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：针对带有峰值模型的近似消息传递的非渐进框架 翻译后的摘要： 近似消息传递（AMP）是解决高维统计问题的有效迭代范式。然而，之前的AMP理论——大多数是基于高维渐进性——无法预测当迭代次数超过$o\big(\frac{\log n}{\log\log n}\big)$（其中$n$为问题维度）时AMP动态的行为。为了解决这个问题，本文在带有峰值矩阵估计的情况下开发了一个非渐进框架来理解AMP。基于AMP更新的新分解以及可控的残差项，我们提出了一种分析方法来表征在独立初始化存在的情况下AMP的有限样本行为，该方法进一步推广以允许谱初始化。作为提出的分析方法的两个具体应用:(i) 当解决$\mathbb{Z}_2$同步问题时，我们预测了谱初始化的AMP的行为，迭代次数可以长达$O\big(\frac{n}{\mathrm{poly}\log n}\big)$，证明了该算法可以成功运行，而无需进行后续的优化阶段（正如\citet{celentano2021local}近期所推测的）；(ii) 我们表征了在稀疏主成分分析中（在带有峰值Wigner模型中），在广泛的信噪比范围内，AMP的非渐近行为。