Analysis of Semi-Supervised Learning on Hypergraphs

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.