Message-passing algorithms for synchronization problems over compact groups

TL;DR

This paper introduces an efficient message-passing algorithm for synchronization problems over compact groups, applicable to various fields, and analyzes its performance and limitations using statistical physics methods.

Contribution

It develops a novel AMP-based iterative algorithm for group synchronization that works with multiple frequency channels and extends to general compact groups.

Findings

Algorithm performs efficiently on large-scale problems.

Analysis reveals phases of easy, hard, and impossible problems.

Evidence suggests the algorithm is often information-theoretically optimal.

Abstract

Various alignment problems arising in cryo-electron microscopy, community detection, time synchronization, computer vision, and other fields fall into a common framework of synchronization problems over compact groups such as Z/L, U(1), or SO(3). The goal of such problems is to estimate an unknown vector of group elements given noisy relative observations. We present an efficient iterative algorithm to solve a large class of these problems, allowing for any compact group, with measurements on multiple 'frequency channels' (Fourier modes, or more generally, irreducible representations of the group). Our algorithm is a highly efficient iterative method following the blueprint of approximate message passing (AMP), which has recently arisen as a central technique for inference problems such as structured low-rank estimation and compressed sensing. We augment the standard ideas of AMP with ideas from representation theory so that the algorithm can work with distributions over compact groups. Using standard but non-rigorous methods from statistical physics we analyze the behavior of our algorithm on a Gaussian noise model, identifying phases where the problem is easy, (computationally) hard, and (statistically) impossible. In particular, such evidence predicts that our algorithm is information-theoretically optimal in many cases, and that the remaining cases show evidence of statistical-to-computational gaps.