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Description

The proximal map has played a significant role in convex analysis and various splitting algorithms. For "simple" functions this proximal map is available in closed-form while in general it needs to be computed by iterative numerical algorithms hence being inefficient. Motivated by applications in machine learning, we study when the proximal map of a sum of functions decomposes into the composition of the (simple) proximal maps of the individual summands. We present a simple sufficient condition and we demonstrate its surprising effectiveness by unifying and extending several results in seemingly unrelated fields. We end our discussion with a few open directions.