Higher-Order Regularization Learning on Hypergraphs

TL;DR

This paper advances the theoretical understanding of Higher-Order Hypergraph Learning (HOHL) by proving the consistency of a truncated version, deriving convergence rates, and demonstrating its empirical effectiveness across various learning scenarios.

Contribution

It extends the theoretical foundation of HOHL by establishing consistency and convergence rates, and empirically validates its robustness in diverse applications.

Findings

Proved the consistency of a truncated HOHL version.

Derived explicit convergence rates for HOHL as a regularizer.

Demonstrated strong empirical performance in active learning and non-geometric datasets.

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.