  We analyze the local convergence of proximal splitting algorithms to solve
optimization problems that are convex besides a rank constraint. For this, we
show conditions under which the proximal operator of a function involving the
rank constraint is locally identical to the proximal operator of its convex
envelope, hence implying local convergence. The conditions imply that the
non-convex algorithms locally converge to a solution whenever a convex
relaxation involving the convex envelope can be expected to solve the
non-convex problem.
