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

*Proceedings of the 35th International Conference on Machine Learning*, PMLR 80:3014-3023, 2018.

#### Abstract

We introduce submodular hypergraphs, a family of hypergraphs that have different submodular weights associated with different cuts of hyperedges. Submodular hypergraphs arise in cluster- ing 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.