  We study the consistency of Lipschitz learning on graphs in the limit of
infinite unlabeled data and finite labeled data. Previous work has conjectured
that Lipschitz learning is well-posed in this limit, but is insensitive to the
distribution of the unlabeled data, which is undesirable for semi-supervised
learning. We first prove that this conjecture is true in the special case of a
random geometric graph model with kernel-based weights. Then we go on to show
that on a random geometric graph with self-tuning weights, Lipschitz learning
is in fact highly sensitive to the distribution of the unlabeled data, and we
show how the degree of sensitivity can be adjusted by tuning the weights. In
both cases, our results follow from showing that the sequence of learned
functions converges to the viscosity solution of an $\infty$-Laplace type
equation, and studying the structure of the limiting equation.
