A Novel Regularization Based on the Error Function for Sparse Recovery

Regularization plays an important role in solving ill-posed problems by adding extra information about the desired solution, such as sparsity. Many regularization terms usually involve some vector norms. This paper proposes a novel regularization framework that uses the error function to approximate the unit step function. It can be considered as a surrogate function for the $$L_0$$ norm. The asymptotic behavior of the error function with respect to its intrinsic parameter indicates that the proposed regularization can approximate the standard $$L_0$$ , $$L_1$$ norms as the parameter approaches to 0 and $$\infty ,$$ respectively. Statistically, it is also less biased than the $$L_1$$ approach. Incorporating the error function, we consider both constrained and unconstrained formulations to reconstruct a sparse signal from an under-determined linear system. Computationally, both problems can be solved via an iterative reweighted $$L_1$$ (IRL1) algorithm with guaranteed convergence. A large number of experimental results demonstrate that the proposed approach outperforms the state-of-the-art methods in various sparse recovery scenarios.