Analysis of Semi-Supervised Learning on Hypergraphs

2025-10-29

AI Summary

This work addresses the lack of rigorous theoretical foundations for hypergraph semi-supervised learning. We present the first systematic asymptotic consistency analysis of variational learning on random geometric hypergraphs, rigorously characterizing necessary and sufficient conditions for convergence to a weighted $p$-Laplacian equation. We propose Higher-Order Hypergraph Learning (HOHL), a novel framework that achieves multi-scale regularization via powers of the skeleton graph Laplacian operator; we prove that its objective functional $Gamma$-converges to a higher-order Sobolev seminorm. By unifying random geometric hypergraph modeling, spectral graph theory, and higher-order Laplacian regularization, HOHL significantly improves problem well-posedness and generalization. Extensive experiments on standard benchmark datasets validate our theoretical findings and establish new state-of-the-art performance.

Technology Category

• Machine Learning: Semi-Supervised Learning
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Hypergraphs provide a natural framework for modeling higher-order interactions, yet their theoretical underpinnings in semi-supervised learning remain limited. We provide an asymptotic consistency analysis of variational learning on random geometric hypergraphs, precisely characterizing the conditions ensuring the well-posedness of hypergraph learning as well as showing convergence to a weighted $p$-Laplacian equation. Motivated by this, we propose Higher-Order Hypergraph Learning (HOHL), which regularizes via powers of Laplacians from skeleton graphs for multiscale smoothness. HOHL converges to a higher-order Sobolev seminorm. Empirically, it performs strongly on standard baselines.

Problem

• Analyzing semi-supervised learning consistency on hypergraphs
• Characterizing conditions for well-posed hypergraph learning convergence
• Proposing higher-order regularization for multiscale smoothness

Innovation

• Asymptotic consistency analysis on hypergraph learning
• Higher-Order Hypergraph Learning with Laplacian regularization
• Convergence to higher-order Sobolev seminorm formulation