Permutation Equivariant Framelet-based Hypergraph Neural Networks

Hypergraphs provide a natural and expressive framework for modeling high-order relationships, enabling the representation of group-wise interactions beyond pairwise connections. While hypergraph neural networks (HNNs) have shown promise for learning on such structures, existing models often rely on shallow message passing and lack the ability to extract multiscale patterns. Framelet-based techniques offer a principled solution by decomposing signals into multiple frequency bands. However, most prior framelet systems, particularly Haar-type ones, are sensitive to node ordering and fail to ensure consistent representations under permutation, leading to instability in hypergraph learning. To address this, we propose Permutation Equivariant Framelet-based Hypergraph Neural Networks (PEF-HNN), a novel framework that integrates multiscale framelet analysis with permutation-consistent learning. We construct a new family of permutation equivariant Haar-type framelets specifically designed for hypergraphs, supported by theoretical analysis of their stability and decomposition properties. Built upon these framelets, PEF-HNN incorporates both low-pass and high-pass components across multiple scales into a unified neural architecture. Extensive experiments on nine benchmark datasets, including three homophilic and four heterophilic hypergraphs, as well as two real-world datasets for visual object classification, demonstrate the effectiveness of our approach, consistently outperforming existing HNN baselines and highlighting the advantages of permutation equivariant framelet design in hypergraph representation learning.