Inexact Proximal Operators for 𝓁p-Quasinorm Minimization
Cian O'Brien, Mark D. Plumbley
Abstract
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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -Lasso. Such problems can be solved using the relevant proximal operator. Here we consider the use of proximal operators for the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> -quasinorm where 0 ≤ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> ≤ 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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> ≤ 1 the algorithm is effective in the high-noise scenario, outperforming the Lasso despite the inexactness of the proximal step.
BibTeX
@inproceedings{icassp2018_inexactproximalo,
title = {Inexact Proximal Operators for 𝓁p-Quasinorm Minimization},
author = {Cian O'Brien and Mark D. Plumbley},
booktitle = {ICASSP 2018},
year = {2018}
}