Epigraphical reformulation for non-proximable mixed norms

This paper proposes an epigraphical reformulation (ER) technique for non-proximable mixed norm regularization. Various regulariza- tion methods using mixed norms have been proposed, where their optimization relies on efficient computation of the proximity opera- tor 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 prox- imable norm and epigraphical constraints. Thus, it can handle a wide range of non-proximable mixed norms as long as the proximal oper- ator 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 de- spite 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.