  In this work, we consider the problem of recovery a planted $k$-densest
sub-hypergraph on $d$-uniform hypergraphs. This fundamental problem appears in
different contexts, e.g., community detection, average-case complexity, and
neuroscience applications as a structural variant of tensor-PCA problem. We
provide tight \emph{information-theoretic} upper and lower bounds for the exact
recovery threshold by the maximum-likelihood estimator, as well as
\emph{algorithmic} bounds based on approximate message passing algorithms. The
problem exhibits a typical statistical-to-computational gap observed in
analogous sparse settings that widen with increasing sparsity of the problem.
The bounds show that the signal structure impacts the location of the
statistical and computational phase transition that the known existing bounds
for the tensor-PCA model do not capture. This effect is due to the generic
planted signal prior that this latter model addresses.
