Circuits, Systems, and Signal Processing

, Volume 37, Issue 5, pp 2164–2178 | Cite as

Sparse Approximate Reconstruction Decomposed by Two Optimization Problems

  • Jun Wang
  • Xing Tao Wang


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.


\(\ell _{0}\) Minimization Smooth function Sparse signal Basis pursuit 



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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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinPeople’s Republic of China

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