Forcing sparsity by projecting with respect to a non-diagonally weighted frobenius norm

This note deals with the computational problem of determining the projection of a given symmetric matrix onto the subspace of symmetric matrices that have a fixed sparsity pattern. This projection is performed with respect to a weighted Frobenius norm involving a metric that is not diagonal. It is shown that the solution to this question is computationally feasible when the metric appearing in the norm is a low rank modification to the identity. Also, generalization to perturbations of higher rank is shown to be increasingly costly in terms of computation.