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

TL;DR

This paper establishes the strong consistency of spectral clustering for sparse degree-corrected hypergraph stochastic block models, demonstrating its effectiveness without preprocessing in a broad parameter range.

Contribution

It proves the first entry-wise eigenvector perturbation bound for degree-corrected hypergraph models, enabling strong consistency results in sparse, non-uniform hypergraphs.

Findings

Spectral clustering is strongly consistent in sparse hypergraphs.

No need for trimming or local refinement in clustering.

First entry-wise eigenvector error bound for degree-corrected hypergraphs.

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.