High-dimensional planted problems, such as finding a hidden dense subgraph within a random graph, often exhibit a gap between statistical and computational feasibility. While recovering the hidden structure may be statistically possible, it is conjectured to be computationally intractable in certain parameter regimes. A powerful approach to understanding this hardness involves proving lower bounds on the efficacy of low-degree polynomial algorithms. We introduce new techniques for establishing such lower bounds, leading to novel results across diverse settings: planted submatrix, planted dense subgraph, the spiked Wigner model, and the stochastic block model. Notably, our results address the estimation task -- whereas most prior work is limited to hypothesis testing -- and capture sharp phase transitions such as the "BBP" transition in the spiked Wigner model (named for Baik, Ben Arous, and Péché) and the Kesten-Stigum threshold in the stochastic block model. Existing work on estimation either falls short of achieving these sharp thresholds or is limited to polynomials of very low (constant or logarithmic) degree. In contrast, our results rule out estimation with polynomials of degree $n^δ$ where $n$ is the dimension and $δ> 0$ is a constant, and in some cases we pin down the optimal constant $δ$. Our work resolves open problems posed by Hopkins & Steurer (2017) and Schramm & Wein (2022), and provides rigorous support within the low-degree framework for conjectures by Abbe & Sandon (2018) and Lelarge & Miolane (2019).

翻译：高维植入问题，例如在随机图中寻找隐藏的稠密子图，通常在统计可行性与计算可行性之间存在差距。尽管恢复隐藏结构在统计上可能可行，但在某些参数区域内被推测为计算上难以处理。理解这种困难性的一种有力方法涉及证明低度多项式算法有效性的下界。我们引入了建立此类下界的新技术，从而在多种场景中取得了新颖结果：植入子矩阵、植入稠密子图、尖峰Wigner模型以及随机块模型。值得注意的是，我们的结果解决了估计任务——而先前的大多数工作仅限于假设检验——并捕捉到了尖锐的相变，例如尖峰Wigner模型中的“BBP”相变（以Baik、Ben Arous和Péché命名）以及随机块模型中的Kesten-Stigum阈值。现有关于估计的工作要么未能达到这些尖锐阈值，要么局限于非常低度（常数或对数级）的多项式。相比之下，我们的结果排除了使用度数为$n^δ$（其中$n$是维度，$δ>0$是常数）的多项式进行估计的可能性，并且在某些情况下我们确定了最优常数$δ$。我们的工作解决了由Hopkins & Steurer（2017）以及Schramm & Wein（2022）提出的开放问题，并在低度框架内为Abbe & Sandon（2018）以及Lelarge & Miolane（2019）的猜想提供了严谨支持。