Continuum Limit of Lipschitz Learning on Graphs

Abstract Tackling semi-supervised learning problems with graph-based methods has become a trend in recent years since graphs can represent all kinds of data and provide suitable framework for studying continuum limits, example, differential operators. A popular strategy here is p -Laplacian learning, which poses smoothness condition on the sought inference function set unlabeled data. For $$p<\infty $$ p < ∞ limits this approach were studied using tools from $$\varGamma Γ -convergence. case $$p=\infty = , referred to as Lipschitz related infinity Laplacian equation concept viscosity solutions. In work, we prove particular, define sequence functionals approximate largest local constant graph -convergence $$L^{\infty }$$ L -topology supremum norm gradient becomes denser. Furthermore, show compactness implies convergence minimizers. our analysis allow varying labeled converges general closed Hausdorff distance. We apply results nonlinear ground states, i.e., minimizers constrained $$L^p$$ -norm, and, by-product, distance functions geodesic functions.