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

TL;DR

This paper introduces L-DEIM, a hybrid method combining leverage scores and DEIM, enabling the selection of more indices than available singular vectors, improving CUR approximation especially in large-scale data scenarios.

Contribution

The paper proposes L-DEIM, a novel variant of DEIM that allows for more index selections than singular vectors, enhancing CUR factorization in big data contexts.

Findings

L-DEIM achieves comparable or better approximation results than existing methods.

L-DEIM is efficient for large data by using lower-rank SVD approximations.

The method extends DEIM's applicability beyond the number of input singular vectors.

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.