Sparse signal recovery via generalized gaussian function

• Published in: Journal of global optimization Vol. 83; no. 4; pp. 783 - 801
• Main Authors: Li, Haiyang, Zhang, Qian, Lin, Shoujin, Peng, Jigen
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2022; Springer; Springer Nature B.V
• Subjects: Algorithms; Analysis; Approximation; Computer Science; Equivalence; Gaussian processes; Iterative methods; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Real Functions; Recovery; Regularization; Signal reconstruction; Sparsity; 49M20; minimization; Regularization minimization; Generalized Gaussian function; 34K29; 90C26; Sparse signal recovery; The DCS algorithm
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-022-01126-2

In this paper, we replace the ℓ 0 norm with the variation of generalized Gaussian function Φ α ( x ) in sparse signal recovery. We firstly show that Φ α ( x ) is a type of non-convex sparsity-promoting function and clearly demonstrate the equivalence among the three minimization models ( P 0 ) : min x ∈ R n ‖ x ‖ 0 subject to A x = b , ( E α ) : min x ∈ R n Φ α ( x ) subject to A x = b and ( E α λ ) : min x ∈ R n 1 2 ‖ A x - b ‖ 2 2 + λ Φ α ( x ) . The established equivalent theorems elaborate that ( P 0 ) can be completely overcome by solving the continuous minimization ( E α ) for some α s, while the latter is computable by solving the regularized minimization ( E α λ ) under certain conditions. Secondly, based on DC algorithm and iterative soft thresholding algorithm, a successful algorithm for the regularization minimization ( E α λ ) , called the DCS algorithm, is given. Finally, plenty of simulations are conducted to compare this algorithm with two classical algorithms which are half algorithm and soft algorithm, and the experiment results show that the DCS algorithm performs well in sparse signal recovery.