Non-local Discrete $$\infty $$ ∞ -Poisson and Hamilton Jacobi Equations

Abstract

In this paper we propose an adaptation of the ∞-Poisson equation on weighted graphs, and propose a finer expression of the ∞-Laplace operator with gradient terms on weighted graphs, by making the link with the biased version of the tug-of-war game. By using this formulation, we propose a hybrid ∞-Poisson Hamilton-Jacobi equation, and we show the link between this version of the ∞-Poisson equation and the adaptation of the eikonal equation on weighted graphs. Our motivation is to use this extension to compute distances on any discrete data that can be represented as a weighted graph. Through experiments and illustrations, we show that this formulation can be used in the resolution of many applications in image, 3D point clouds, and high dimensional data processing using a single framework.

Limits and consistency of nonlocal and graph approximations to the Eikonal equation

“…For instance, (H.11) holds when ψ = 0 and P ≥ 0. This example, when considered on weighted graphs (see Section 4), corresponds to computing distances on discrete images, meshes, point clouds, or any data that can be represented as a weighted graph; see [52,20] and references therein.…”

“…In recent years, nonlinear partial differential equations (PDEs) on graphs and networks have attracted increasing interest since they naturally arise in many practical problems in mathematics, physics, biology, economy and data science (e.g., internet and vehicular traffic, social networks, population dynamics, image processing and computer vision, machine learning); see [6,22,24,40] and references therein. Among those PDEs, Hamilton-Jacobi equations, including Eikonal-type equations, have been considered in [20,21,32,50,51,52] on weighted graphs for data processing, and in [1,9,10,29,44] on topological networks or other very special types of networks. From a different motivation, Hamilton-Jacobi equations on graphs were also studied in [48] to derive discrete versions of some functional inequalities.…”

“…Adaptation of the Eikonal equation on graphs for discrete data processing has been proposed in [50,21]; see (1). This has led to many applications including semisupervised clustering and classification on meshes, point clouds, or images [50,21,52,20]; see also [36] which proposed a framework dedicated to solve the Eikonal equation on point clouds. Despite availability of compelling numerical evidence for the efficiency of (1) for these tasks, a clear theoretical understanding of its solutions is lacking and no results on its consistency are available to the best of our knowledge.…”

“…The stationary solution of (P ε ) would satisfy a corresponding Eikonal equation. There are many applications motivating (1), for instance in data processing, analysis and machine learning on graphs, on unstructured meshes or grids, and on point clounds; see [5,12,20,21,33,32,36,50,51,52] with different choices made for the potential P , boundary points and data ( Γ, ψ).…”