Detection of Dense Subhypergraphs by Low-Degree Polynomials

Detection of a planted dense subgraph in a random graph is a fundamental statistical and computational problem that has been extensively studied in recent years. We study a hypergraph version of the problem. Let $G^r(n,p)$ denote the $r$-uniform Erdős-Rényi hypergraph model with $n$ vertices and edge density $p$. We consider detecting the presence of a planted $G^r(n^γ, n^{-α})$ subhypergraph in a $G^r(n, n^{-β})$ hypergraph, where $0< α< β< r-1$ and $0 < γ< 1$. Focusing on tests that are degree-$n^{o(1)}$ polynomials of the entries of the adjacency tensor, we determine the threshold between the easy and hard regimes for the detection problem. More precisely, for $0 < γ< 1/2$, the threshold is given by $α= βγ$, and for $1/2 \le γ< 1$, the threshold is given by $α= β/2 + r(γ- 1/2)$. Our results are already new in the graph case $r=2$, as we consider the subtle log-density regime where hardness based on average-case reductions is not known. Our proof of low-degree hardness is based on a conditional variant of the standard low-degree likelihood calculation.