  We consider a modification of the OMM energy functional which contains an
$\ell^1$ penalty term in order to find a sparse representation of the low-lying
eigenspace of self-adjoint operators. We analyze the local minima of the
modified functional as well as the convergence of the modified functional to
the original functional. Algorithms combining soft thresholding with gradient
descent are proposed for minimizing this new functional. Numerical tests
validate our approach. As an added bonus, we also prove the unanticipated and
remarkable property that every local minimum the OMM functional without the
$\ell^1$ term is also a global minimum.
