Sparse signal recovery via generalized gaussian function

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.