# Multi-Frequency Phase Synchronization

## TL;DR

This paper introduces a multi-frequency approach to phase synchronization, formulating it as a nonconvex optimization problem and developing an efficient algorithm that outperforms existing methods by leveraging harmonic retrieval techniques.

## Contribution

The paper presents a novel multi-frequency formulation and a simple two-stage algorithm for phase synchronization, extending to general synchronization over compact Lie groups.

## Key findings

- Algorithm significantly outperforms state-of-the-art methods
- Utilizes multi-frequency information for improved accuracy
- Achieves these results with only mild additional computational costs

## Abstract

We propose a novel formulation for phase synchronization -- the statistical problem of jointly estimating alignment angles from noisy pairwise comparisons -- as a nonconvex optimization problem that enforces consistency among the pairwise comparisons in multiple frequency channels. Inspired by harmonic retrieval in signal processing, we develop a simple yet efficient two-stage algorithm that leverages the multi-frequency information. We demonstrate in theory and practice that the proposed algorithm significantly outperforms state-of-the-art phase synchronization algorithms, at a mild computational costs incurred by using the extra frequency channels. We also extend our algorithmic framework to general synchronization problems over compact Lie groups.