  Consider the community detection problem in random hypergraphs under the
non-uniform hypergraph stochastic block model (HSBM), where each hyperedge
appears independently with some given probability depending only on the labels
of its vertices. We establish, for the first time in the literature, a sharp
threshold for exact recovery under this non-uniform case, subject to minor
constraints; in particular, we consider the model with multiple communities ($K
\geq 2$). One crucial point here is that by aggregating information from all
the uniform layers, we may obtain exact recovery even in cases when this may
appear impossible if each layer were considered alone. Two efficient algorithms
that successfully achieve exact recovery above the threshold are provided. The
theoretical analysis of our algorithms relies on the concentration and
regularization of the adjacency matrix for non-uniform random hypergraphs,
which could be of independent interest. We also address some open problems
regarding parameter knowledge and estimation.
