Sharp Impossibility Results for Hyper-graph Testing

Jiashun Jin, Tracy Ke, Jiajun Liang

Keywords: community detection, social network, Degree-corrected Mix-membership model, non-uniform hypergraph, global testing, minimax lower bound, region of impossibility, tensor scaling, degree matching, SBM, tensor

TL;DR: Propose to use a degree matching strategy to derive sharp lower bounds for hypergraph global testing in a broad degree-corrected mixed-membership (DCMM) non-uniform hypergraph setting.

Abstract:

In a broad Degree-Corrected Mixed-Membership (DCMM) setting, we test whether a non-uniform hypergraph has only one community or has multiple communities. Since both the null and alternative hypotheses have many unknown parameters,
the challenge is, given an alternative, how to identify the null that is hardest to separate from the alternative. We approach this by proposing a degree matching strategy where the main idea is leveraging the theory for tensor scaling to create a least favorable pair of hypotheses. We present a result on standard minimax lower bound theory and a result on Region of Impossibility (which is more informative than the minimax lower bound). We show that our lower bounds are tight by introducing a new test that attains the lower bound up to a logarithmic factor. We also discuss the case where the hypergraphs may have mixed-memberships.

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