Recovery of a Planted k-Densest Sub-Hypergraph

Recovery of a planted k-densest sub-hypergraph is a fundamental problem that appears in different contexts, e.g. community detection, average case complexity, and neuroscience applications. The underlying hypergraph parameters determine the geometry of the solution space and the statistical dependency between solutions. This captures whether the structured signal is highly localized, naturally suggesting a criterion to determine the boundary conditions for which the recovery is possible. In this work, we provide new information-theoretic upper and lower bounds for the recovery problem. These bounds apply to the whole spectrum of the hypergraph parameters, ranging from complex combinatorial search problems with high statistical dependency between solutions, to an extremely localized solution space, equivalent to the random energy model. The new bounds improve significantly prior bounds on most of the interesting regimes, and also provide the first results on partial recovery.