Analysis of Semi-Supervised Learning on Hypergraphs

TL;DR

This work develops a rigorous discrete-to-continuum framework for semi-supervised learning on hypergraphs. It proves pointwise convergence of discrete hypergraph operators to a weighted $p$-Laplacian and uses $ ext{Γ}$-convergence to characterize well- and ill-posed regimes as the interaction length-scale $psilon_n$ shrinks, revealing that classical hypergraph learning is inherently first-order. To overcome limitations, the authors introduce Higher-Order Hypergraph Learning (HOHL), which penalizes higher-order derivatives via powers of skeleton graph Laplacians across multiple scales, and they show HOHL converges to a higher-order Sobolev-type energy in the continuum. They also establish a formal link between HOHL and multiscale Laplacian learning, providing both theoretical guarantees and empirical evidence of improved performance in standard SSL benchmarks. The framework enables principled comparison of SSL methods through their continuum limits and offers design principles for higher-order regularization in hypergraph-based learning.

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.

Figures (5)

• Figure 1: From graphs to hypergraphs. Left: In the graph, the vertices $v_1$, $v_2$, and $v_3$ are all connected pairwise. Right: A single hyperedge is added connecting all three vertices, transitioning from a graph to a hypergraph representation.
• Figure 2: Classification of several algorithms based on their continuum limit. Edges indicate convergence to a Sobolev-type seminorm in the continuum limit, with each algorithm linked to its associated $\mathrm{W}^{(k,p)}$ space. Methods from left-to-right: calderSlepcev, LapRef, scholkopfHyper2006, elalaoui16, shi2025hypergraphplaplacianequationsdata, Stuart, Merkurjev, this work.
• Figure 3: Illustration of the HOHL energy with $p_k = k$. Left: For $q=2$, the energy imposes hierarchical regularization by penalizing $v^\top L^{(1)} v$ on skeleton edges $E^{(1)}$ and $v^\top (L^{(2)})^2 v$ on $E^{(2)}$. Right: With the random hypergraph model of \ref{['eq:main:hypergraphWeights']}, in high-density regions, hyperedges of large size capture finer structural details, and HOHL imposes stronger smoothness to exploit this local structure.
• Figure 4: Well- and Ill-posedness characterization of hypergraph learning as a function of the length-scale $\varepsilon_n$.
• Figure 5: Well- and Ill-posedness characterization of HOHL as a function of the length-scale $\varepsilon_n^{(q)}$. The striped regions are conjectured results.

Theorems & Definitions (56)

• Definition 2.1
• Proposition 2.2
• Theorem 2.3: Existence of transport maps
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Proposition 3.1: Discrete Euler-Lagrange equations of hypergraph learning
• Theorem 3.2: Pointwise consistency
• Theorem 3.3: Asymptotic consistency analysis of hypergraph learning
• Theorem 3.4: Asymptotic consistency analysis of higher-order hypergraph learning