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.

翻译：我们建立了植入式-植入式测试中低次多项式检验的首个尖锐阈值，其目标是以渐近可忽略的错误率判断两个结构化植入机制中哪一个生成了观测数据。我们证明了植入子矩阵和植入稠密子图模型中计数社区的低次上下界匹配。所得测试阈值在精确常数级别上与已知低次恢复阈值一致。相比之下，弱测试（其目标是优于随机猜测）不存在尖锐阈值，而是呈现我们识别出的平滑过渡。为证明我们的结论，我们开发了一个基于低次恢复中潜变量展开的植入式-植入式测试框架，并采用新方法识别并剪除非信号贡献。