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, 56:1 | Cite as

A new generalized shrinkage conjugate gradient method for sparse recovery

  • Hamid EsmaeiliEmail author
  • Shima Shabani
  • Morteza Kimiaei
Article
  • 139 Downloads

Abstract

In this paper, a new procedure, called generalized shrinkage conjugate gradient (GSCG), is presented to solve the \(\ell _1\)-regularized convex minimization problem. In GSCG, we present a new descent condition. If such a condition holds, an efficient descent direction is presented by an attractive combination of a generalized form of the conjugate gradient direction and the ISTA descent direction. Otherwise, ISTA is improved by a new step-size of the shrinkage operator. The global convergence of GSCG is established under some assumptions and its sublinear (R-linear) convergence rate in the convex (strongly convex) case. In numerical results, the suitability of GSCG is evaluated for compressed sensing and image debluring problems on the set of randomly generated test problems with dimensions \(n\in \{2^{10},\ldots ,2^{17}\}\) and some images, respectively, in Matlab. These numerical results show that GSCG is efficient and robust for these problems in terms of the speed and ability of the sparse reconstruction in comparison with several state-of-the-art algorithms.

Keywords

\(\ell _1\)-Minimization Compressed sensing Image debluring Shrinkage operator Generalized conjugate gradient method Nonmonotone technique Line search method Global convergence 

Mathematics Subject Classification

65K05 90C25 90C06 94A08 

Notes

Acknowledgements

We would like to thank the high performance computing (HPC) center, a branch of institute for research in fundamental Physics and Mathematics (IPM), to help us to use HPC’s cluster for computing numerical results. The third author acknowledges the financial support of the Doctoral Program “Vienna Graduate School on Computational Optimization” funded by Austrian Science Foundation under Project No. W1260-N35.

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Copyright information

© Istituto di Informatica e Telematica del Consiglio Nazionale delle Ricerche 2018

Authors and Affiliations

  • Hamid Esmaeili
    • 1
    Email author
  • Shima Shabani
    • 1
  • Morteza Kimiaei
    • 2
  1. 1.Department of MathematicsBu-Ali Sina UniversityHamedanIran
  2. 2.Faculty of MathematicsUniversity of ViennaViennaAustria

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