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Stable Recovery of Sparse Signals with Non-Convex Weighted $r$-Norm Minus 1-Norm

Stable Recovery of Sparse Signals with Non-Convex Weighted $r$-Norm Minus 1-Norm

Year:    2025

Author:    Jianwen Huang, Feng Zhang, Xinling Liu, Jianjun Wang, Jinping Jia, Runke Wang

Journal of Computational Mathematics, Vol. 43 (2025), Iss. 1 : pp. 43–62

Abstract

Given the measurement matrix $A$ and the observation signal $y,$ the central purpose of compressed sensing is to find the most sparse solution of the underdetermined linear system $y = Ax+z,$ where $x$ is the $s$-sparse signal to be recovered and $z$ is the noise vector. Zhou and Yu [Front. Appl. Math. Stat., 5 (2019), Article 14] recently proposed a novel non-convex weighted $ℓ_r−ℓ_1$ minimization method for effective sparse recovery. In this paper, under newly coherence-based conditions, we study the non-convex weighted $ℓ_r −ℓ_1$ minimization in reconstructing sparse signals that are contaminated by different noises. Concretely, the results reveal that if the coherence $\mu$ of measurement matrix $A$ fulfills $$\mu < \kappa (s; r, α, N), s > 1, α^{\frac{1}{r}} N^{\frac{1}{2}} < 1,$$ then any $s$-sparse signals in the noisy scenarios could be ensured to be reconstructed robustly by solving weighted $ℓ_r − ℓ_1$ minimization non-convex optimization problem. Furthermore, some central remarks are presented to clear that the reconstruction assurance is much weaker than the existing ones. To the best of our knowledge, this is the first mutual coherence-based sufficient condition for such approach.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.2307-m2022-0225

Journal of Computational Mathematics, Vol. 43 (2025), Iss. 1 : pp. 43–62

Published online:    2025-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    20

Keywords:    Compressed sensing Sparse recovery Mutual coherence Sufficient condition.

Author Details

Jianwen Huang

Feng Zhang

Xinling Liu

Jianjun Wang

Jinping Jia

Runke Wang

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