A Unified Random Walk, Its Induced Laplacians and Spectral Convolutions for Deep Hypergraph Learning

Area of Science:

• Graph Theory
• Machine Learning
• Network Science

Background:

• Hypergraph modeling captures complex higher-order interactions.
• Existing random walks on hypergraphs have limitations in utilizing edge-dependent vertex weights (EDVWs) and expressiveness.
• Advanced hypergraph analysis requires more robust modeling techniques.

Purpose of the Study:

• To propose a unified random walk framework for hypergraph modeling that integrates hyperedge degrees and vertex weights.
• To develop a novel hypergraph Laplacian incorporating EDVWs for enhanced expressiveness and spectral properties.
• To introduce the General Hypergraph Spectral Convolution (GHSC) framework for effective deep hypergraph learning.

Main Methods:

• Developed a unified random walk framework for hypergraphs.
• Established equivalence conditions between hypergraph and graph random walks.
• Introduced a novel unified random-walk-based hypergraph Laplacian with EDVWs.
• Proposed the General Hypergraph Spectral Convolution (GHSC) framework, extending Graph Convolutional Neural Networks (GCNNs).

Main Results:

• The proposed framework integrates hyperedge degrees and vertex weights for robust hypergraph modeling.
• The unified hypergraph Laplacian exhibits desirable spectral properties and incorporates EDVWs.
• The GHSC framework demonstrates state-of-the-art performance across diverse datasets, including citation networks, visual objects, and protein modeling.
• Significant improvements were observed in protein structure modeling using EDVW-hypergraphs.

Conclusions:

• The unified random walk framework advances hypergraph modeling and spectral theory.
• The GHSC framework provides a versatile and effective approach for deep hypergraph learning.
• The integration of EDVWs enhances the performance of hypergraph learning tasks, particularly in complex domains like protein structure modeling.