Sparse signal recovery via generalized gaussian function

Haiyang Li (), Qian Zhang (), Shoujin Lin () and Jigen Peng ()
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Haiyang Li: Guangzhou University
Qian Zhang: University of Electronic Science and Technology
Shoujin Lin: Zhongshan MLTOR Intelligent Equipment Co., Ltd
Jigen Peng: Guangzhou University

Journal of Global Optimization, 2022, vol. 83, issue 4, No 6, 783-801

Abstract: Abstract In this paper, we replace the $$\ell _0$$ ℓ 0 norm with the variation of generalized Gaussian function $$\Phi _\alpha (x)$$ Φ α ( x ) in sparse signal recovery. We firstly show that $$\Phi _\alpha (x)$$ Φ α ( x ) is a type of non-convex sparsity-promoting function and clearly demonstrate the equivalence among the three minimization models $$(\mathrm{P}_0):\min \limits _{x\in {\mathbb {R}}^n}\Vert x\Vert _0$$ ( P 0 ) : min x ∈ R n ‖ x ‖ 0 subject to $$ Ax=b$$ A x = b , $${\mathrm{(E}_\alpha )}:\min \limits _{x\in {\mathbb {R}}^n}\Phi _\alpha (x)$$ ( E α ) : min x ∈ R n Φ α ( x ) subject to $$Ax=b$$ A x = b and $$(\mathrm{E}^{\lambda }_\alpha ):\min \limits _{x\in {\mathbb {R}}^n}\frac{1}{2}\Vert Ax-b\Vert ^{2}_{2}+\lambda \Phi _\alpha (x).$$ ( E α λ ) : min x ∈ R n 1 2 ‖ A x - b ‖ 2 2 + λ Φ α ( x ) . The established equivalent theorems elaborate that $$(\mathrm{P}_0)$$ ( P 0 ) can be completely overcome by solving the continuous minimization $$(\mathrm{E}_\alpha )$$ ( E α ) for some $$\alpha $$ α s, while the latter is computable by solving the regularized minimization $$(\mathrm{E}^{\lambda }_\alpha )$$ ( E α λ ) under certain conditions. Secondly, based on DC algorithm and iterative soft thresholding algorithm, a successful algorithm for the regularization minimization $$(\mathrm{E}^{\lambda }_\alpha )$$ ( 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.

Keywords: Sparse signal recovery; $$\ell _{0}$$ ℓ 0 minimization; Generalized Gaussian function; Regularization minimization; The DCS algorithm; 90C26; 34K29; 49M20 (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10898-022-01126-2

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