Submodular Hypergraphs: p-Laplacians, Cheeger Inequalities and Spectral Clustering
Machine Learning
2018-10-12 v4 Discrete Mathematics
Data Structures and Algorithms
Social and Information Networks
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.
Keywords
Cite
@article{arxiv.1803.03833,
title = {Submodular Hypergraphs: p-Laplacians, Cheeger Inequalities and Spectral Clustering},
author = {Pan Li and Olgica Milenkovic},
journal= {arXiv preprint arXiv:1803.03833},
year = {2018}
}
Comments
A short version of this paper is presented in ICML 2018. This version includes the definition of a sequence of eigenvalues for 1-Laplacian