Approximate message passing from random initialization with applications to Z2 synchronization

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 is intractable in general due to the requirement of computing high-dimensional integrations/summations, 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 Z(2) synchronization, we characterize the finite-sample dynamics of AMP from random initialization, uncovering its rapid global convergence. Our theory-which is nonasymptotic in nature-in this model unveils the non -necessity of a careful initialization for the success of AMP.