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.

通过添加有关所需解决方案的额外信息（例如稀疏性），正则化在解决不适定的问题中起着重要作用。许多正则化项通常涉及一些向量范数。本文提出了一种新颖的正则化框架，该框架使用误差函数来近似单位步长函数。它可以被视为\（L_0 \）规范的替代函数。误差函数相对于其固有参数的渐近行为表明，当参数分别接近0和\（\ infty，\）时，拟议的正则化可以逼近标准\（L_0 \）和\（L_1 \）范数。从统计上讲，它也比\（L_1 \）偏少方法。结合误差函数，我们考虑了受约束和不受约束的公式，以从欠定线性系统中重建稀疏信号。计算上，这两个问题都可以通过具有保证收敛性的迭代重新加权\（L_1 \）（IRL1）算法来解决。大量实验结果表明，在各种稀疏恢复方案中，所提出的方法均优于最新方法。