Low-degree polynomials have emerged as a powerful paradigm for providing evidence of statistical-computational gaps across a variety of high-dimensional statistical models [Wein25]. For detection problems -- where the goal is to test a planted distribution $\mathbb{P}'$ against a null distribution $\mathbb{P}$ with independent components -- the standard approach is to bound the advantage using an $\mathbb{L}^2(\mathbb{P})$-orthonormal family of polynomials. However, this method breaks down for estimation tasks or more complex testing problems where $\mathbb{P}$ has some planted structures, so that no simple $\mathbb{L}^2(\mathbb{P})$-orthogonal polynomial family is available. To address this challenge, several technical workarounds have been proposed [SW22,SW25], though their implementation can be delicate. In this work, we propose a more direct proof strategy. Focusing on random graph models, we construct a basis of polynomials that is almost orthonormal under $\mathbb{P}$, in precisely those regimes where statistical-computational gaps arise. This almost orthonormal basis not only yields a direct route to establishing low-degree lower bounds, but also allows us to explicitly identify the polynomials that optimize the low-degree criterion. This, in turn, provides insights into the design of optimal polynomial-time algorithms. We illustrate the effectiveness of our approach by recovering known low-degree lower bounds, and establishing new ones for problems such as hidden subcliques, stochastic block models, and seriation models.

翻译：低次多项式已成为一种强大的范式，为各种高维统计模型中的统计-计算间隙提供证据[Wein25]。对于检测问题——其目标是检验具有独立分量的植入分布$\mathbb{P}'$与零分布$\mathbb{P}$——标准方法是使用$\mathbb{L}^2(\mathbb{P})$-正交多项式族来界定优势度。然而，当$\mathbb{P}$本身具有某些植入结构时（例如在估计任务或更复杂的检验问题中），由于缺乏简单的$\mathbb{L}^2(\mathbb{P})$-正交多项式族，该方法将失效。为应对这一挑战，已有若干技术性解决方案被提出[SW22,SW25]，但其实现过程可能较为复杂。本文提出一种更直接的证明策略。聚焦于随机图模型，我们在统计-计算间隙出现的精确机制中，构造了在$\mathbb{P}$下近似正交的多项式基。该近似正交基不仅为建立低次多项式下界提供了直接路径，还允许我们显式识别优化低次准则的多项式。这进一步为设计最优多项式时间算法提供了理论洞见。通过恢复已知的低次多项式下界，并为隐子团、随机块模型及序列排序模型等问题建立新的下界，我们验证了该方法的有效性。