Low-degree estimation thresholds in planted hypergraphs and tensor PCA

Abstract

A central question in high-dimensional statistics is to understand statistical--computational gaps: regimes in which recovering a hidden signal is information-theoretically possible but conjectured to be computationally intractable. The low-degree framework offers a concrete way to study this gap by restricting attention to estimators that are polynomials of degree at most $D$ in the observed data. In this paper, we study low-degree estimation in planted dense subhypergraph, sparse tensor PCA, and tensor PCA with a general prior. For the planted dense subhypergraph model on $n$ vertices, we identify two regimes depending on whether the planted set is larger or smaller than $\sqrt{n}$. Above this scale, we identify a sharp threshold for low-degree estimation. Below this scale, we establish hardness in the regimes predicted by prior work, thereby resolving a question of Schramm and Wein (2022) and Sohn and Wein (2025). For sparse tensor PCA, we identify an analogous sharp phase transition. For tensor PCA with a general prior, we prove a low-degree estimation lower bound at the critical signal scale, matching the degree--signal tradeoff suggested by prior work. Our lower bounds apply to degree $D=n^δ$, where $n$ is the dimension and $δ>0$ is a constant, and we complement them with corresponding low-degree upper bounds. In addition, for planted dense subhypergraph and sparse tensor PCA above the $\sqrt{n}$ scale, we convert our upper bounds into polynomial-time algorithms that achieve almost exact recovery above the sharp threshold, yielding polynomial-time algorithms succeeding up to this threshold. Our proofs extend the framework of Sohn and Wein (2025) through a conditional variant that yields the correct signal-to-noise ratio in settings where the unconditional approach is insufficient.