Epigraphical Reformulation for Non-Proximable Mixed Norms

This paper proposes an epigraphical reformulation (ER) technique for non-proximable mixed norm regularization. Various regularization methods using mixed norms have been proposed, where their optimization relies on efficient computation of the proximity operator of the mixed norms. Although the sophisticated design of mixed norms significantly improves the performance of regularization, the proximity operator of such a mixed norm is often unavailable. Our ER decouples a non-proximable mixed norm function into a proximable norm and epigraphical constraints. Thus, it can handle a wide range of non-proximable mixed norms as long as the proximal operator of the outermost norm, and the projection onto the epigraphical constraints can be efficiently computed. Moreover, we prove that our ER does not change the minimizer of the original problem despite using a certain inequality approximation. We also provide a new structure-tensor-based regularization as an application of our framework, which illustrates the utility of ER.