices, L-DEIM is particularly attractive for example in big data problems when computing a rank-$k$-SVD approximation is expensive even for moderately small $k$ since it uses a lower-rank SVD approximation instead of the full rank-$k$ SVD. We empirically demonstrate the performance of L-DEIM, which despite its efficiency, may achieve comparable results to the original DEIM and even better approximations than some state-of-the-art methods.