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Spectral methods offer a tractable, global framework for clustering in graphsvia eigenvector computations on graph matrices. Hypergraph data, in whichentities interact on edges of arbitrary size, poses challenges for matrixrepresentations and therefore for spectral clustering. We study spectralclustering for nonuniform hypergraphs based on the hypergraph nonbacktrackingoperator. After reviewing the definition of this operator and its basicproperties, we prove a theorem of Ihara-Bass type which allows eigenpaircomputations to take place on a smaller matrix, often enabling fastercomputation. We then propose an alternating algorithm for inference in ahypergraph stochastic blockmodel via linearized belief-propagation whichinvolves a spectral clustering step again using nonbacktracking operators. Weprovide proofs related to this algorithm that both formalize and extend severalprevious results. We pose several conjectures about the limits of spectralmethods and detectability in hypergraph stochastic blockmodels in general,supporting these with in-expectation analysis of the eigeinpairs of our studiedoperators. We perform experiments in real and synthetic data that demonstratethe benefits of hypergraph methods over graph-based ones when interactions ofdifferent sizes carry different information about cluster structure.

Nonbacktracking spectral clustering of nonuniform hypergraphs