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Hypergraph p-Laplacians and Scale Spaces.

Authors :
Fazeny, Ariane
Tenbrinck, Daniel
Lukin, Kseniia
Burger, Martin
Source :
Journal of Mathematical Imaging & Vision; Aug2024, Vol. 66 Issue 4, p529-549, 21p
Publication Year :
2024

Abstract

The aim of this paper is to revisit the definition of differential operators on hypergraphs, which are a natural extension of graphs in systems based on interactions beyond pairs. In particular, we focus on the definition of Laplacian and p-Laplace operators for oriented and unoriented hypergraphs, their basic properties, variational structure, and their scale spaces. We illustrate that diffusion equations on hypergraphs are possible models for different applications such as information flow on social networks or image processing. Moreover, the spectral analysis and scale spaces induced by these operators provide a potential method to further analyze complex data and their multiscale structure. The quest for spectral analysis and suitable scale spaces on hypergraphs motivates in particular a definition of differential operators with trivial first eigenfunction and thus more interpretable second eigenfunctions. This property is not automatically satisfied in existing definitions of hypergraph p-Laplacians, and we hence provide a novel axiomatic approach that extends previous definitions and can be specialized to satisfy such (or other) desired properties. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
09249907
Volume :
66
Issue :
4
Database :
Complementary Index
Journal :
Journal of Mathematical Imaging & Vision
Publication Type :
Academic Journal
Accession number :
179068653
Full Text :
https://doi.org/10.1007/s10851-024-01183-0