A constructive approach for computing the proximity operator of the p-th power of the ℓ1 norm
A constructive approach for computing the proximity operator of the p-th power of the ℓ1 norm
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DOI：
10.1016/j.acha.2023.06.007
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发表时间：
2023
影响因子：
2.5
通讯作者：
Ashley Prater-Bennette;Lixin Shen;Erin E. Tripp
中科院分区：
文献类型：
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作者：
Ashley Prater-Bennette;Lixin Shen;Erin E. Tripp
本注是研究h1范数的幂函数p=‖⋅‖1p的邻近算子。对于一般p，计算接近算子需要求解一个潜在的高度非线性包含的系统。对于p= 1， h 1的接近算子是众所周知的软阈值算子。当p= 2时，函数h 2作为惩罚函数，促进对感兴趣的优化问题的结构化解决；最近的文献讨论了h2接近算子的计算。通过研究1范数幂函数的接近算子的性质，我们将开发一种简单而合理的方法来计算p与p> 1的接近算子。特别地，对于平方1范数函数，我们的方法提供了一种替代的，但明确的方法来找到它的接近算子。我们还讨论了h p的结构如何表示一类相对稀疏性促进函数。
This note is to study the proximity operator of h p=‖⋅‖ 1 p, the power function of the ℓ 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 ℓ 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 ℓ 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.