Consistency of Lipschitz learning with infinite unlabeled data and finite labeled data

We prove that Lipschitz learning on graphs is consistent with the absolutely minimal Lipschitz extension problem in the limit of infinite unlabeled data and finite labeled data. In particular, we show that the continuum limit is independent of the distribution of the unlabeled data, which suggests the algorithm is fully supervised (and not semisupervised) in this setting. We also present some new ideas for modifying Lipschitz learning to incorporate the distribution of the unlabeled data.