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Primal and dual alternating direction algorithms for ℓ 1 -ℓ 1 -norm minimization problems in compressive sensing

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  • Yunhai Xiao
  • Hong Zhu
  • Soon-Yi Wu

Abstract

In this paper, we propose, analyze and test primal and dual versions of the alternating direction algorithm for the sparse signal reconstruction from its major noise contained observation data. The algorithm minimizes a convex non-smooth function consisting of the sum of ℓ 1 -norm regularization term and ℓ 1 -norm data fidelity term. We minimize the corresponding augmented Lagrangian function alternatively from either primal or dual forms. Both of the resulting subproblems admit explicit solutions either by using a one-dimensional shrinkage or by an efficient Euclidean projection. The algorithm is easily implementable and it requires only two matrix-vector multiplications per-iteration. The global convergence of the proposed algorithm is established under some technical conditions. The extensions to the non-negative signal recovery problem and the weighted regularization minimization problem are also discussed and tested. Numerical results illustrate that the proposed algorithm performs better than the state-of-the-art algorithm YALL1. Copyright Springer Science+Business Media, LLC 2013

Suggested Citation

  • Yunhai Xiao & Hong Zhu & Soon-Yi Wu, 2013. "Primal and dual alternating direction algorithms for ℓ 1 -ℓ 1 -norm minimization problems in compressive sensing," Computational Optimization and Applications, Springer, vol. 54(2), pages 441-459, March.
    • Handle: RePEc:spr:coopap:v:54:y:2013:i:2:p:441-459
    DOI: 10.1007/s10589-012-9475-x
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    References listed on IDEAS

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    1. B. S. He & H. Yang & S. L. Wang, 2000. "Alternating Direction Method with Self-Adaptive Penalty Parameters for Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 106(2), pages 337-356, August.
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