  We consider the exact recovery problem in the hypergraph stochastic block
model (HSBM) with $k$ blocks of equal size. More precisely, we consider a
random $d$-uniform hypergraph $H$ with $n$ vertices partitioned into $k$
clusters of size $s = n / k$. Hyperedges $e$ are added independently with
probability $p$ if $e$ is contained within a single cluster and $q$ otherwise,
where $0 \leq q < p \leq 1$. We present a spectral algorithm which recovers the
clusters exactly with high probability, given mild conditions on $n, k, p, q$,
and $d$. Our algorithm is based on the adjacency matrix of $H$, which is a
symmetric $n \times n$ matrix whose $(u, v)$-th entry is the number of
hyperedges containing both $u$ and $v$. To the best of our knowledge, our
algorithm is the first to guarantee exact recovery when the number of clusters
$k=\Theta(\sqrt{n})$.
