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

• Published in: European journal of applied mathematics Vol. 28; no. 6; pp. 922 - 948
• Main Authors: ELMOATAZ, A., DESQUESNES, X., TOUTAIN, M.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.12.2017; Cambridge University Press (CUP)
• Subjects: Adaptation; Applied mathematics; Clustering; Computer Science; Dirichlet problem; Game theory; Graphs; Image Processing; Iterative methods; Machine learning; Mathematical analysis; Operators (mathematics); Signal processing; Uniqueness; image processing; Game p-Laplacian; machine learning; graph signal processing; Tug-of-War games
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792517000122

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 Journal 145(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.