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Higher-Order Regularization Learning on Hypergraphs

Adrien Weihs, Andrea L. Bertozzi, Matthew Thorpe

TL;DR

This work advances HOHL, a multiscale, higher-order hypergraph regularization framework, by establishing explicit convergence rates for fully supervised minimizers and proving the consistency of truncated HOHL energies with the same continuum limit. It shows HOHL preserves the quadratic form of Laplace learning and can be interpreted as Laplace learning on a constructed graph, enabling standard solvers and strong active-learning performance. The analysis bridges discrete hypergraph energies and continuum Sobolev-type seminorms via the TL^p topology and Gamma-convergence, and extends HOHL to non-geometric hypergraphs with state-of-the-art results on benchmark datasets. Practically, HOHL delivers robust, efficient semi-supervised learning with improved sample efficiency, and its non-geometric extension broadens applicability to a wide range of data modalities.

Abstract

Higher-Order Hypergraph Learning (HOHL) was recently introduced as a principled alternative to classical hypergraph regularization, enforcing higher-order smoothness via powers of multiscale Laplacians induced by the hypergraph structure. Prior work established the well- and ill-posedness of HOHL through an asymptotic consistency analysis in geometric settings. We extend this theoretical foundation by proving the consistency of a truncated version of HOHL and deriving explicit convergence rates when HOHL is used as a regularizer in fully supervised learning. We further demonstrate its strong empirical performance in active learning and in datasets lacking an underlying geometric structure, highlighting HOHL's versatility and robustness across diverse learning settings.

Higher-Order Regularization Learning on Hypergraphs

TL;DR

This work advances HOHL, a multiscale, higher-order hypergraph regularization framework, by establishing explicit convergence rates for fully supervised minimizers and proving the consistency of truncated HOHL energies with the same continuum limit. It shows HOHL preserves the quadratic form of Laplace learning and can be interpreted as Laplace learning on a constructed graph, enabling standard solvers and strong active-learning performance. The analysis bridges discrete hypergraph energies and continuum Sobolev-type seminorms via the TL^p topology and Gamma-convergence, and extends HOHL to non-geometric hypergraphs with state-of-the-art results on benchmark datasets. Practically, HOHL delivers robust, efficient semi-supervised learning with improved sample efficiency, and its non-geometric extension broadens applicability to a wide range of data modalities.

Abstract

Higher-Order Hypergraph Learning (HOHL) was recently introduced as a principled alternative to classical hypergraph regularization, enforcing higher-order smoothness via powers of multiscale Laplacians induced by the hypergraph structure. Prior work established the well- and ill-posedness of HOHL through an asymptotic consistency analysis in geometric settings. We extend this theoretical foundation by proving the consistency of a truncated version of HOHL and deriving explicit convergence rates when HOHL is used as a regularizer in fully supervised learning. We further demonstrate its strong empirical performance in active learning and in datasets lacking an underlying geometric structure, highlighting HOHL's versatility and robustness across diverse learning settings.

Paper Structure

This paper contains 33 sections, 16 theorems, 139 equations, 4 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related works
  4. Background
  5. The TLp Topology
  6. Gamma-Convergence of Functionals
  7. Main results
  8. Hypergraphs
  9. HOHL
  10. Fully supervised problem with HOHL regularization.
  11. Truncated HOHL energies.
  12. Assumptions
  13. Main results
  14. Fully supervised problem with HOHL regularization
  15. Truncated energies
  16. ...and 18 more sections

Key Result

Proposition 2.2

Let $(\mu_n, u_n) \in \mathrm{TL}^{p}$ be a sequence and $(\mu, u) \in \mathrm{TL}^{p}$. Assume that $\mu$ is absolutely continuous with respect to the Lebesgue measure. Then the following are equivalent:

Figures (4)

  • Figure 1: From graphs to hypergraphs (from weihs2025Hypergraphs). 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: Skeleton graphs with $q = 2$ (from weihs2025Hypergraphs).
  • Figure 3: Accuracy in active learning using Laplacian and HOHL priors. We use $k^{(1)} = 50$, $k^{(2)} = 30$, $\lambda_1 = 1$, $\lambda_2 = 4$, $p_1 = 1$, $p_2 = 2$. Left: MNIST dataset. Right: fashionMNIST dataset.
  • Figure 4: Hyperedge size distributions for all datasets. Zoo and Mushroom exhibit nearly uniform distributions; Cora and Citeseer are bimodal, with both large and small hyperedges. $H$ denotes the total number of hyperedges in each case.

Theorems & Definitions (32)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3: Existence of transport maps
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6: Convergence of minimizers
  • Proposition 2.7: Convergence of minimizers
  • Theorem 3.1: Rates between discrete minimizers and labelling function
  • Remark 3.2: Optimal regularization parameter $\tau$
  • Theorem 3.3: Consistency of the truncated sum of Laplacians
  • ...and 22 more