p-Laplacians for Manifold-valued Hypergraphs

2025/07/14 by Jo Andersson Stokke, Stokke, Jo Andersson, Ronny Bergmann +5
Computer Science · Mathematics · #Topological and Geometric Data Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2507.10335

Abstract

Hypergraphs extend traditional graphs by enabling the representation of N-ary relationships through higher-order edges. Akin to a common approach of deriving graph Laplacians, we define function spaces and corresponding symmetric products on the nodes and edges to derive hypergraph Laplacians. While this has been done before for Euclidean features, this work generalizes previous hypergraph Laplacian approaches to accommodate manifold-valued hypergraphs for many commonly encountered manifolds.

Citations

• Manifold GCN: Diffusion-based Convolutional Neural Network for Manifold-valued Graphs
• Hyperbolic Neural Collaborative Recommender
• Hyperbolic Deep Neural Networks: A Survey
• Submodular Hypergraphs: p-Laplacians, Cheeger Inequalities and Spectral Clustering
• Normalized graph Laplacians for directed graphs

Related

• p-Laplacian Operators on Hypergraphs
• High-ordered Random Walks and Generalized Laplacians on Hypergraphs
• Re-imagining Spectral Graph Theory
• Hyperwalk Formulae for Even and Odd Laplacians in Finite CW-Hypergraphs
• Improved graph Laplacian via geometric self-consistency