# The Total Variation on Hypergraphs - Learning on Hypergraphs Revisited

arXiv: https://arxiv.org/abs/1312.5179 · HTML: https://mlsanity.com/paper/1312.5179/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Dec 2013.

## Built on by (most cited first, 30 of 35)

- [You are AllSet: A Multiset Function Framework for Hypergraph Neural Networks](https://mlsanity.com/paper/2106.13264/) (2021)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Equivariant Hypergraph Diffusion Neural Operators](https://mlsanity.com/paper/2207.06680/) (2022)
- [Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks](https://mlsanity.com/paper/2304.10031/) (2023)
- [Hypergraph Transformer for Skeleton-based Action Recognition](https://mlsanity.com/paper/2211.09590/) (2022)
- [Sheaf Hypergraph Networks](https://mlsanity.com/paper/2309.17116/) (2023)
- [I'm Me, We're Us, and I'm Us: Tri-directional Contrastive Learning on Hypergraphs](https://mlsanity.com/paper/2206.04739/) (2022)
- [Hypergraph Convolutional Networks via Equivalency between Hypergraphs and Undirected Graphs](https://mlsanity.com/paper/2203.16939/) (2022)
- [Hypergraph Neural Networks through the Lens of Message Passing: A Common Perspective to Homophily and Architecture Design](https://mlsanity.com/paper/2310.07684/) (2023)
- [A nonlinear diffusion method for semi-supervised learning on hypergraphs](https://mlsanity.com/paper/2103.14867/) (2021)
- [Adaptive Neural Message Passing for Inductive Learning on Hypergraphs](https://mlsanity.com/paper/2109.10683/) (2021)
- [Multi-Modal Hypergraph Diffusion Network with Dual Prior for Alzheimer Classification](https://mlsanity.com/paper/2204.02399/) (2022)
- [Local Hyper-Flow Diffusion](https://mlsanity.com/paper/2102.07945/) (2021)
- [Hypergraph Self-supervised Learning with Sampling-efficient Signals](https://mlsanity.com/paper/2404.11825/) (2024)
- [Nothing Stands Alone: Relational Fake News Detection with Hypergraph Neural Networks](https://mlsanity.com/paper/2212.12621/) (2022)
- [CAT-Walk: Inductive Hypergraph Learning via Set Walks](https://mlsanity.com/paper/2306.11147/) (2023)
- [A Versatile Framework for Attributed Network Clustering via K-Nearest Neighbor Augmentation](https://mlsanity.com/paper/2408.05459/) (2024)
- [Demystifying Higher-Order Graph Neural Networks](https://mlsanity.com/paper/2406.12841/) (2024)
- [Hypergraphs as Weighted Directed Self-Looped Graphs: Spectral Properties, Clustering, Cheeger Inequality](https://mlsanity.com/paper/2411.03331/) (2024)
- [Hypergraph Neural Diffusion: A PDE-Inspired Framework for Hypergraph Message Passing](https://mlsanity.com/paper/2604.10955/) (2026)
- [Multi-class Graph Clustering via Approximated Effective p-Resistance](https://mlsanity.com/paper/2306.08617/) (2023)
- [Modeling Edge-Specific Node Features through Co-Representation Neural Hypergraph Diffusion](https://mlsanity.com/paper/2405.14286/) (2024)
- [Hypergraph Modeling via Spectral Embedding Connection: Hypergraph Cut, Weighted Kernel k-means, and Heat Kernel](https://mlsanity.com/paper/2203.09888/) (2022)
- [Hypergraphs with Edge-Dependent Vertex Weights: p-Laplacians and Spectral Clustering](https://mlsanity.com/paper/2208.07457/) (2022)
- [Hypergraphs with Edge-Dependent Vertex Weights: Spectral Clustering based on the 1-Laplacian](https://mlsanity.com/paper/2305.00462/) (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)
- [Hypergraph p-Laplacian equations for data interpolation and semi-supervised learning](https://mlsanity.com/paper/2411.12601/) (2024)
- [Hypergraph clustering using Ricci curvature: an edge transport perspective](https://mlsanity.com/paper/2412.15695/) (2024)
- [Scalable Hypergraph Structure Learning with Diverse Smoothness Priors](https://mlsanity.com/paper/2504.03583/) (2025)

## Abstract (arXiv)

Hypergraphs allow one to encode higher-order relationships in data and are thus a very flexible modeling tool. Current learning methods are either based on approximations of the hypergraphs via graphs or on tensor methods which are only applicable under special conditions. In this paper, we present a new learning framework on hypergraphs which fully uses the hypergraph structure. The key element is a family of regularization functionals based on the total variation on hypergraphs.