Density matrix minimization with ${\mathrm{\ell }}_1$ regularization

We propose a convex variational principle to find sparse representation of low-lying eigenspace of symmetric matrices. In the context of electronic structure calculation, this corresponds to a sparse density matrix minimization algorithm with ${\mathrm{\ell }}_1$ regularization. The minimization problem can be efficiently solved by a split Bergman iteration type algorithm. We further prove that from any initial condition, the algorithm converges to a minimizer of the variational principle.