On Decomposing the Proximal Map

The proximal map is the key step in gradient-type algorithms, which have become prevalent in large-scale high-dimensional problems. For simple functions this proximal map is available in closed-form while for more complicated functions it can become highly nontrivial. Motivated by the need of combining regularizers to simultaneously induce different types of structures, this paper initiates a systematic investigation of when the proximal map of a sum of functions decomposes into the composition of the proximal maps of the individual summands. We not only unify a few known results scattered in the literature but also discover several new decompositions obtained almost effortlessly from our theory.