Hypergraphs as Weighted Directed Self-Looped Graphs: Spectral Properties, Clustering, Cheeger Inequality

Hypergraphs naturally arise when studying group relations and have been widely used in the field of machine learning. To the best of our knowledge, the recently proposed edge-dependent vertex weights (EDVW) modeling is one of the most generalized modeling methods of hypergraphs, i.e., most existing hypergraph conceptual modeling methods can be generalized as EDVW hypergraphs without information loss. However, the relevant algorithmic developments on EDVW hypergraphs remain nascent: compared to the spectral theories for graphs, its formulations are incomplete, the spectral clustering algorithms are not well-developed, and the hypergraph Cheeger Inequality is not well-defined. To this end, deriving a unified random walk-based formulation, we propose our definitions of hypergraph Rayleigh Quotient, NCut, boundary/cut, volume, and conductance, which are consistent with the corresponding definitions on graphs. Then, we prove that the normalized hypergraph Laplacian is associated with the NCut value, which inspires our proposed HyperClus-G algorithm for spectral clustering on EDVW hypergraphs. Finally, we prove that HyperClus-G can always find an approximately linearly optimal partitioning in terms of both NCut and conductance. Additionally, we provide extensive experiments to validate our theoretical findings from an empirical perspective. Code of HyperClus-G is available at https://github.com/iDEA-iSAIL-Lab-UIUC/HyperClus-G.