  This paper introduces a new framework for constructing the Discrete Empirical
Interpolation Method DEIM projection operator. The interpolation node selection
procedure is formulated using the QR factorization with column pivoting, and it
enjoys a sharper error bound for the DEIM projection error. Furthermore, for a
subspace $\mathcal{U}$ given as the range of an orthonormal $U$, the DEIM
projection does not change if $U$ is replaced by $U \Omega$ with arbitrary
unitary matrix $\Omega$. In a large-scale setting, the new approach allows
modifications that use only randomly sampled rows of $U$, but with the
potential of producing good approximations with corresponding probabilistic
error bounds. Another salient feature of the new framework is that robust and
efficient software implementation is easily developed, based on readily
available high performance linear algebra packages.
