Local Convergence of Proximal Splitting Methods for Rank Constrained Problems
This work addresses convergence guarantees for non-convex optimization in machine learning and signal processing, but it is incremental as it builds on existing proximal methods and convex relaxation theory.
The paper tackles the problem of analyzing local convergence for proximal splitting algorithms applied to optimization problems with rank constraints, showing that under certain conditions, these non-convex algorithms converge locally to a solution when a convex relaxation is effective.
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.