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Sparse Approximate Reconstruction Decomposed by Two Optimization Problems

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Abstract

We propose a novel sparse signal reconstruction method aiming to directly minimize \(\ell _{0}\)-quasinorm. Based on the smoothed \(\ell _{0}\)-quasinorm, we show that there exists an unconstrained optimization problem such that both this problem and the basis pursuit problem are subproblems of the \(\ell _{0}\) minimization problem. Moreover, we can obtain a sparse solution to the \(\ell _{0}\) minimization by solving these two subproblems. In addition, we establish the relation between solutions to the \(\ell _{0}\) minimization and the least square solutions of a linear system. Finally, we present some numerical experiments to illustrate our results.

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Acknowledgements

We thank the anonymous referees, associate editors, and editor-in-chief for their thoughtful and insightful comments, which improved the paper greatly. The research was supported partially by National Natural Science Foundation of China (Grant Nos. 10871056 and 10971150).

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  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Jun Wang & Xing Tao Wang

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  1. Jun Wang
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  2. Xing Tao Wang
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Correspondence to Xing Tao Wang.

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Wang, J., Wang, X.T. Sparse Approximate Reconstruction Decomposed by Two Optimization Problems. Circuits Syst Signal Process 37, 2164–2178 (2018). https://doi.org/10.1007/s00034-017-0667-6

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