Implicit Hypergraph Neural Networks: A Stable Framework for Higher-Order Relational Learning with Provable Guarantees

TL;DR

This paper introduces Implicit Hypergraph Neural Networks (IHGNN), a stable and efficient framework for modeling higher-order relations in hypergraphs using implicit equilibrium formulations, with provable guarantees and superior performance.

Contribution

The paper proposes IHGNN, an implicit hypergraph neural network that overcomes limitations of layer-based models, providing stability, efficiency, and theoretical guarantees for higher-order relational learning.

Findings

IHGNN outperforms traditional hypergraph neural networks in accuracy.

IHGNN demonstrates robustness to initialization and hyperparameter changes.

The model achieves better generalization and stability in experiments.

Abstract

Many real-world interactions are group-based rather than pairwise such as papers with multiple co-authors and users jointly engaging with items. Hypergraph neural networks have shown great promise at modeling higher-order relations, but their reliance on a fixed number of explicit message-passing layers limits long-range dependency capture and can destabilize training as depth grows. In this work, we introduce Implicit Hypergraph Neural Networks (IHGNN), which bring the implicit equilibrium formulation to hypergraphs: instead of stacking layers, IHGNN computes representations as the solution to a nonlinear fixed-point equation, enabling stable and efficient global propagation across hyperedges without deep architectures. We develop a well-posed training scheme with provable convergence, analyze the oversmoothing conditions and expressivity of the model, and derive a transductive generalization bound on hypergraphs. We further present an implicit-gradient training procedure coupled with a projection-based stabilization strategy. Extensive experiments on citation benchmarks show that IHGNN consistently outperforms strong traditional graph/hypergraph neural network baselines in both accuracy and robustness. Empirically, IHGNN is resilient to random initialization and hyperparameter variation, highlighting its strong generalization and practical value for higher-order relational learning.