Hypergraph p -Laplacian regularization on point clouds for data interpolation

Shi, Kehan; Burger, Martin

doi:10.1016/j.na.2025.113807

Journal Article

PUBDB-2026-01165

Hypergraph p -Laplacian regularization on point clouds for data interpolation

Shi, K. (Corresponding author)Extern* ; Burger, M.DESY*

2025
Elsevier, Pergamon Press Amsterdam [u.a.]

Nonlinear analysis / Theory, methods & applications 257, 113807 - (2025) [10.1016/j.na.2025.113807]

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Please use a persistent id in citations: doi:10.1016/j.na.2025.113807

Report No.: arXiv:2405.01109

Abstract: As a generalization of graphs, hypergraphs are widely used to model higher-order relationsin data. This paper explores the benefit of the hypergraph structure for the interpolationof point cloud data that contain no explicit structural information. We define the 𝜀𝑛-ballhypergraph and the 𝑘𝑛-nearest neighbor hypergraph on a point cloud and study the 𝑝-Laplacian regularization on the hypergraphs. We prove the variational consistency betweenthe hypergraph 𝑝-Laplacian regularization and the continuum 𝑝-Laplacian regularization in asemisupervised setting when the number of points 𝑛 goes to infinity while the number oflabeled points remains fixed. A key improvement compared to the graph case is that the resultsrely on weaker assumptions on the upper bound of 𝜀𝑛 and 𝑘𝑛. To solve the convex but non-differentiable large-scale optimization problem, we utilize the stochastic primal–dual hybridgradient algorithm. Numerical experiments on data interpolation verify that the hypergraph𝑝-Laplacian regularization outperforms the graph 𝑝-Laplacian regularization in preventing thedevelopment of spikes at the labeled points.

Classification:

ddc:510

Note: 34 pages

Contributing Institute(s):

Computational Imaging (FS-CI)

Research Program(s):

623 - Data Management and Analysis (POF4-623) (POF4-623)

Experiment(s):

No specific instrument

Database coverage:
Medline

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