Sharp Low-Degree Thresholds for Planted-vs-Planted Testing

2026-06-03

AI Summary

This work investigates the problem of distinguishing between two structured generative models under the planted-vs-planted setting with vanishing error probability, focusing on community-counting tasks in the planted submatrix and dense subgraph models. We propose a unified analytical framework based on low-degree polynomial tests, introducing a latent-variable expansion technique inspired by low-degree recovery and incorporating signal–noise separation and pruning strategies. For the first time, we establish sharp thresholds for low-degree testing in such problems: strong detection exhibits a sharp phase transition, whereas weak detection displays a smooth transition. Our derived upper and lower bounds match precisely, and the resulting detection threshold aligns—up to constant factors—with the known low-degree recovery threshold.

Abstract

We establish the first sharp thresholds for low-degree polynomial tests in planted-vs-planted settings, where the goal is to determine with vanishing error which of two structured planted mechanisms generated the observed data. We prove matching low-degree upper and lower bounds for counting communities in the planted submatrix and planted dense subgraph models. The resulting testing threshold coincides, down to the sharp constant, with the known low-degree recovery threshold. In contrast, the task of weak testing, where the goal is to outperform random guessing, does not have a sharp threshold but rather a smooth transition, which we identify. To prove our results, we develop a framework for planted-vs-planted testing that builds on a latent-variable expansion originating in low-degree recovery and employs new methods to identify and prune non-signal contributions.

Problem

• planted-vs-planted testing
• low-degree polynomial tests
• sharp thresholds
• structured planted mechanisms
• weak testing

Innovation

• sharp threshold
• low-degree polynomial
• planted-vs-planted testing
• latent-variable expansion
• non-signal pruning