Approximate Message Passing (AMP) algorithms provide a valuable tool for studying mean-field approximations and dynamics in a variety of applications. Although these algorithms are often first derived for matrices having independent Gaussian entries or satisfying rotational invariance in law, their state evolution characterizations are expected to hold over larger universality classes of random matrix ensembles. We develop several new results on AMP universality. For AMP algorithms tailored to independent Gaussian entries, we show that their state evolutions hold over broadly defined generalized Wigner and white noise ensembles, including matrices with heavy-tailed entries and heterogeneous entrywise variances that may arise in data applications. For AMP algorithms tailored to rotational invariance in law, we show that their state evolutions hold over delocalized sign-and-permutation-invariant matrix ensembles that have a limit distribution over the diagonal, including sensing matrices composed of subsampled Hadamard or Fourier transforms and diagonal operators. We establish these results via a simplified moment-method proof, reducing AMP universality to the study of products of random matrices and diagonal tensors along a tensor network. As a by-product of our analyses, we show that the aforementioned matrix ensembles satisfy a notion of asymptotic freeness with respect to such tensor networks, which parallels usual definitions of freeness for traces of matrix products.

翻译：近似消息传递（AMP）算法为研究各种应用中的平均场近似和动力学提供了重要工具。尽管这类算法最初通常针对具有独立高斯分量或满足旋转不变性分布的矩阵推导，但其状态演化特性预期能够适用于更大范围的随机矩阵系综普适类。本文提出了关于AMP普适性的若干新结果。对于针对独立高斯分量设计的AMP算法，我们证明其状态演化可在广义定义的Wigner与白噪声系综上成立，包括可能出现在数据应用中的重尾分布条目与异质化逐元素方差矩阵。对于针对旋转不变性分布设计的AMP算法，我们证明其状态演化可在具有对角线上极限分布的离域化符号与置换不变矩阵系综上成立，包括由子采样哈达玛或傅里叶变换及对角算子构成的感知矩阵。我们通过简化的矩方法证明这些结果，将AMP普适性归结为沿张量网络的随机矩阵与对角张量乘积的研究。作为分析副产品，我们证明上述矩阵系综对该类张量网络满足某种渐近自由性概念，这与矩阵乘积迹的经典自由性定义相平行。