Continuum limits of nonlocal $p$-Laplacian variational problems on graphs

Yosra Hafiene (1) , Jalal M. Fadili (1, 2, 3, 4) , Abderrahim Elmoataz (1)

1 Equipe Image - Laboratoire GREYC - UMR6072
2 NU - Normandie Université
3 ENSICAEN - École Nationale Supérieure d'Ingénieurs de Caen
4 GREYC - Groupe de Recherche en Informatique, Image et Instrumentation de Caen

In this paper, we study a nonlocal variational problem which consists of minimizing in $L^2$ the sum of a quadratic data fidelity and a regularization term corresponding to the $L^p$-norm of the nonlocal gradient. In particular, we study convergence of the numerical solution to a discrete version of this nonlocal variational problem to the unique solution of the continuum one. To do so, we derive an error bound and highlight the role of the initial data and the kernel governing the nonlocal interactions. When applied to variational problems on graphs, this error bound allows us to show the consistency of the discretized variational problem as the number of vertices goes to infinity. More precisely, for networks in convergent graph sequences (simple and weighted deterministic dense graphs as well as random inhomogeneous graphs), we prove convergence and provide rates of convergence of solutions for the discrete models to the solution of the continuum problem as the number of vertices grows.

Mots clés: nonlocal p-Laplacian; Variational problems; error bounds; graph limits; discrete solutions

Domaines: Traitement du signal et de l'image [eess.SP]; Théorie de l'information [cs.IT]; Théorie [stat.TH]; Machine Learning [stat.ML]; Analyse numérique [math.NA]; Analyse fonctionnelle [math.FA]; Statistiques [math.ST]; Optimisation et contrôle [math.OC]

Yosra Hafiene, Jalal M. Fadili, Abderrahim Elmoataz. Continuum limits of nonlocal $p$-Laplacian variational problems on graphs. SIAM Journal on Imaging Sciences, 2019. ⟨hal-02268163⟩