Approximate message passing from random initialization with applications to $\mathbb{Z}_{2}$ synchronization

TL;DR

This paper analyzes the behavior of Approximate Message Passing (AMP) algorithms starting from random initialization in the context of $ ext{Z}_2$ synchronization, providing the first finite-sample, non-asymptotic theoretical guarantees for this setting.

Contribution

It offers the first non-asymptotic analysis of AMP from random initialization in $ ext{Z}_2$ synchronization, demonstrating rapid global convergence without special initializations.

Findings

AMP converges rapidly from random initialization in $ ext{Z}_2$ synchronization.

Theoretical characterization holds for finite samples without spectral initialization.

Provides insights into AMP's practical utility and theoretical foundations.

Abstract

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.