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.

中文翻译：

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一种新的基于误差函数的稀疏恢复正则化

正则化通过添加有关所需解决方案的额外信息（例如稀疏性），在解决不适定问题方面发挥着重要作用。许多正则化项通常涉及一些向量范数，例如 $L_1$ 和 $L_2$ 范数。在本文中，我们提出了一种新颖的正则化框架，该框架使用误差函数来逼近单位阶跃函数。它可以被视为 $L_0$ 范数的替代函数。误差函数相对于其内在参数的渐近行为表明，当参数分别接近 $0$ 和 $\infty,$ 时，所提出的正则化可以近似于标准的 $L_0$、$L_1$ 范数。从统计上讲，它也比 $L_1$ 方法偏差小。然后，当从欠定线性系统恢复稀疏信号时，我们将误差函数合并到约束或无约束模型中。在计算上，这两个问题都可以通过迭代重加权 $L_1$ (IRL1) 算法解决，并保证收敛。大量实验结果表明，所提出的方法在各种稀疏恢复场景中优于最先进的方法。