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On Decomposing the Proximal Map

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On Decomposing the Proximal Map
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30
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CC Attribution - NonCommercial - NoDerivatives 4.0 International:
You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
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Abstract
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.