UNIVERSALITY OF APPROXIMATE MESSAGE PASSING ALGORITHMS AND TENSOR NETWORKS

• 成果类型: Article
• 署名作者: Wang, Tianhao; Zhong, Xinyi; Fan, Zhou
• 署名单位: Yale University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2056
• 发表日期: 2024
• 页码: 3943-3994
• 关键词: phase-transitions; state evolution; INFORMATION; unitary; graphs

摘要

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 de- localized sign-and-permutation-invariant matrix ensembles that have a limit distribution over the diagonal, including sensing matrices composed of sub- sampled 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.