Strong Consistency of Spectral Clustering for the Sparse Degree-Corrected Hypergraph Stochastic Block Model

Abstract

We prove strong consistency of spectral clustering under the degree-corrected hypergraph stochastic block model in the sparse regime where the maximum expected hyperdegree is as small as $\Omega (\log n)$ with $n$ denoting the number of nodes. We show that the basic spectral clustering without preprocessing or postprocessing is strongly consistent in an even wider range of the model parameters, in contrast to previous studies that either trim high-degree nodes or perform local refinement. At the heart of our analysis is the entry-wise eigenvector perturbation bound derived by the “leave-one-out”technique. To the best of our knowledge, this is the first entry-wise error bound for degree-corrected hypergraph models, resulting in the strong consistency for clustering non-uniform hypergraphs with heterogeneous hyperdegrees.