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p-Laplacians for Manifold-valued Hypergraphs

Jo Andersson Stokke, Ronny Bergmann, Martin Hanik, Christoph von Tycowicz

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.

p-Laplacians for Manifold-valued Hypergraphs

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.

Paper Structure

This paper contains 10 sections, 2 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Oriented Hypergraphs
  4. Differential Geometry
  5. The Manifold Hypergraph Laplacian
  6. Function spaces on hypergraphs
  7. Gradients and Laplacians
  8. Experiment
  9. Conclusion
  10. Acknowledgments.

Key Result

lemma 1

The Fréchet gradient has the following two properties: The pairwise gradient has the following two properties:

Figures (1)

  • Figure 1: Random initial hypergraphs (first column) along equilibria of heat diffusion for $\Delta_2^{F}, \Delta_2^{P}$, and graph Laplacian (columns 2, 3, and 4, respectively).

Theorems & Definitions (5)

  • definition 1
  • definition 2: Gradients
  • lemma 1
  • definition 3: $p$-Laplacians
  • proposition 1