  Community detection in hypergraphs is explored. Under a generative hypergraph
model called "d-wise hypergraph stochastic block model" (d-hSBM) which
naturally extends the Stochastic Block Model from graphs to d-uniform
hypergraphs, the asymptotic minimax mismatch ratio is characterized. For
proving the achievability, we propose a two-step polynomial time algorithm that
achieves the fundamental limit. The first step of the algorithm is a hypergraph
spectral clustering method which achieves partial recovery to a certain
precision level. The second step is a local refinement method which leverages
the underlying probabilistic model along with parameter estimation from the
outcome of the first step. To characterize the asymptotic performance of the
proposed algorithm, we first derive a sufficient condition for attaining weak
consistency in the hypergraph spectral clustering step. Then, under the
guarantee of weak consistency in the first step, we upper bound the worst-case
risk attained in the local refinement step by an exponentially decaying
function of the size of the hypergraph and characterize the decaying rate. For
proving the converse, the lower bound of the minimax mismatch ratio is set by
finding a smaller parameter space which contains the most dominant error
events, inspired by the analysis in the achievability part. It turns out that
the minimax mismatch ratio decays exponentially fast to zero as the number of
nodes tends to infinity, and the rate function is a weighted combination of
several divergence terms, each of which is the Renyi divergence of order 1/2
between two Bernoulli's. The Bernoulli's involved in the characterization of
the rate function are those governing the random instantiation of hyperedges in
d-hSBM. Experimental results on synthetic data validate our theoretical finding
that the refinement step is critical in achieving the optimal statistical
limit.
