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 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 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 l₀, l_1/2, l_2/3, 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.