Feishe Chen, Lixin Shen, Bruce W. Suter
Abstract: Sparse modelling with the $l_p$ norm of $0 \le p \le 1$ requires the availability of the proximity operator of the lp 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 lp 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 $0 \le p \le 1$ is applied to the $l_p$-regularisation for compressive sensing and image restoration.
Journal: IET Signal Processing, 10(5), 2016, 557-565.
DOI: 10.1049/iet-spr.2015.0244