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 Gr(n,p) denote the r-uniform ErdHos-R'enyi hypergraph model with n vertices and edge density p. We consider detecting the presence of a planted Gr(nγ, n-α) subhypergraph in a Gr(n, n-β) hypergraph, where 0< α < β < r-1 and 0 < γ < 1. Focusing on tests that are degree-no(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 = β γ, and for 1/2 ≤ 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.