Analysis of Semi-Supervised Learning on Hypergraphs
This work addresses a theoretical gap in hypergraph learning for researchers, though it appears incremental as it builds on existing variational and Laplacian methods.
The paper tackles the limited theoretical understanding of semi-supervised learning on hypergraphs by providing an asymptotic consistency analysis and proposing Higher-Order Hypergraph Learning (HOHL), which regularizes with powers of Laplacians for multiscale smoothness and shows strong empirical performance on standard baselines.
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.