A Unified Optimization Framework for Low-Rank Inducing Penalties

In this paper we study the convex envelopes of a new class of functions. Using this approach, we are able to unify two important classes of regularizers from unbiased non-convex formulations and weighted nuclear norm penalties. This opens up for possibilities of combining the best of both worlds, and to leverage each methods contribution to cases where simply enforcing one of the regularizers are insufficient. We show that the proposed regularizers can be incorporated in standard splitting schemes such as Alternating Direction Methods of Multipliers (ADMM), and other sub-gradient methods. This can be implemented efficiently since the the proximal operator can be computed fast. Furthermore, we show on real non-rigid structure from motion datasets, the issues that arise from using weighted nuclear norm penalties, and how this can be remedied using our proposed prior-free method.