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On the game p-Laplacian on weighted graphs with applications in image processing and data clustering†

Published online by Cambridge University Press:  03 July 2017

A. ELMOATAZ ,
X. DESQUESNES  and
M. TOUTAIN
A. ELMOATAZ
Affiliation:
Université de Caen Normandie and the ENSICAEN in the GREYC Laboratory, Image Team, 6 Boulevard Maréchal Juin, F-14050 Caen Cedex, France email: abderrahim.elmoataz-billah@unicaen.fr
X. DESQUESNES
Affiliation:
University of ORLEANS, Orleans, France email: xavier.desquesnes@univ-orleans.fr
M. TOUTAIN
Affiliation:
Datexim SAS, 51 Avenue de la Côte de Nacre, 14000 Caen Cedex, France email: matthieu.toutain@unicaen.fr
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Abstract

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Game-theoretic p-Laplacian or normalized p-Laplacian operator is a version of classical variational p-Laplacian which was introduced recently in connection with stochastic games called Tug-of-War with noise (Peres et al. 2008, Tug-of-war with noise: A game-theoretic view of the p-laplacian. Duke Mathematical Journal145(1), 91–120). In this paper, we propose an adaptation and generalization of this operator on weighted graphs for 1 ≤ p ≤ ∞. This adaptation leads to a partial difference operator which is a combination between 1-Laplace, infinity-Laplace and 2-Laplace operators on graphs. Then we consider the Dirichlet problem associated to this operator and we prove the uniqueness and existence of the solution. We show that the solution leads to an iterative non-local average operator on graphs. Finally, we propose to use this operator as a unified framework for interpolation problems in signal processing on graphs, such as image processing and machine learning.

Keywords

Game p-Laplaciangraph signal processingimage processingmachine learningTug-of-War games
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Special Issue 6: Big Data and Partial Differential Equations , December 2017 , pp. 922 - 948
Copyright
Copyright © Cambridge University Press 2017 

Footnotes

†

This work was funded under a doctoral grant supported by the Coeur et Cancer association, the regional council of Normandy, the European FEDER Grant (PLANUCA Project) and the project ANR GRAPHSIP.

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