  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 $\infty,$ 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.
