Over the last decades, two distinct approaches have been instrumental to our understanding of the computational complexity of statistical estimation. The statistical physics literature predicts algorithmic hardness through local stability and monotonicity properties of the Franz--Parisi (FP) potential \cite{franz1995recipes,franz1997phase}, while the mathematically rigorous literature characterizes hardness via the limitations of restricted algorithmic classes, most notably low-degree polynomial estimators \cite{hopkins2017efficient}. For many inference models, these two perspectives yield strikingly consistent predictions, giving rise to a long-standing open problem of establishing a precise mathematical relationship between them. In this work, we show that for estimation problems the power of low-degree polynomials is equivalent to the monotonicity of the annealed FP potential for a broad family of Gaussian additive models (GAMs) with signal-to-noise ratio $λ$. In particular, subject to a low-degree conjecture for GAMs, our results imply that the polynomial-time limits of these models are directly implied by the monotonicity of the annealed FP potential, in conceptual agreement with predictions from the physics literature dating back to the 1990s.

翻译：近几十年来，两种不同的方法对理解统计估计的计算复杂度起到了关键作用。统计物理学文献通过Franz-Parisi(FP)势的局部稳定性和单调性来预测算法难度\cite{franz1995recipes,franz1997phase}，而数学严格文献则通过受限算法类（尤其是低次多项式估计器）的局限性来描述难度\cite{hopkins2017efficient}。对于许多推断模型，这两种视角得出了高度一致的预测，由此引发了一个长期未解决的开放问题：如何在两者之间建立精确的数学关系。本研究证明，对于具有信噪比$λ$的广泛高斯加性模型族，估计问题的低次多项式能力等价于退火FP势的单调性。特别地，在满足高斯加性模型低次猜想的前提下，我们的结果表明这些模型的多项式时间极限直接由退火FP势的单调性决定，这与物理学文献可追溯至20世纪90年代的预测在概念上一致。