Angular Synchronization by Eigenvectors and Semidefinite Programming

TL;DR

This paper introduces a robust eigenvector-based algorithm for angular synchronization that accurately estimates unknown angles from noisy, outlier-contaminated measurements, with applications in various fields.

Contribution

The paper presents a novel eigenvector method for angular synchronization that is highly stable and effective even with a large proportion of outliers, and connects it to semidefinite programming relaxations.

Findings

Successfully estimated 400 angles with 90% outliers in under a second.

Analyzed the method's performance using random matrix theory and information theory.

Extended the approach to other group synchronization problems with practical applications.

Abstract

The angular synchronization problem is to obtain an accurate estimation (up to a constant additive phase) for a set of unknown angles $\theta_1,...,\theta_n$ from $m$ noisy measurements of their offsets $\theta_i-\theta_j \mod 2\pi$. Of particular interest is angle recovery in the presence of many outlier measurements that are uniformly distributed in $[0,2\pi)$ and carry no information on the true offsets. We introduce an efficient recovery algorithm for the unknown angles from the top eigenvector of a specially designed Hermitian matrix. The eigenvector method is extremely stable and succeeds even when the number of outliers is exceedingly large. For example, we successfully estimate $n=400$ angles from a full set of $m={400 \choose 2}$ offset measurements of which 90% are outliers in less than a second on a commercial laptop. The performance of the method is analyzed using random matrix theory and information theory. We discuss the relation of the synchronization problem to the combinatorial optimization problem \textsc{Max-2-Lin mod} $L$ and present a semidefinite relaxation for angle recovery, drawing similarities with the Goemans-Williamson algorithm for finding the maximum cut in a weighted graph. We present extensions of the eigenvector method to other synchronization problems that involve different group structures and their applications, such as the time synchronization problem in distributed networks and the surface reconstruction problems in computer vision and optics.