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.

中文翻译：

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尖峰模型中近似消息传递的非渐近框架

近似消息传递 (AMP) 成为解决高维统计问题的有效迭代范式。然而，当迭代次数超过 $o\big(\frac{\log n}{\log\log n}\ big)$ ($n$ 是问题维度)。为了解决这一不足，本文开发了一个非渐近框架来理解尖峰矩阵估计中的 AMP。基于 AMP 更新和可控残差项的新分解，我们制定了一个分析方法来表征存在独立初始化时 AMP 的有限样本行为，进一步推广以允许谱初始化。作为提议的分析方法的两个具体结果：(i) 在求解 $\mathbb{Z}_2$ 同步时，我们预测频谱初始化 AMP 的行为高达 $O\big(\frac{n}{\mathrm{poly}\log n}\big)$迭代，表明该算法无需后续细化阶段即可成功（正如 \citet{celentano2021local} 最近推测的那样）；(ii) 我们在广泛的信噪比范围内表征稀疏 PCA（在尖峰 Wigner 模型中）中 AMP 的非渐近行为。