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Smoothed \(\ell _1\)-regularization-based line search for sparse signal recovery

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Abstract

\(\ell _1\)-regularization-based sparse recovery has received a considerable attention over the last decade. In this paper, a solver called SQNSR is proposed to recover signals with high dynamic range. SQNSR utilizes linear search strategy and quasi-Newton step to the solve composite objective function for the sparse recovery problem. Since \(\ell _1\)-norm-regularized item is nonsmooth, smoothing technique is introduced to obtain an approximate smoothed function. The sufficient and necessary condition is also derived for the feasible smoothed objective function. By limiting the step length in each iteration, the convergence of SQNSR is guaranteed to obtain the sparsest solution. Numerical simulations are implemented to test the performance of the proposed approach and verify the theoretical analysis.

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Notes

  1. At http://www.stanford.edu/~boyd/l1_ls/.

  2. At http://www.lx.it.pt/~mtf/SpaRSA/.

  3. At http://statweb.stanford.edu/~candes/nesta/.

  4. At http://www.math.ucdavis.edu/~mpf/software.html.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 61374135 and 61633005. The authors would like to thank Prof. Hong Zhu for his helpful suggestions in improving the performance of the proposed method. The code for FISTA, NESTA and \(\ell _1\_ls\) has helped the authors greatly to implement SQNSR which is presented by Prof. Stephen Becker, Kwangmoo Koh, etc. We are are grateful to the editor and the anonymous reviewers for their thoughtful and constructive comments on this article.

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  1. State Key Laboratory of Power Transmission Equipment and System Security and New Technology, College of Automation, Chongqing University, Chongqing, 400044, China

    Zhimin Yang, Yi Chai & Jianfeng Qu

  2. Division of Civil, Chemical and Environmental Engineering, University of Surrey, Guildford, GU2 7XH, UK

    Tao Chen

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  1. Zhimin Yang
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Correspondence to Yi Chai.

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Yang, Z., Chai, Y., Chen, T. et al. Smoothed \(\ell _1\)-regularization-based line search for sparse signal recovery. Soft Comput 21, 4813–4828 (2017). https://doi.org/10.1007/s00500-016-2423-4

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