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Operator Splitting Methods in Compressive Sensing and Sparse Approximation

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Splitting Methods in Communication, Imaging, Science, and Engineering

Part of the book series: Scientific Computation ((SCIENTCOMP))

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Abstract

Compressive sensing and sparse approximation have many emerging applications, and are a relatively new driving force for the development of splitting methods in optimization. Many sparse coding problems are well described by variational models with ℓ 1-norm penalties and constraints that are designed to promote sparsity. Successful algorithms need to take advantage of the separable structure of potentially complicated objectives by “splitting” them into simpler pieces and iteratively solving a sequence of simpler convex minimization problems. In particular, isolating ℓ 1 terms from the rest of the objective leads to simple soft thresholding updates or ℓ 1 ball projections. A few basic splitting techniques can be used to design a huge variety of relevant algorithms. This chapter will focus on operator splitting strategies that are based on proximal operators, duality, and alternating direction methods. These will be explained in the context of basis pursuit variants and through compressive sensing applications.

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Notes

  1. 1.

    It is more correct to say light diverted towards/away from the detector results in a 0∕1 coefficient. In practice, 0∕1 measurements are converted to + 1∕ − 1 measurements by subtracting the average image intensity. Measurement matrices with + 1∕ − 1 coefficients are more well conditioned than their 0∕1 counterparts, and are easier to handle numerically.

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Authors and Affiliations

  1. Department of Computer Science, University of Maryland, College Park, MD, USA

    Tom Goldstein

  2. Shanghai Jiao Tong University, Institute of Natural Sciences and School of Mathematical Sciences, Shanghai, China

    Xiaoqun Zhang

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  1. Tom Goldstein
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  2. Xiaoqun Zhang
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Correspondence to Tom Goldstein .

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Editors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, Texas, USA

    Roland Glowinski

  2. Department of Mathematics, UCLA, Los Angeles, California, USA

    Stanley J. Osher

  3. Department of Mathematics, UCLA, Los Angeles, California, USA

    Wotao Yin

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Goldstein, T., Zhang, X. (2016). Operator Splitting Methods in Compressive Sensing and Sparse Approximation. In: Glowinski, R., Osher, S., Yin, W. (eds) Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-41589-5_9

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