A non-convex piecewise quadratic approximation of $$\ell _{0}$$ ℓ 0 regularization: theory and accelerated algorithm

Non-convex regularization has been recognized as an especially important approach in recent studies to promote sparsity. In this paper, we study the non-convex piecewise quadratic approximation (PQA) regularization for sparse solutions of the linear inverse problem. It is shown that exact recovery of sparse signals and stable recovery of compressible signals are possible through local optimum of this regularization. After developing a thresholding representation theory for PQA regularization, we propose an iterative PQA thresholding algorithm (PQA algorithm) to solve this problem. The PQA algorithm converges to a local minimizer of the regularization, with an eventually linear convergence rate. Furthermore, we adopt the idea of accelerated gradient method to design the accelerated iterative PQA thresholding algorithm (APQA algorithm), which is also linearly convergent, but with a faster convergence rate. Finally, we carry out a series of numerical experiments to assess the performance of both algorithms for PQA regularization. The results show that PQA regularization outperforms $$\ell _1$$ ℓ 1 and $$\ell _{1/2}$$ ℓ 1 / 2 regularizations in terms of accuracy and sparsity, while the APQA algorithm is demonstrated to be significantly better than the PQA algorithm.