This paper is concerned with the problem of reconstructing an unknown rank-one matrix with prior structural information from noisy observations. While computing the Bayes-optimal estimator seems intractable in general due to its nonconvex nature, Approximate Message Passing (AMP) emerges as an efficient first-order method to approximate the Bayes-optimal estimator. However, the theoretical underpinnings of AMP remain largely unavailable when it starts from random initialization, a scheme of critical practical utility. Focusing on a prototypical model called $\mathbb{Z}_{2}$ synchronization, we characterize the finite-sample dynamics of AMP from random initialization, uncovering its rapid global convergence. Our theory provides the first non-asymptotic characterization of AMP in this model without requiring either an informative initialization (e.g., spectral initialization) or sample splitting.

翻译：本文研究从含噪观测中重建具有先验结构信息的未知秩一矩阵问题。虽然贝叶斯最优估计器因其非凸性质通常难以计算，但近似消息传递（AMP）作为一种高效的一阶方法能够逼近贝叶斯最优估计。然而，当AMP从随机初始化（一种具有关键实用价值的方案）开始时，其理论基础仍大多缺失。聚焦于名为$\mathbb{Z}_{2}$同步的原型模型，我们刻画了从随机初始化出发的AMP的有限样本动态，揭示了其快速全局收敛的特性。我们的理论首次在该模型下给出了AMP无需信息性初始化（如谱初始化）或样本分割的非渐近刻画。