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

Gen Li; Wei Fan; Yuting Wei

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arXiv · 2302.03682

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

Gen Li·Wei Fan·Yuting Wei

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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.

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BibTeX RIS

Gen Li, Wei Fan, Yuting Wei. 2023-02-07. Approximate message passing from random initialization with applications to $\mathbb{Z}_{2}$ synchronization. https://arxiv.org/abs/2302.03682

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