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arXiv · 2510.25354

Analysis of Semi-Supervised Learning on Hypergraphs

Adrien Weihs·Andrea L. Bertozzi·Matthew Thorpe
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Abstract

Hypergraphs provide a natural framework for modeling multiway interactions. We analyze a class of variational semi-supervised learning problems posed on random geometric hypergraphs and establish asymptotic consistency in the large-data limit. In particular, we identify scaling regimes that ensure well-posedness--yielding nontrivial label propagation rather than collapse to a constant labeling--and show that discrete minimizers converge, in the continuum, to solutions of a density-weighted p-Laplacian equation. We also propose Higher-Order Hypergraph Learning (HOHL), a multiscale regularization scheme based on powers of Laplacians associated with hypergraph-induced subgraphs. For geometric point clouds, we analyze an efficient multiscale Laplacian surrogate for HOHL and prove convergence to a higher-order Sobolev-type seminorm. Numerical experiments on standard benchmarks support the practical utility of the resulting higher-order regularization.

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Adrien Weihs, Andrea L. Bertozzi, Matthew Thorpe. 2025-10-29. Analysis of Semi-Supervised Learning on Hypergraphs. https://arxiv.org/abs/2510.25354

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