Tug of War games and PDEs on graphs with applications in image and high dimensional data processing

Abstract

The aim of this note is to revisit the connections between some stochastic games, namely Tug-of-War games, and a class of nonlocal PDEs on graphs. We consider a general formulation of Tug-of-War games which is shown to be related to many classical PDEs in the continuous setting. We transcribe these equations on graphs using ad hoc differential operators and we show that it covers several nonlocal PDEs on graphs such as $$\infty $$ ∞ -Laplacian, game p-Laplacian and the eikonal equation. This unifying mathematical framework allows us to easily design simple algorithms to solve several inverse problems in imaging and data science, with a particular focus on cultural heritage and medical imaging.