Local Convergence of Proximal Splitting Methods for Rank Constrained Problems

TL;DR

This paper investigates the local convergence behavior of proximal splitting algorithms applied to rank-constrained convex optimization problems, showing conditions under which these algorithms behave like convex relaxations.

Contribution

It establishes conditions where non-convex proximal algorithms locally match convex relaxations, ensuring convergence near solutions for rank-constrained problems.

Findings

Proximal operators of rank-constrained functions can match those of their convex envelopes locally.

Local convergence of non-convex algorithms is guaranteed under specific conditions.

Convex relaxations can predict the behavior of non-convex algorithms near solutions.

Abstract

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.