# Analysis of p-Laplacian Regularization in Semi-Supervised Learning

arXiv: https://arxiv.org/abs/1707.06213 · HTML: https://mlsanity.com/paper/1707.06213/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jul 2017.

## Built on by (most cited first, 30 of 39)

- [Graph-based Semi-supervised Learning: A Comprehensive Review](https://mlsanity.com/paper/2102.13303/) (2021)
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- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [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)
- [Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise](https://mlsanity.com/paper/2206.11386/) (2022)
- [Poisson Reweighted Laplacian Uncertainty Sampling for Graph-based Active Learning](https://mlsanity.com/paper/2210.15786/) (2022)
- [Rates of Convergence for Regression with the Graph Poly-Laplacian](https://mlsanity.com/paper/2209.02305/) (2022)
- [Consistency of semi-supervised learning, stochastic tug-of-war games, and the p-Laplacian](https://mlsanity.com/paper/2401.07463/) (2024)
- [Laplace Learning in Wasserstein Space](https://mlsanity.com/paper/2511.13229/) (2025)
- [Multi-class Graph Clustering via Approximated Effective p-Resistance](https://mlsanity.com/paper/2306.08617/) (2023)
- [Implicit Graph Neural Diffusion Networks: Convergence, Generalization, and Over-Smoothing](https://mlsanity.com/paper/2308.03306/) (2023)
- [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)
- [Eikonal depth: an optimal control approach to statistical depths](https://mlsanity.com/paper/2201.05274/) (2022)
- [Joint Graph Learning and Model Fitting in Laplacian Regularized Stratified Models](https://mlsanity.com/paper/2305.02573/) (2023)
- [Continuum limit of p-biharmonic equations on graphs](https://mlsanity.com/paper/2404.19689/) (2024)
- [Hypergraph p-Laplacian regularization on point clouds for data interpolation](https://mlsanity.com/paper/2405.01109/) (2024)
- [Nonparametric regression on random geometric graphs sampled from submanifolds](https://mlsanity.com/paper/2405.20909/) (2024)
- [Graph Laplacian-based Bayesian Multi-fidelity Modeling](https://mlsanity.com/paper/2409.08211/) (2024)
- [Hypergraph p-Laplacian equations for data interpolation and semi-supervised learning](https://mlsanity.com/paper/2411.12601/) (2024)
- [Higher-Order Regularization Learning on Hypergraphs](https://mlsanity.com/paper/2510.26533/) (2025)
- [Finsler Geometry, Graph Neural Networks, and You](https://mlsanity.com/paper/2606.17185/) (2026)
- [Laplacian-based Semi-Supervised Learning in Multilayer Hypergraphs by Coordinate Descent](https://mlsanity.com/paper/2301.12184/) (2023)
- [A few-shot graph Laplacian-based approach for improving the accuracy of low-fidelity data](https://mlsanity.com/paper/2304.04862/) (2023)
- [p-Laplacian Transformer](https://mlsanity.com/paper/2311.03235/) (2023)
- [Interface Laplace Learning: Learnable Interface Term Helps Semi-Supervised Learning](https://mlsanity.com/paper/2408.05419/) (2024)
- [Enhancing Robustness of Graph Neural Networks through p-Laplacian](https://mlsanity.com/paper/2409.19096/) (2024)
- [Identifying Memorization of Diffusion Models through p-Laplace Analysis: Estimators, Bounds and Applications](https://mlsanity.com/paper/2505.08246/) (2025)

## Abstract (arXiv)

We investigate a family of regression problems in a semi-supervised setting. The task is to assign real-valued labels to a set of n sample points, provided a small training subset of N labeled points. A goal of semi-supervised learning is to take advantage of the (geometric) structure provided by the large number of unlabeled data when assigning labels. We consider random geometric graphs, with connection radius ε(n), to represent the geometry of the data set. Functionals which model the task reward the regularity of the estimator function and impose or reward the agreement with the training data. Here we consider the discrete p-Laplacian regularization. We investigate asymptotic behavior when the number of unlabeled points increases, while the number of training points remains fixed. We uncover a delicate interplay between the regularizing nature of the functionals considered and the nonlocality inherent to the graph constructions. We rigorously obtain almost optimal ranges on the scaling of ε(n) for the asymptotic consistency to hold. We prove that the minimizers of the discrete functionals in random setting converge uniformly to the desired continuum limit. Furthermore we discover that for the standard model used there is a restrictive upper bound on how quickly ε(n) must converge to zero as n → ∞. We introduce a new model which is as simple as the original model, but overcomes this restriction.