Inexact Proximal Operators for $\ell_{p}$-Quasinorm Minimization

Proximal methods are an important tool in signal processing applications, where many problems can be characterized by the minimization of an expression involving a smooth fitting term and a convex regularization term - for example the classic $\ell_{1}$ -Lasso. Such problems can be solved using the relevant proximal operator. Here we consider the use of proximal operators for the $\ell_{p}$ -quasinorm where $0\leq p\leq 1$. Rather than seek a closed form solution, we develop an iterative algorithm using a Majorization-Minimization procedure which results in an inexact operator. Experiments on image denoising show that for $p\leq 1$ the algorithm is effective in the high-noise scenario, outperforming the Lasso despite the inexactness of the proximal step.