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

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 ℓ 0 and ℓ 1 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 ℓ 1/2 norm and ℓ 2/3 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 ℓ 0 , ℓ 1/2 , ℓ 2/3 , and ℓ 1 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.