Almost Sharp Equivalence between Approximate Message Passing and Low-Degree Polynomials

Abstract

We prove a sharp lower bound for growing-degree polynomial estimation in the symmetric rank-one matrix estimation problem. The observation is \[ \boldsymbol{Y}=\frac{1}{\sqrt{n}} \boldsymbolθ \boldsymbolθ^{\top}+\boldsymbol{W}, \] where the coordinates of $\mathbf{\theta }$ are independent with a common, fixed sub-Gaussian distribution ${\pi }_{\Theta }$ satisfying $\mathbb{E}[\Theta ]\ne 0$, and $\mathbf{W}$ is symmetric with independent standard Gaussian upper-triangular entries. For every such prior, 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 $\mathbb{E}[{\Theta }^2]-q_{\mathsf{amp}}$, the limiting error of Bayes approximate message passing (AMP). This extends the constant-degree equivalence of Montanari and Wein~\cite{montanari2025equivalence} to growing degree under the same prior assumptions. 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 thereby answers the growing-degree AMP-equivalence question discussed in~\cite{wein2025computational, maleki2026high} for nonzero-mean sub-Gaussian priors in the stated degree range. We also obtain a moment-dependent finite-dimensional lower bound for any nonzero-mean prior with all moments finite. The proof bounds low-degree MMSE using \emph{conditional} joint cumulants and a refined application of Jensen's inequality. The conditional cumulants are defined using an auxiliary Gaussian channel $\mathbf{R}$ 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.

arXiv comment29 pages, 2 figures; extended the results and added a detailed description of AI usage