论文标题
基于稀疏恢复的误差函数的新颖正则化
A Novel Regularization Based on the Error Function for Sparse Recovery
论文作者
论文摘要
正规化通过添加有关所需解决方案（例如稀疏性）的额外信息来解决不适合问题的重要作用。许多正规化条款通常涉及某些矢量规范，例如$ l_1 $和$ l_2 $ norms。在本文中，我们提出了一个新颖的正则化框架，该框架使用误差函数来近似单位步骤函数。它可以被视为$ L_0 $ NORM的替代功能。误差函数相对于其内在参数的渐近行为表明，随着参数接近$ 0 $和$ \ infty $，提议的正则化可以近似标准的$ l_0 $，$ l_1 $ norms。从统计学上讲，它也不是$ L_1 $方法的偏差。然后，在从不确定的线性系统中恢复稀疏信号时，我们将误差函数合并到受约束或无约束的模型中。在计算上，可以通过保证收敛的迭代重新加权$ L_1 $（IRL1）算法来解决这两个问题。大量实验结果表明，在各种稀疏恢复方案中，所提出的方法的表现优于最先进的方法。
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.
