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 ℓ
₁
-Lasso. Such problems can be solved using the relevant proximal operator. Here we consider the use of proximal operators for the ℓ
_p
-quasinorm where 0 ≤
p
≤ 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
≤ 1 the algorithm is effective in the high-noise scenario, outperforming the Lasso despite the inexactness of the proximal step.