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 Erd\H{o}s-R\'enyi hypergraph model with $n$ vertices and edge density $p$. We consider detecting the presence of a planted $G^r(n^\gamma, n^{-\alpha})$ subhypergraph in a $G^r(n, n^{-\beta})$ hypergraph, where $0< \alpha < \beta < r-1$ and $0 < \gamma < 1$. 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 $0 < \gamma < 1/2$, the threshold is given by $\alpha = \beta \gamma$, and for $1/2 \le \gamma < 1$, the threshold is given by $\alpha = \beta/2 + r(\gamma - 1/2)$. 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.