发表机构
Harvard University（哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明秩一感知矩阵下AMP算法的状态演化与高斯系综定量一致，误差为O(d^{-1/2})，从而确立其均方误差和谱分布的普适性。
AI 中文摘要
近似消息传递（AMP）算法因其计算效率高，并能通过低维的“状态演化”递归获得精确刻画而颇具吸引力。在本工作中，我们针对具有中心化秩一感知矩阵 $Z_i=(x_ix_i^\top-I_d)/\sqrt d$ 的AMP建立了定量普适性，其中 $x_i$ 是独立的标准高斯向量。这些矩阵出现在二次回归和二次神经网络学习中。它们的归一化向量化在维度 $p=d(d+1)/2$ 上具有各向同性的协方差，但坐标间存在强依赖。对于带高斯噪声的线性观测和指定的光滑有界谱去噪器，我们将秩一感知与协方差匹配的各向同性高斯系综进行比较。在固定的迭代范围内，当 $n\leq Cp$ 且归一化信号能量、初始化、系数以及去噪器前四阶导数具有一致界时，归一化信号重叠、跨时间重叠以及光滑线性谱统计量在每个固定 $L^m$ 中相差 $O(d^{-1/2})$。这产生了归一化均方误差、固定谱矩和经验谱分布的普适性。每当高斯观测量允许联合状态演化极限时，相同的预测在秩一感知下也成立。证明使用了逐列Lindeberg替换、留一法分析、光滑谱展开和高阶矩集中。对于指定的光滑谱去噪器，该结果确立了Maillard等人（arXiv:2408.03733）所 conjectured 的AMP普适性。
英文摘要
Approximate Message Passing (AMP) algorithms are attractive as they are computationally efficient and simultaneously admit a precise characterization in terms of the low-dimensional "state-evolution" recursion. In this work, we establish quantitative universality for AMP with centered rank-one sensing matrices $Z_i=(x_ix_i^\top-I_d)/\sqrt d$, where $x_i$ are independent standard Gaussian vectors. These matrices arise in quadratic regression and learning quadratic neural networks. Their normalized vectorizations have isotropic covariance in dimension $p=d(d+1)/2$, but strongly dependent coordinates. For linear observations with Gaussian noise and prescribed smooth, bounded spectral denoisers, we compare rank-one sensing with a covariance-matched isotropic Gaussian ensemble. At a fixed iteration horizon, with $n\leq Cp$ and uniform bounds on the normalized signal energy, initialization, coefficients, and first four denoiser derivatives, normalized signal overlaps, cross-time overlaps, and smooth linear spectral statistics differ by $O(d^{-1/2})$ in every fixed $L^m$. This yields universality of the normalized mean-squared error, fixed spectral moments, and empirical spectral distributions. Whenever the Gaussian observables admit a joint state-evolution limit, the same predictions hold under rank-one sensing. The proof uses columnwise Lindeberg replacement, leave-one-out analysis, smooth spectral expansions, and high-moment concentration. For prescribed smooth spectral denoisers, the result establishes the AMP universality conjectured by Maillard et al. (arXiv:2408.03733).
