近似消息传递与低次多项式之间的几乎尖锐等价性
Almost Sharp Equivalence between Approximate Message Passing and Low-Degree Polynomials
- School of Mathematical Sciences, Peking University（北京大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明高斯植入子矩阵模型中增长次数多项式估计的尖锐下界，与贝叶斯AMP误差匹配，解决伯努利秩一情形的AMP等价性问题。
AI中文摘要：
我们证明了高斯植入子矩阵模型中增长次数多项式估计的尖锐下界。观测模型为 $$ \boldsymbol{Y}= \frac{\lambda}{\sqrt{n}} \boldsymbol{\theta} \boldsymbol{\theta}^{\top}+\boldsymbol{W}, $$ 其中 $\boldsymbol{\theta}$ 的坐标是独立的 $\mathsf{Ber}(\rho)$ 变量，$\boldsymbol{W}$ 是对称矩阵，其严格上三角元素为独立的标准高斯变量。对于每个固定的 $\lambda>0$ 和 $\rho\in(0,1)$，我们给出了一个显式的有限维界，该界蕴含：任何次数为 $D(n)=o(n^{1/60})$ 的多项式估计器序列，其归一化均方误差的下极限至少为 $\rho-q_{\mathsf{amp}}/\lambda$，即贝叶斯近似消息传递（AMP）的极限误差。这扩展了 Montanari 和 Wein~\cite{montanari2025equivalence} 针对伯努利先验的常次数结果。结合他们对固定迭代 AMP 的多项式逼近，该界确定了当 $D(n)\to\infty$ 在此范围内时的精确极限低次 MMSE。因此，它解决了~\cite{wein2025computational, maleki2026high} 中讨论的增长次数 AMP 等价性问题的伯努利秩一情形。证明利用信号坐标及其乘积的\emph{条件}联合累积量构造了一个低次证书。具体而言，我们以校准到 AMP 不动点的辅助高斯信道 $\boldsymbol{R}$ 为条件。这保留了在无条件累积量界中丢失的信号依赖性，并产生了随次数增长所需的定量控制所需的抵消。本文中的大部分论证是使用 GPT-6 Astra 生成的。
英文摘要：
We prove a sharp lower bound for growing-degree polynomial estimation in the Gaussian planted submatrix model. The observation is $$ \boldsymbol{Y}= \fracλ{\sqrt{n}} \boldsymbolθ \boldsymbolθ^{\top}+\boldsymbol{W}, $$ where the coordinates of $\boldsymbolθ$ are independent $\mathsf{Ber}(ρ)$ variables and $\boldsymbol{W}$ is symmetric with independent standard Gaussian upper-triangular entries. For every fixed $λ>0$ and $ρ\in(0,1)$, we give an explicit finite-dimensional bound implying that every sequence of polynomial estimators of degree $D(n)=o(n^{1/60})$ has normalized mean-square error with limit inferior at least $ρ-q_{\mathsf{amp}}/λ$, the limiting error of Bayes approximate message passing (AMP). This extends the constant-degree result of Montanari and Wein~\cite{montanari2025equivalence} for the Bernoulli prior. Combined with their polynomial approximation of fixed-iteration AMP, the bound identifies the exact limiting low-degree MMSE whenever $D(n)\to\infty$ within this range. It therefore resolves the Bernoulli rank-one case of the growing-degree AMP-equivalence question discussed in~\cite{wein2025computational, maleki2026high}. The proof constructs a low-degree certificate using \emph{conditional} joint cumulants of the signal coordinates and their products. Specifically, we condition on an auxiliary Gaussian channel $\boldsymbol{R}$ calibrated to the AMP fixed point. This retains signal dependence that is lost in unconditional cumulant bounds and produces the cancellations needed for quantitative control as the degree grows. Most of the arguments in this paper were generated using GPT-6 Astra.
