The angular synchronization problem aims to accurately estimate (up to a constant additive phase) a set of unknown angles $\theta_1, \dots, \theta_n\in[0, 2\pi)$ from $m$ noisy measurements of their offsets $\theta_i-\theta_j \;\mbox{mod} \; 2\pi.$ Applications include, for example, sensor network localization, phase retrieval, and distributed clock synchronization. An extension of the problem to the heterogeneous setting (dubbed $k$-synchronization) is to estimate $k$ groups of angles simultaneously, given noisy observations (with unknown group assignment) from each group. Existing methods for angular synchronization usually perform poorly in high-noise regimes, which are common in applications. In this paper, we leverage neural networks for the angular synchronization problem, and its heterogeneous extension, by proposing GNNSync, a theoretically-grounded end-to-end trainable framework using directed graph neural networks. In addition, new loss functions are devised to encode synchronization objectives. Experimental results on extensive data sets demonstrate that GNNSync attains competitive, and often superior, performance against a comprehensive set of baselines for the angular synchronization problem and its extension, validating the robustness of GNNSync even at high noise levels.

翻译：角度同步问题旨在从一组未知角度$\theta_1, \dots, \theta_n\in[0, 2\pi)$的偏移量$\theta_i-\theta_j \;\mbox{mod} \; 2\pi$的$m$个含噪测量值中，精确估计这些角度（相差一个常数加法相位）。其应用包括传感器网络定位、相位检索和分布式时钟同步等。该问题的异质扩展（称为$k$-同步）是在每组观测信号（分组分配未知）含噪的情况下，同时估计$k$组角度。现有角度同步方法在高噪声环境中通常表现不佳，而高噪声在应用中普遍存在。本文通过提出GNNSync——一种基于有向图神经网络、具有理论保障的端到端可训练框架——将神经网络应用于角度同步问题及其异质扩展。此外，我们设计了新的损失函数以编码同步目标。在大量数据集上的实验结果表明，在角度同步问题及其扩展中，GNNSync相对于全面的基线方法取得了具有竞争力且通常更优的性能，验证了GNNSync在高噪声水平下的鲁棒性。