Hypergraph 1-Spectral Clustering with General Submodular Weights

We develop a spectral clustering algorithm for general submodular hypergraphs based on their 1-Laplacians. More precisely, we utilize the eigenvector associated with the second smallest eigenvalue of the hypergraph 1-Laplacian to cluster the vertices. The computation of the eigenvector is based on the inverse power method, the key of which is to solve its inner-loop optimization problem. Efficient solutions to this inner problem exist when submodular hypergraphs are equipped with hyperedge splitting functions with special structures such as when they are cardinality based or graph reducible. In this paper, we present a solution to the inner problem for general submodular splitting functions by adopting a random coordinate descent method together with the Fujishige-Wolfe algorithm. Numerical experiments using real-world data demonstrate the effectiveness of the proposed clustering algorithm.