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

Abstract

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.

Models for information propagation on graphs

“…From modelling and algorithmic developments, we can apply these results to larger datasets, extending the work of [24] to look at new robust methods for label propagation problems in semi-supervised learning. Our approach may be applied to propagation phenomena such as opinion formation on social networks or consensus in multi-agent systems, where methods such as graph Laplacians, [14,16,20] and other distance metrics are typically used. Our framework can also encapsulate control problems such as using s (or w) as a control, thus could be used in applications such as evacuation planning [1].…”

Partial differential equations and variational methods for geometric processing of images

“…We have not aimed to be exhaustive. Other PDE operators on graphs have found applications in image analysis, such as the p-Laplacian [21,32,33,34,54,74,80]. Other theoretical investigations include [57] and recently these methods have been used to study artificial neural networks [69,79].…”

Gradient and Lipschitz Estimates for Tug-of-War Type Games

“…The analytic and probabilistic results we obtain also apply to this limiting case. Moreover, let us also point out that the game theoretic p-Laplacian has gained interest in image processing and machine learning [Does11,ETT15,EDT17,ELT16]. It has also applications in economic modeling [NP17].…”