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

Abderrahim Elmoataz (1) , Xavier Desquesnes (2) , Zakaria Lakhdari (1) , Olivier Lezoray (1)

1 Equipe Image - Laboratoire GREYC - UMR6072
2 image

In this paper, an adaptation of the infinity Laplacian equation to weighted graphs is proposed. This adaptation leads to a nonlocalpartial 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 anduniqueness of the solution of this equation. Our motivation stems from the extension of the nonlocal infinity Laplacian equationfrom image processing to machine learning fields, with proposed illustrations for image inpainting and semi-supervised clustering.

Mots clés: Nonlocal infinity Laplacian; Partial difference equations; Tug-of-war game; Weighted graphs; Image processing; Semi-supervised data clustering

Domaines: Traitement des images [eess.IV]; Apprentissage [cs.LG]

Abderrahim Elmoataz, Xavier Desquesnes, Zakaria Lakhdari, Olivier Lezoray. Nonlocal infinity Laplacian equation on graphs with applications in image processing and machine learning. Mathematics and Computers in Simulation, 2014, 102, pp.153--163. ⟨10.1016/j.matcom.2014.01.007⟩. ⟨hal-01018137⟩