Least-Square-Based Three-Term Conjugate Gradient Projection Method for ? 1 -Norm Problems with Application to Compressed Sensing

Abdulkarim Hassan Ibrahim, Poom Kumam, Auwal Bala Abubakar, Jamilu Abubakar and Abubakar Bakoji Muhammad
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Abdulkarim Hassan Ibrahim: KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
Poom Kumam: KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
Auwal Bala Abubakar: KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
Jamilu Abubakar: KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
Abubakar Bakoji Muhammad: Faculty of Natural Sciences II, Institute of Mathematics, Martin Luther University Halle-Wittenberg, 06099 Halle, Germany

Mathematics, 2020, vol. 8, issue 4, 1-21

Abstract: In this paper, we propose, analyze, and test an alternative method for solving the ? 1 -norm regularization problem for recovering sparse signals and blurred images in compressive sensing. The method is motivated by the recent proposed nonlinear conjugate gradient method of Tang, Li and Cui [Journal of Inequalities and Applications, 2020(1), 27] designed based on the least-squares technique. The proposed method aims to minimize a non-smooth minimization problem consisting of a least-squares data fitting term and an ? 1 -norm regularization term. The search directions generated by the proposed method are descent directions. In addition, under the monotonicity and Lipschitz continuity assumption, we establish the global convergence of the method. Preliminary numerical results are reported to show the efficiency of the proposed method in practical computation.

Keywords: image processing; compressed sensing; ? 1 -norm regularization; nonlinear equations; conjugate gradient method; projection method; global convergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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