Statistical and computational thresholds for the planted $k$-densest sub-hypergraph problem
This work addresses the fundamental problem of recovering dense sub-hypergraphs, relevant for community detection and neuroscience, by establishing precise recovery thresholds and algorithmic limits.
This paper investigates the planted k-densest sub-hypergraph problem, a structural variant of tensor-PCA, providing tight information-theoretic bounds for exact recovery via maximum-likelihood estimation and algorithmic bounds using approximate message passing. The problem exhibits a statistical-to-computational gap that widens with sparsity, and the signal structure significantly influences phase transitions, a factor not captured by existing tensor-PCA models.
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.