Nonbacktracking spectral clustering of nonuniform hypergraphs

TL;DR

This paper introduces a spectral clustering method for nonuniform hypergraphs using the nonbacktracking operator, providing theoretical insights, an efficient algorithm, and demonstrating its advantages over traditional graph-based methods.

Contribution

It develops a spectral clustering framework for nonuniform hypergraphs based on the nonbacktracking operator, including theoretical analysis and an efficient inference algorithm.

Findings

Hypergraph spectral clustering outperforms graph-based methods in mixed interaction data.

The nonbacktracking operator enables faster eigenpair computations.

Experimental results validate the benefits of hypergraph methods in real and synthetic data.

Abstract

Spectral methods offer a tractable, global framework for clustering in graphs via eigenvector computations on graph matrices. Hypergraph data, in which entities interact on edges of arbitrary size, poses challenges for matrix representations and therefore for spectral clustering. We study spectral clustering for nonuniform hypergraphs based on the hypergraph nonbacktracking operator. After reviewing the definition of this operator and its basic properties, we prove a theorem of Ihara-Bass type which allows eigenpair computations to take place on a smaller matrix, often enabling faster computation. We then propose an alternating algorithm for inference in a hypergraph stochastic blockmodel via linearized belief-propagation which involves a spectral clustering step again using nonbacktracking operators. We provide proofs related to this algorithm that both formalize and extend several previous results. We pose several conjectures about the limits of spectral methods and detectability in hypergraph stochastic blockmodels in general, supporting these with in-expectation analysis of the eigeinpairs of our studied operators. We perform experiments in real and synthetic data that demonstrate the benefits of hypergraph methods over graph-based ones when interactions of different sizes carry different information about cluster structure.