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Fast Moreau envelope computation I: numerical algorithms

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Abstract

The present article summarizes the state of the art algorithms to compute the discrete Moreau envelope, and presents a new linear-time algorithm, named NEP for NonExpansive Proximal mapping. Numerical comparisons between the NEP and two existing algorithms: The Linear-time Legendre Transform (LLT) and the Parabolic Envelope (PE) algorithms are performed. Worst-case time complexity, convergence results, and examples are included. The fast Moreau envelope algorithms first factor the Moreau envelope as several one-dimensional transforms and then reduce the brute force quadratic worst-case time complexity to linear time by using either the equivalence with Fast Legendre Transform algorithms, the computation of a lower envelope of parabolas, or, in the convex case, the non expansiveness of the proximal mapping.

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  1. Computer Science, I. K. Barber School of Arts and Sciences-Unit 4, The University of British Columbia Okanagan, 3333 University Way, Kelowna, BC V1V 1V7, Canada

    Yves Lucet

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Lucet, Y. Fast Moreau envelope computation I: numerical algorithms. Numer Algor 43, 235–249 (2006). https://doi.org/10.1007/s11075-006-9056-0

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