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A Convergent Fixed-Point Proximity Algorithm Accelerated by FISTA for the ℓ0 Sparse Recovery Problem

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Imaging, Vision and Learning Based on Optimization and PDEs (IVLOPDE 2016)

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

We propose an approximation model of the original ℓ 0 minimization model arising from various sparse signal recovery problems. The objective function of the proposed model uses the Moreau envelope of the ℓ 0 norm to promote the sparsity of the signal in a tight framelet system . This leads to a non-convex optimization problem involved the ℓ 0 norm. We identify a local minimizer of the proposed non-convex optimization problem with a global minimizer of a related convex optimization problem. Based on this identification, we develop a two stage algorithm for solving the proposed non-convex optimization problem and study its convergence. Moreover, we show that FISTA can be employed to speed up the convergence rate of the proposed algorithm to reach the optimal convergence rate of \(\mathcal {O}(1/k^2)\). We present numerical results to confirm the theoretical estimate.

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Acknowledgements

The author Xueying Zeng is supported by the Natural Science Foundation of China (No. 11701538, 11771408) and the Fundamental Research Funds for the Central Universities (No. 201562012). Both Lixin Shen and Yuesheng Xu were supported in part by the US National Science Foundation under Grant DMS-1522332. The author Yuesheng Xu is supported in part by the Special Project on High-performance Computing under the National Key R&D Program (No. 2016YFB0200602), and by the Natural Science Foundation of China under grants 11471013 and 11771464.

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Authors and Affiliations

  1. School of Mathematical Sciences, Ocean University of China, Qingdao, People’s Republic of China

    Xueying Zeng

  2. Department of Mathematics, Syracuse University, Syracuse, NY, USA

    Lixin Shen

  3. School of Data and Computer Science, Guangdong Province Key Lab of Computational Science, Sun Yat-sen University, Guangzhou, People’s Republic of China

    Yuesheng Xu

  4. Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA, USA

    Yuesheng Xu

Authors
  1. Xueying Zeng
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  2. Lixin Shen
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  3. Yuesheng Xu
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Corresponding author

Correspondence to Yuesheng Xu .

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Editors and Affiliations

  1. Department of Mathematics, University of Bergen, Bergen, Norway

    Xue-Cheng Tai

  2. Norwegian Defence Research Establishment, Kjeller, Norway

    Egil Bae

  3. University of South-Eastern Norway, Porsgrunn, Norway

    Marius Lysaker

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Zeng, X., Shen, L., Xu, Y. (2018). A Convergent Fixed-Point Proximity Algorithm Accelerated by FISTA for the ℓ0 Sparse Recovery Problem. In: Tai, XC., Bae, E., Lysaker, M. (eds) Imaging, Vision and Learning Based on Optimization and PDEs. IVLOPDE 2016. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-91274-5_2

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