  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 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 $p<\infty$ continuum limits of this approach were
studied using tools from $\Gamma$-convergence. For the case $p=\infty$, 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
$\Gamma$-convergence. In particular, we define a sequence of functionals which
approximate the largest local Lipschitz constant of a graph function and prove
$\Gamma$-convergence in the $L^\infty$-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, i.e.,
minimizers with constrained $L^p$-norm, and, as a by-product, prove convergence
of graph distance functions to geodesic distance functions.
