This paper presents a novel regularization with a non-convex, non-smooth term of the form with parameters to solve ill-posed linear problems with sparse solutions. We investigate the existence, stability and convergence of the regularized solution. It is shown that this type of regularization is well-posed and yields sparse solutions. Under an appropriate source condition, we get the convergence rate in the -norm for a priori and a posteriori parameter choice rules, respectively. A numerical algorithm is proposed and analyzed based on an iterative threshold strategy with the generalized conditional gradient method. We prove the convergence even though the regularization term is non-smooth and non-convex. The algorithm can easily be implemented because of its simple structure. Some numerical experiments are performed to test the efficiency of the proposed approach. The experiments show that regularization with performs better in comparison with the classical sparsity regularization and can be used as an alternative to the regularizer.

The non-convex α ‖ ⋅ ‖ ℓ 1 − β ‖ ⋅ ‖ ℓ 2 ( α ⩾ β ⩾ 0 ) regularization is a new approach for sparse recovery. A minimizer of the α ‖ ⋅ ‖ ℓ 1 − β ‖ ⋅ ‖ ℓ 2 regularized function can be computed by applying the ST-( αℓ ₁ − βℓ ₂ ) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. However, the conventional PG method is limited to solve problems with the classical ℓ ₁ sparsity regularization. In this paper, we present two accelerated alternatives to the ST-( αℓ ₁ − βℓ ₂ ) algorithm by extending the PG method to the non-convex α ‖ ⋅ ‖ ℓ 1 − β ‖ ⋅ ‖ ℓ 2 sparsity regularization. Moreover, we discuss a strategy to determine the radius R of the ℓ ₁ -ball constraint by Morozov’s discrepancy principle. Numerical results are reported to illustrate the efficiency of the proposed approach.

Image normally has both dots-like and curve structures. But the traditional wavelet or multidirectional wave (ridgelet, contourlet, curvelet, etc.) could only restore one of these structures efficiently so that the restoration results for complex images are unsatisfactory. For the image restoration, this paper adopted a strategy of combined shearlet and wavelet frame and proposed a new restoration method. Theoretically, image sparse representation of dots-like and curve structures could be achieved by shearlet and wavelet, respectively. Under theL1regularization, the two frame-sparse structures could show their respective advantages and efficiently restore the two structures. In order to achieve superlinear convergence, this paper applied semismooth Newton method based on subgradient to solve objective functional without differentiability. Finally, through numerical results, the effectiveness of this strategy was validated, which presented outstanding advantages for any individual frame alone. Some detailed information that could not be restored in individual frame could be clearly demonstrated with this strategy.

In this paper, Morozov's discrepancy principle is considered for the non-convex $ \alpha\ell_1-\beta\ell_2 $ sparsity regularization ($ \alpha>\beta>0 $). It is shown that if $ \tau>1 $ satisfies some conditions, there exists a regularization parameter $ \alpha $ such that $ \delta\leq \|A(x_{\alpha,\beta}^{\delta})-y^{\delta}\|_Y\leq \tau \delta $ holds. Furthermore, it is shown that $ \alpha $ converges to 0 as $ \delta\rightarrow 0 $. In addition, well-posedness and convergence rate results are presented for the regularized solution under Morozov's discrepancy principle. Numerical simulation results are reported to illustrate the efficiency of the proposed approach.