Non-Local Morphological PDEs and $p$-Laplacian Equation on Graphs With Applications in Image Processing and Machine Learning

In this paper, we introduce a new class of non-local
p
-Laplacian operators that interpolate between non-local Laplacian and infinity Laplacian. These operators are discrete analogous of the game
p
-laplacian operators on Euclidean spaces, and involve discrete morphological gradient on graphs. We study the Dirichlet problem associated with the new
p
-Laplacian equation and prove existence and uniqueness of it's solution. We also consider non-local diffusion on graphs involving these operators. Finally, we propose to use these operators as a unified framework for solution of many inverse problems in image processing and machine learning.