# Continuum Limit of Lipschitz Learning on Graphs

arXiv: https://arxiv.org/abs/2012.03772 · HTML: https://mlsanity.com/paper/2012.03772/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Dec 2020.

## Built on by (most cited first, 6 of 6)

- [Gamma-convergence of a nonlocal perimeter arising in adversarial machine learning](https://mlsanity.com/paper/2211.15223/) (2022)
- [Uniform Convergence Rates for Lipschitz Learning on Graphs](https://mlsanity.com/paper/2111.12370/) (2021)
- [Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise consistency and global lower bounds](https://mlsanity.com/paper/2307.02378/) (2023)
- [Hypergraph p-Laplacian regularization on point clouds for data interpolation](https://mlsanity.com/paper/2405.01109/) (2024)
- [Higher-Order Regularization Learning on Hypergraphs](https://mlsanity.com/paper/2510.26533/) (2025)
- [Planner-Admissible Graph-PDE Value Extensions for Sparse Goal-Conditioned Planning](https://mlsanity.com/paper/2605.19185/) (2026)

## Abstract (arXiv)

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<∞ continuum limits of this approach were studied using tools from Gamma-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 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^∞-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 Lp-norm, and, as a by-product, prove convergence of graph distance functions to geodesic distance functions.