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 norm, e.g., L1 and L2 norms. In this paper, we propose 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 L0 norm. The asymptotic behavior of the error function with respect to its intrinsic parameter indicates that the proposed regularization can approximate the standard L0, L1 norms as the parameter approaches to 0 and ∞, respectively. Statistically, it is also less biased than the L1 approach. We then incorporate the error function into either a constrained or an unconstrained model when recovering a sparse signal from an under-determined linear system. Computationally, both problems can be solved via an iterative reweighted L1 (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.