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 hp=‖⋅‖1p, 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 hp 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 hp represents a class of relative sparsity promoting functions.