Submodular Hypergraphs: p-Laplacians, Cheeger Inequalities and Spectral Clustering

TL;DR

This paper introduces submodular hypergraphs with weighted cuts, defines p-Laplacians, and develops spectral clustering algorithms, advancing higher-order clustering techniques for complex data structures.

Contribution

It presents the novel concept of submodular hypergraphs, defines p-Laplacians for them, and develops spectral clustering algorithms based on these operators.

Findings

Derived p-Laplacians for submodular hypergraphs

Established Cheeger inequalities and nodal domain theorems

Developed algorithms for spectral clustering using 1- and 2-Laplacians

Abstract

We introduce submodular hypergraphs, a family of hypergraphs that have different submodular weights associated with different cuts of hyperedges. Submodular hypergraphs arise in clustering applications in which higher-order structures carry relevant information. For such hypergraphs, we define the notion of p-Laplacians and derive corresponding nodal domain theorems and k-way Cheeger inequalities. We conclude with the description of algorithms for computing the spectra of 1- and 2-Laplacians that constitute the basis of new spectral hypergraph clustering methods.