Efficient Sparse Recovery and Demixing Using Nonconvex Regularization

Sparse demixing aims to separate signals that are sparse in some general dictionary, which has wide applications in signal and image processing, such as in super-resolution, image inpainting, robust sparse recovery, source separation, interference cancellation, saturation, and clipping restoration. For sparsity promotion in sparse demixing, the convex ℓ
₁
norm is of the most popular but it has a bias problem. In comparison, nonconvex regularization can mitigate the bias problem and can be expected to yield significantly better performance. In this paper, we employ the nonconvex ℓ
_q
-norm (0 ≤ q <; 1) for sparsity promotion and consider a linearly constrained ℓ
_q
-minimization formulation for the sparse demixing problem. Since the ℓ
_q
-minimization formulation is nonconvex and nonsmoothing, the standard alternative direction method of multipliers (ADMM) often fails to converge. To address this problem, we develop an iteratively reweighted ADMM algorithm which solves convex subproblems in each iteration and is convergent. Further, for the application of color image inpainting, we extend the new algorithm for multi-channel (RGB) joint recovery. The experimental results showed that the new algorithms can achieve significantly better performance than the ℓ
₁
algorithm.