This note is to study the proximity operator of $h_p={\Vert \cdot \Vert }_1^p$, the power function of the ${\ell }_1$ norm. For general p, computing the proximity operator requires solving a system of potentially highly nonlinear inclusions. For $p=1$, the proximity operator of $h_1$ is the well known soft-thresholding operator. For $p=2$, the function $h_2$ serves as a penalty function that promotes structured solutions to optimization problems of interest; the computation of the proximity operator of $h_2$ has been discussed in recent literature. By examining the properties of the proximity operator of the power function of the ${\ell }_1$ norm, we will develop a simple and well-justified approach to compute the proximity operator of $h_p$ with $p>1$. In particular, for the squared ${\ell }_1$ norm function, our approach provides an alternative, yet explicit way to finding its proximity operator. We also discuss how the structure of $h_p$ represents a class of relative sparsity promoting functions.

本笔记是为了研究邻近算子$H_p={\Vert \cdot \Vert }_1^p$, 的幂函数${\ell }_1$规范。对于一般的p，计算邻近算子需要求解潜在高度非线性包含物的系统。为了$p=1$，邻近算子$H_1$是众所周知的软阈值算子。为了$p=2$， 功能$H_2$作为惩罚函数，促进感兴趣的优化问题的结构化解决方案；的邻近算子的计算$H_2$最近的文献中已经讨论过。通过检查幂函数的邻近算子的性质${\ell }_1$规范，我们将开发一种简单且合理的方法来计算$H_p$和$p>1$。特别地，对于平方${\ell }_1$范数函数，我们的方法提供了一种替代但明确的方法来查找其邻近运算符。我们还讨论了如何构建$H_p$代表一类相对稀疏性促进函数。