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Proximal Splitting Methods in Signal Processing

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Fixed-Point Algorithms for Inverse Problems in Science and Engineering

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 49))

Abstract

The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of inverse problems and, especially, in signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework. Applications of proximal methods in signal recovery and synthesis are discussed.

AMS 2010 Subject Classification: 90C25, 65K05, 90C90, 94A08

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Acknowledgements

This work was supported by the Agence Nationale de la Recherche under grants ANR-08-BLAN-0294-02 and ANR-09-EMER-004-03.

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions – UMR CNRS 7598, UPMC Université Paris 06, 75005, Paris, France

    Patrick L. Combettes

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  1. Patrick L. Combettes
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  2. Jean-Christophe Pesquet
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Correspondence to Patrick L. Combettes .

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Editors and Affiliations

  1. Okanagan Campus, Department of Mathematics and Statistic, University of British Columbia, Kelowna, V1V 1V7, British Columbia, Canada

    Heinz H. Bauschke

  2. School of Mathematics & Statistics, Division of Information Technology, University of South Australia, Mawson Lakes Blvd., Mawson Lakes, 5095, South Australia, Australia

    Regina S. Burachik

  3. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Place Jussieu 4, Paris, 75005, France

    Patrick L. Combettes

  4. Dept. Physics, Lab. Atomic & Solid State, Cornell University, Clark Hall, Ithaca, 14853-2501, New York, USA

    Veit Elser

  5. Institut für Numerische und Angewandte M, Universität Göttingen, Lotzestr. 16-18, Göttingen, 37073, Germany

    D. Russell Luke

  6. Faculty of Mathematics, Dept. Combinatorics & Optimization, University of Waterloo, Waterloo, N2L 3G1, Ontario, Canada

    Henry Wolkowicz

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Combettes, P.L., Pesquet, JC. (2011). Proximal Splitting Methods in Signal Processing. In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications(), vol 49. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9569-8_10

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