Nonlocal infinity Laplacian equation on graphs with applications in image processing and machine learning

In this paper, an adaptation of the infinity Laplacian equation to weighted graphs is proposed. This adaptation leads to a nonlocal partial difference equation on graphs, which is an extension of the well-known approximations of the infinity Laplacian equation. To do so, we study the limit as p tends to infinity of minimizers of p-harmonic function on graphs. We also prove the existence and uniqueness of the solution of this equation. Our motivation stems from the extension of the nonlocal infinity Laplacian equation from image processing to machine learning fields, with proposed illustrations for image inpainting and semi-supervised clustering.