Article: Signal recovery adapted to a dictionary from non-convex compressed sensing Journal: International Journal of Computing Science and Mathematics (IJCSM) 2023 Vol.18 No.3 pp.224 - 234 Abstract: This paper studies the reconstruction of signals that are sparse or nearly sparse with respect to a tight frame D from underdetermined linear systems. In the paper, we propose a non-convex relaxed ℓq(0<q ≤1) minimisation for sparse dictionary recovery. Based on the ℓq robust D-Null Space Property, we derive the non-sparse solution to the non-convex relaxed ℓq minimization problem and the associating performance bound in which the ∥D*D∥1,1 in the noise bound term is removed. Additionally, we show that our method can stably recover sparse or approximately sparse signals with respect to a tight frame provided that the measurement matrix A fulfils a properly adapted restricted isometry property. As a byproduct, when we choose ρ → ∞, we obtain the recovery guarantee and the corresponding error estimation via the unconstrained non-convex ℓq minimisation. Inderscience Publishers - linking academia, business and industry through research

Title: Signal recovery adapted to a dictionary from non-convex compressed sensing

Authors: Jianwen Huang; Feng Zhang; Xinling Liu; Jinping Jia; Runke Wang

Addresses: School of Mathematical Sciences, Chongqing Normal University, Chongqing, 400047, China; School of Mathematics and Statistics, Tianshui Normal University, Tianshui, 741001, China ' School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China ' Key Laboratory of Optimization Theory and Applications at China West Normal University of Sichuan Province, School of Mathematics and Information, China West Normal University, Nanchong, 637009, China ' School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu, 741001, China ' College of Resources and Environmental Engineering, Tianshui Normal University, Tianshui, Gansu, 741001, China

Abstract: This paper studies the reconstruction of signals that are sparse or nearly sparse with respect to a tight frame D from underdetermined linear systems. In the paper, we propose a non-convex relaxed ℓ_q(0<q ≤1) minimisation for sparse dictionary recovery. Based on the ℓ_q robust D-Null Space Property, we derive the non-sparse solution to the non-convex relaxed ℓ_q minimization problem and the associating performance bound in which the ∥D^*D∥_1,1 in the noise bound term is removed. Additionally, we show that our method can stably recover sparse or approximately sparse signals with respect to a tight frame provided that the measurement matrix A fulfils a properly adapted restricted isometry property. As a byproduct, when we choose ρ → ∞, we obtain the recovery guarantee and the corresponding error estimation via the unconstrained non-convex ℓ_q minimisation.

Keywords: compressed sensing; ℓ_q robust D-Null Space Property; non-convex relaxed ℓ_q minimisation method; restricted isometry property adapted D; sparse recovery.

DOI: 10.1504/IJCSM.2023.134569

International Journal of Computing Science and Mathematics, 2023 Vol.18 No.3, pp.224 - 234

Received: 03 Mar 2022
Accepted: 11 Apr 2023
Published online: 27 Oct 2023 *

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Title: Signal recovery adapted to a dictionary from non-convex compressed sensing

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