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

arXiv: https://arxiv.org/abs/1710.10364 · HTML: https://mlsanity.com/paper/1710.10364/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Oct 2017.

## Built on by (most cited first, 15 of 15)

- [Uniform Convergence Rates for Lipschitz Learning on Graphs](https://mlsanity.com/paper/2111.12370/) (2021)
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- [Improved Graph-based semi-supervised learning Schemes](https://mlsanity.com/paper/2407.00760/) (2024)
- [Robust Certification for Laplace Learning on Geometric Graphs](https://mlsanity.com/paper/2104.10837/) (2021)
- [Hypergraph p-Laplacian regularization on point clouds for data interpolation](https://mlsanity.com/paper/2405.01109/) (2024)
- [Graph-Based Semi-Supervised Segregated Lipschitz Learning](https://mlsanity.com/paper/2411.03273/) (2024)
- [Higher-Order Regularization Learning on Hypergraphs](https://mlsanity.com/paper/2510.26533/) (2025)
- [Computing Approximate ℓp Sensitivities](https://mlsanity.com/paper/2311.04158/) (2023)
- [Topology-Aware Active Learning on Graphs](https://mlsanity.com/paper/2510.25892/) (2025)
- [Planner-Admissible Graph-PDE Value Extensions for Sparse Goal-Conditioned Planning](https://mlsanity.com/paper/2605.19185/) (2026)

## Abstract (arXiv)

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 ∞-Laplace type equation, and studying the structure of the limiting equation.