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