Iteratively weighted thresholding homotopy method for the sparse solution of underdetermined linear equations

Abstract

Recently, iteratively reweighted methods have attracted much interest in compressed sensing, outperforming their unweighted counterparts in most cases. In these methods, decision variables and weights are optimized alternatingly, or decision variables are optimized under heuristically chosen weights. In this paper, we present a novel weighted l1-norm minimization problem for the sparsest solution of underdetermined linear equations. We propose an iteratively weighted thresholding method for this problem, wherein decision variables and weights are optimized simultaneously. Furthermore, we prove that the iteration process will converge eventually. Using the homotopy technique, we enhance the performance of the iteratively weighted thresholding method. Finally, extensive computational experiments show that our method performs better in terms of both running time and recovery accuracy compared with some state-of-the-art methods.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 61672005 and 11571074). The authors thank the anonymous reviewers for their helpful suggestions and comments, which helped improve the quality of this manuscript.

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Correspondence to Wenxing Zhu.

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Zhu, W., Huang, Z., Chen, J. et al. Iteratively weighted thresholding homotopy method for the sparse solution of underdetermined linear equations. Sci. China Math. 64, 639–664 (2021). https://doi.org/10.1007/s11425-018-9467-7

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Keywords

  • sparse optimization
  • weighted thresholding method
  • homotopy method

MSC(2010)

  • 15A06
  • 15A29
  • 65K05
  • 90C25
  • 90C26
  • 90C59