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A CUR factorization is often utilized as a substitute for the singular valuedecomposition (SVD), especially when a concrete interpretation of the singularvectors is challenging. Moreover, if the original data matrix possessesproperties like nonnegativity and sparsity, a CUR decomposition can betterpreserve them compared to the SVD. An essential aspect of this approach is themethodology used for selecting a subset of columns and rows from the originalmatrix. This study investigates the effectiveness of \emph{one-round sampling}and iterative subselection techniques and introduces new iterative subselectionstrategies based on iterative SVDs. One provably appropriate technique forindex selection in constructing a CUR factorization is the discrete empiricalinterpolation method (DEIM). Our contribution aims to improve the approximationquality of the DEIM scheme by iteratively invoking it in several rounds, in thesense that we select subsequent columns and rows based on the previouslyselected ones. That is, we modify $A$ after each iteration by removing theinformation that has been captured by the previously selected columns and rows.We also discuss how iterative procedures for computing a few singular vectorsof large data matrices can be integrated with the new iterative subselectionstrategies. We present the results of numerical experiments, providing acomparison of one-round sampling and iterative subselection techniques, anddemonstrating the improved approximation quality associated with using thelatter.

A DEIM-CUR factorization with iterative SVDs