# Hypergraph p-Laplacian: A Differential Geometry View

arXiv: https://arxiv.org/abs/1711.08171 · HTML: https://mlsanity.com/paper/1711.08171/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Nov 2017.

## Built on by (most cited first, 11 of 11)

- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Sheaf Hypergraph Networks](https://mlsanity.com/paper/2309.17116/) (2023)
- [A nonlinear diffusion method for semi-supervised learning on hypergraphs](https://mlsanity.com/paper/2103.14867/) (2021)
- [A Comprehensive Survey on Spectral Clustering with Graph Structure Learning](https://mlsanity.com/paper/2501.13597/) (2025)
- [Multi-Modal Hypergraph Diffusion Network with Dual Prior for Alzheimer Classification](https://mlsanity.com/paper/2204.02399/) (2022)
- [Gaussian Processes on Hypergraphs](https://mlsanity.com/paper/2106.01982/) (2021)
- [Multi-class Graph Clustering via Approximated Effective p-Resistance](https://mlsanity.com/paper/2306.08617/) (2023)
- [Heterogeneous Temporal Hypergraph Neural Network](https://mlsanity.com/paper/2506.17312/) (2025)
- [Hypergraph Modeling via Spectral Embedding Connection: Hypergraph Cut, Weighted Kernel k-means, and Heat Kernel](https://mlsanity.com/paper/2203.09888/) (2022)
- [Two-dimensional Taxonomy for N-ary Knowledge Representation Learning Methods](https://mlsanity.com/paper/2506.05626/) (2025)
- [Higher-Order Regularization Learning on Hypergraphs](https://mlsanity.com/paper/2510.26533/) (2025)

## Abstract (arXiv)

The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph p-Laplacian. Unlike the existing two-node graph Laplacians, this generalization makes it possible to analyze hypergraphs, where the edges are allowed to connect any number of nodes. Moreover, we propose a semi-supervised learning method based on the proposed hypergraph p-Laplacian, and formalize them as the analogue to the Dirichlet problem, which often appears in physics. We further explore theoretical connections to normalized hypergraph cut on a hypergraph, and propose normalized cut corresponding to hypergraph p-Laplacian. The proposed p-Laplacian is shown to outperform standard hypergraph Laplacians in the experiment on a hypergraph semi-supervised learning and normalized cut setting.