Hypergraphs with Edge-Dependent Vertex Weights: p-Laplacians and Spectral Clustering

TL;DR

This paper introduces a novel hypergraph model with edge-dependent vertex weights, extending spectral clustering methods via p-Laplacians, and demonstrates improved clustering accuracy through efficient algorithms and real-world data experiments.

Contribution

It develops a framework for hypergraphs with edge-dependent vertex weights, enabling the extension of spectral theory and efficient eigenvector computation for enhanced clustering.

Findings

Higher clustering accuracy with EDVW-based spectral clustering

Effective eigenvector computation for hypergraph 1-Laplacian

Validation on real-world datasets shows improved performance

Abstract

We study p-Laplacians and spectral clustering for a recently proposed hypergraph model that incorporates edge-dependent vertex weights (EDVW). These weights can reflect different importance of vertices within a hyperedge, thus conferring the hypergraph model higher expressivity and flexibility. By constructing submodular EDVW-based splitting functions, we convert hypergraphs with EDVW into submodular hypergraphs for which the spectral theory is better developed. In this way, existing concepts and theorems such as p-Laplacians and Cheeger inequalities proposed under the submodular hypergraph setting can be directly extended to hypergraphs with EDVW. For submodular hypergraphs with EDVW-based splitting functions, we propose an efficient algorithm to compute the eigenvector associated with the second smallest eigenvalue of the hypergraph 1-Laplacian. We then utilize this eigenvector to cluster the vertices, achieving higher clustering accuracy than traditional spectral clustering based on the 2-Laplacian. More broadly, the proposed algorithm works for all submodular hypergraphs that are graph reducible. Numerical experiments using real-world data demonstrate the effectiveness of combining spectral clustering based on the 1-Laplacian and EDVW.