Approximate message passing (AMP) emerges as an effective iterative paradigm for solving high-dimensional statistical problems. However, prior AMP theory -- which focused mostly on high-dimensional asymptotics -- fell short of predicting the AMP dynamics when the number of iterations surpasses $o\big(\frac{\log n}{\log\log n}\big)$ (with $n$ the problem dimension). To address this inadequacy, this paper develops a non-asymptotic framework for understanding AMP in spiked matrix estimation. Built upon new decomposition of AMP updates and controllable residual terms, we lay out an analysis recipe to characterize the finite-sample behavior of AMP in the presence of an independent initialization, which is further generalized to allow for spectral initialization. As two concrete consequences of the proposed analysis recipe: (i) when solving $\mathbb{Z}_2$ synchronization, we predict the behavior of spectrally initialized AMP for up to $O\big(\frac{n}{\mathrm{poly}\log n}\big)$ iterations, showing that the algorithm succeeds without the need of a subsequent refinement stage (as conjectured recently by \citet{celentano2021local}); (ii) we characterize the non-asymptotic behavior of AMP in sparse PCA (in the spiked Wigner model) for a broad range of signal-to-noise ratio.

翻译：近似信息传递( AMP) 是解决高维统计问题的有效迭代模式。 然而, 先前的 AMP 理论( 大多侧重于高维无症状学), 却在迭代数量超过美元( 问题维度为美元) ( 问题维度为美元) 时没有预测 AMP 动态 。 为解决这一不足, 本文开发了一个非隐性框架, 用于在加压的矩阵估测中理解 AMP 。 在对 AMP 更新和控制性剩余术语进行新的分解后, 我们推出一种分析配方, 用以在独立初始化时描述 AMP 的有限抽样行为, 进一步推广, 以允许光谱初始化 。 作为拟议分析配方的两个具体结果 :( 一) 在解决 $\ mathb2 同步时, 我们预测光度初始化的 AMP AMP 行为, 最高至 $O\ big (\ mag ( matthry{ pol) 和 可控制的信号残留性 剩余条件 。