  We consider the community detection problem in sparse random hypergraphs
under the non-uniform hypergraph stochastic block model (HSBM), a general model
of random networks with community structure and higher-order interactions. When
the random hypergraph has bounded expected degrees, we provide a spectral
algorithm that outputs a partition with at least a $\gamma$ fraction of the
vertices classified correctly, where $\gamma\in (0.5,1)$ depends on the
signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the
number of vertices goes to infinity, our algorithm achieves weak consistency,
which improves the previous results in Ghoshdastidar and Dukkipati (2017) for
non-uniform HSBMs.
  Our spectral algorithm consists of three major steps: (1) Hyperedge
selection: select hyperedges of certain sizes to provide the maximal
signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition:
construct a regularized adjacency matrix and obtain an approximate partition
based on singular vectors; (3) Correction and merging: incorporate the
hyperedge information from adjacency tensors to upgrade the error rate
guarantee. The theoretical analysis of our algorithm relies on the
concentration and regularization of the adjacency matrix for sparse non-uniform
random hypergraphs, which can be of independent interest.
