p-Laplacians for Manifold-valued Hypergraphs

TL;DR

This paper generalizes hypergraph Laplacians to manifold-valued hypergraphs, enabling analysis of complex relationships in higher-order data structures across various manifolds.

Contribution

It introduces a novel framework for defining hypergraph Laplacians on manifold-valued data, extending previous Euclidean-based methods to a broader class of manifolds.

Findings

Framework for manifold-valued hypergraph Laplacians

Applicable to various common manifolds

Enables advanced analysis of higher-order relationships

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.