A constructive approach for computing the proximity operator of the p-th power of the ℓ₁ norm

This note is to study the proximity operator of h_p=‖⋅‖₁^p, the power function of the ℓ₁ norm. For general p, computing the proximity operator requires solving a system of potentially highly nonlinear inclusions. For p=1, the proximity operator of h₁ is the well known soft-thresholding operator. For p=2, the function h₂ serves as a penalty function that promotes structured solutions to optimization problems of interest; the computation of the proximity operator of h₂ has been discussed in recent literature. By examining the properties of the proximity operator of the power function of the ℓ₁ 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 ℓ₁ 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.