# The game theoretic p-Laplacian and semi-supervised learning with few labels

arXiv: https://arxiv.org/abs/1711.10144 · HTML: https://mlsanity.com/paper/1711.10144/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Nov 2017.

## Built on by (most cited first, 22 of 22)

- [Uniform Convergence Rates for Lipschitz Learning on Graphs](https://mlsanity.com/paper/2111.12370/) (2021)
- [Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depth](https://mlsanity.com/paper/2202.08789/) (2022)
- [Poisson Reweighted Laplacian Uncertainty Sampling for Graph-based Active Learning](https://mlsanity.com/paper/2210.15786/) (2022)
- [Graph Laplacian for Semi-Supervised Learning](https://mlsanity.com/paper/2301.04956/) (2023)
- [Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications](https://mlsanity.com/paper/2111.03217/) (2021)
- [Rates of Convergence for Regression with the Graph Poly-Laplacian](https://mlsanity.com/paper/2209.02305/) (2022)
- [Transferability of Graph Neural Networks using Graphon and Sampling Theories](https://mlsanity.com/paper/2307.13206/) (2023)
- [Consistency of semi-supervised learning, stochastic tug-of-war games, and the p-Laplacian](https://mlsanity.com/paper/2401.07463/) (2024)
- [Numerical solution of a PDE arising from prediction with expert advice](https://mlsanity.com/paper/2406.05754/) (2024)
- [Laplace Learning in Wasserstein Space](https://mlsanity.com/paper/2511.13229/) (2025)
- [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)
- [Learning Label Initialization for Time-Dependent Harmonic Extension](https://mlsanity.com/paper/2205.01358/) (2022)
- [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)
- [Laplacian-based Semi-Supervised Learning in Multilayer Hypergraphs by Coordinate Descent](https://mlsanity.com/paper/2301.12184/) (2023)
- [p-Laplacian Transformer](https://mlsanity.com/paper/2311.03235/) (2023)
- [Convergence of graph Dirichlet energies and graph Laplacians on intersecting manifolds of varying dimensions](https://mlsanity.com/paper/2509.24458/) (2025)
- [Energy Approach from varepsilon-Graph to Continuum Diffusion Model with Connectivity Functional](https://mlsanity.com/paper/2510.25114/) (2025)
- [Planner-Admissible Graph-PDE Value Extensions for Sparse Goal-Conditioned Planning](https://mlsanity.com/paper/2605.19185/) (2026)
- [A Near-Linear-Time Solver for Graph p-Laplacian Semi-Supervised Learning via Continuation in p](https://mlsanity.com/paper/2607.03503/) (2026)
- [Target-Aware State-Adaptive p-Dirichlet Graph Neural Regression for Non-Invasive Body-Composition Estimation](https://mlsanity.com/paper/2608.29496/) (2026)

## Abstract (arXiv)

We study the game theoretic p-Laplacian for semi-supervised learning on graphs, and show that it is well-posed in the limit of finite labeled data and infinite unlabeled data. In particular, we show that the continuum limit of graph-based semi-supervised learning with the game theoretic p-Laplacian is a weighted version of the continuous p-Laplace equation. We also prove that solutions to the graph p-Laplace equation are approximately Holder continuous with high probability. Our proof uses the viscosity solution machinery and the maximum principle on a graph.