Computing the proximity operator of the ℓp norm with 0 < p < 1

Abstract

Sparse modelling with the l p norm of 0 ≤ p ≤ 1 requires the availability of the proximity operator of the l p norm. The proximity operators of the l0 and l1 norms are the well-known hard- and soft-thresholding estimators, respectively. In this study, the authors give a complete study on the properties of the proximity operator of the l p norm. Based on these properties, explicit formulas of the proximity operators of the l1/2 norm and l2/3 norm are derived with simple proofs; for other values of p, an iterative Newton's method is developed to compute the proximity operator of the l p norm by fully exploring the available proximity operators of the l0, l1/2, l2/3, and l1 norms. As applications, the proximity operator of the l p norm with 0 ≤ p ≤ 1 is applied to the l p -regularisation for compressive sensing and image restoration.