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Signal reconstruction by conjugate gradient algorithm based on smoothing \(l_1\)-norm

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Abstract

The \(l_1\)-norm regularized minimization problem is a non-differentiable problem and has a wide range of applications in the field of compressive sensing. Many approaches have been proposed in the literature. Among them, smoothing \(l_1\)-norm is one of the effective approaches. This paper follows this path, in which we adopt six smoothing functions to approximate the \(l_1\)-norm. Then, we recast the signal recovery problem as a smoothing penalized least squares optimization problem, and apply the nonlinear conjugate gradient method to solve the smoothing model. The algorithm is shown globally convergent. In addition, the simulation results not only suggest some nice smoothing functions, but also show that the proposed algorithm is competitive in view of relative error.

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Acknowledgements

The authors would like to thank the anonymous referee and the editor for their valuable comments, viewpoints, and suggestions, which help improve the manuscript a lot.

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Correspondence to Jein-Shan Chen.

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Caiying Wu: The research is supported by the Natural Science Foundation of Inner Mongolia Autonomous Region (2018MS01016)

Jiaming Zhan: The research is supported by the Natural Science Foundation of Inner Mongolia Autonomous Region (2018MS01016)

Yue Lu: The research is supported by National Natural Science Foundation of China (Grant No. 11601389), Doctoral Foundation of Tianjin Normal University (Grant No. 52XB1513), 2017-Outstanding Young Innovation Team Cultivation Program of Tianjin Normal University (Grant No. 135202TD1703) and Program for Innovative Research Team in Universities of Tianjin (Grant No. TD13-5078).

Jein-Shan Chen: The research is supported by Ministry of Science and Technology, Taiwan.

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Wu, C., Zhan, J., Lu, Y. et al. Signal reconstruction by conjugate gradient algorithm based on smoothing \(l_1\)-norm. Calcolo 56, 42 (2019). https://doi.org/10.1007/s10092-019-0340-5

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