Statistical and computational thresholds for the planted k-densest sub-hypergraph problem

Abstract

In this work, we consider the problem of recovery a planted k-densest sub-hypergraph on a d-uniform hypergraph. 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 statistical upper and lower bounds for the exact recovery threshold by the maximum-likelihood estimator, as well as algorithmic bounds based on approximate message passing algorithms. The problem exhibits a typical statistical-computational gap observed in analogous sparse settings that widens with increasing sparsity of the problem. The bounds show that the signal structure impacts the location of the statistical and computational phase transition not captured by the known existing bounds for the tensor PCA model. This effect is due to the generic planted signal prior that this latter model addresses.