  A CUR factorization is often utilized as a substitute for the singular value
decomposition (SVD), especially when a concrete interpretation of the singular
vectors is challenging. Moreover, if the original data matrix possesses
properties like nonnegativity and sparsity, a CUR decomposition can better
preserve them compared to the SVD. An essential aspect of this approach is the
methodology used for selecting a subset of columns and rows from the original
matrix. This study investigates the effectiveness of \emph{one-round sampling}
and iterative subselection techniques and introduces new iterative subselection
strategies based on iterative SVDs. One provably appropriate technique for
index selection in constructing a CUR factorization is the discrete empirical
interpolation method (DEIM). Our contribution aims to improve the approximation
quality of the DEIM scheme by iteratively invoking it in several rounds, in the
sense that we select subsequent columns and rows based on the previously
selected ones. Thus, we modify $A$ after each iteration by removing the
information that has been captured by the previously selected columns and rows.
We also discuss how iterative procedures for computing a few singular vectors
of large data matrices can be integrated with the new iterative subselection
strategies. We present the results of numerical experiments, providing a
comparison of one-round sampling and iterative subselection techniques, and
demonstrating the improved approximation quality associated with using the
latter.
