The low-degree polynomial framework has emerged as a powerful tool for providing evidence of statistical-computational gaps in high-dimensional inference. For detection problems, the standard approach bounds the low-degree advantage through an explicit orthonormal basis. However, this method does not extend naturally to estimation tasks, and thus fails to capture the \emph{detection-recovery gap phenomenon} that arises in many high-dimensional problems. Although several important advances have been made to overcome this limitation \cite{SW22, SW25, CGGV25+}, the existing approaches often rely on delicate, model-specific combinatorial arguments. In this work, we develop a general approach for obtaining \emph{conditional computational lower bounds} for recovery problems from mild bounds on low-degree testing advantage. Our method combines the notion of algorithmic contiguity in \cite{Li25} with a cross-validation reduction in \cite{DHSS25} that converts successful recovery into a hypothesis test with lopsided success probabilities. In contrast to prior unconditional lower bounds, our argument is conceptually simple, flexible, and largely model-independent. We apply this framework to several canonical inference problems, including planted submatrix, planted dense subgraph, stochastic block model, multi-frequency angular synchronization, orthogonal group synchronization, and multi-layer stochastic block model. In the first three settings, our method recovers existing low-degree lower bounds for recovery in \cite{SW22, SW25} via a substantially simpler argument. In the latter three, it gives new evidence for conjectured computational thresholds including the persistence of detection-recovery gaps. Together, these results suggest that mild control of low-degree advantage is often sufficient to explain computational barriers for recovery in high-dimensional statistical models.

翻译：低度多项式框架已成为高维推断中证明统计-计算差距的有力工具。对于检测问题，标准方法通过显式正交基约束低度优势。然而，该方法无法自然扩展到估计任务，因而无法捕捉许多高维问题中出现的检测-恢复差距现象。尽管已有若干重要进展克服这一局限[22,25,+25]，现有方法常依赖于特定模型、细节繁复的组合论证。本文中，我们发展了一种通用方法，能从低度测试优势的弱约束出发，推导恢复问题的条件性计算下界。我们的方法将[25]中算法邻接性的概念与[25]中通过交叉验证降维将成功恢复转化为偏向性假设检验的技术相结合。与先前的无条件下界不同，我们的论证在概念上简洁、灵活且基本与模型无关。我们将此框架应用于若干经典推断问题，包括植入子矩阵、植入稠密子图、随机块模型、多频角同步、正交群同步及多层随机块模型。在前三个场景中，我们的方法通过显著简化的论证复现了[22,25]中已有的恢复低度下界；在后三个场景中，它为包括检测-恢复差距持续存在的推测性计算阈值提供了新证据。这些结果共同表明，低度优势的弱约束通常足以解释高维统计模型中恢复问题的计算障碍。