  We derive a CUR 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 $A \approx C U R$,
where $C$ and $R$ are subsets of the columns and rows of $A$, and $U$ is
constructed to make $CUR$ a good approximation. Given a low-rank singular value
decomposition $A \approx V S W^T$, 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 large-scale problems, $V$
and $W$ can be approximated using an incremental QR algorithm that makes one
pass through $A$. Numerical examples illustrate the favorable performance of
the DEIM-CUR method, compared to CUR approximations based on leverage scores.
