k-sparse vector recovery via Truncated ℓ1-ℓ2 local minimization

• Published in: Optimization letters Vol. 18; no. 1; pp. 291 - 305
• Main Authors: Xie, Shaohua, Li, Jia
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2024
• Subjects: Computational Intelligence; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Operations Research/Decision Theory; Optimization; Original Paper; Simulation; sparse vector; minimization; Locally recovery; Compressed sensing; Truncated
• ISSN: 1862-4472; 1862-4480
• DOI: 10.1007/s11590-023-01991-0

This article mainly solves the following model, m i n ‖ x Γ x , t C ‖ 1 - ‖ x Γ x , t C ‖ 2 s u b j e c t t o A x = y , where Γ x , t ⊂ [ n ] represents the index of the maximum number of t elements in x after taking the absolute value. We call this model Truncated ℓ 1 - ℓ 2 model. We mainly deal with the recovery of unknown signals under the condition of | s u p p ( x ) | > t , σ t ( x ) > σ t + 1 ( x ) , where σ t ( x ) represents the t largest number in | x |. Firstly, we give the necessary and sufficient condition for recovering the fixed unknown signal satisfying the above two conditions via Truncated ℓ 1 - ℓ 2 local minimization. Then, according to this condition, we give the necessary and sufficient conditions to recovering for all unknown signals satisfying the above two conditions via Truncated ℓ 1 - ℓ 2 local minimization. Compared with N. Bi’s recent proposed condition in Bi and Tang (Appl Comput Harmon Anal 56:337–350, 2022), we will show that our condition is weaker and the detail of such discussion is in Remark 3 of the manuscript. Then, we give the algorithm of Truncated ℓ 1 - ℓ 2 model. According to this algorithm, we do data experiments and the data experiments show that the recovery rate of Truncated ℓ 1 - ℓ 2 is better than that of model ℓ 1 - ℓ 2 .