The paper deals with an interesting theoretical question concerning the proximity operator. It investigates when the proximity of the sum of two convex functions decomposes into the composition of the corresponding proximity operators. The problem is interesting since in the applications there is a growing interest in building complex regularizers by adding several simple terms.
They pursues a quite complete study. After proving a simple sufficient condition (Theorem 1), they gives the main result of the paper (Theorem 4): it is a complete characterization of the property (for a function) of being radial versus the property of being "well-coupled" with positively homogeneous functions (where well-coupled means that the prox of the sum of the couple decomposes into the composition of the two individual prox map). They also consider the case of polyhedral gauge functions, deriving a sufficient condition which is expressed by means of a cone invariance property. Examples are provided which show several prox-decomposition results, recovering known facts (in a simpler way) but also proving new ones.
The value of the paper is mainly on the theoretical side. It sheds light on the mechanism of composing proximity operators and unifies several particular results that were spread in the literature. The article is well written and technically sound. The only fault I see is that perhaps some times is not completely rigorous as I explain in the following.