A Novel Regularization Based on the Error Function for Sparse Recovery
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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., L_1 and L_2 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 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 ∞, respectively. Statistically, it is also less biased than the L_1 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 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.
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