Orbital minimization method with ℓ1 regularization

We consider a modification of the orbital minimization method (OMM) energy functional which contains an ℓ¹ 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. In addition, we also prove the unanticipated and remarkable property that every local minimum of the OMM functional without the ℓ¹ term is also a global minimum.