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：角同步问题旨在从$m$个带有噪声的偏移量测量值$\theta_i-\theta_j \;\mbox{mod} \; 2\pi$中，精确估计一组未知角度$\theta_1, \dots, \theta_n\in[0, 2\pi)$（直至一个常数附加相位）。其应用场景包括传感器网络定位、相位检索和分布式时钟同步等。该问题在异质环境下的扩展（称为$k$-同步）则是在考虑每组未知组分配的情况下，同时估计$k$组角度。现有角同步方法在高噪声场景（实际应用中普遍存在）下通常表现不佳。本文通过提出GNNSync——一个基于有向图神经网络的理论完备端到端可训练框架，将神经网络应用于角同步问题及其异质扩展。此外，我们设计了新的损失函数来编码同步目标。在广泛数据集上的实验结果表明，GNNSync在角同步问题及其扩展任务中，相较于全面的基线方法取得了有竞争力甚至更优的性能，验证了GNNSync在高噪声水平下的鲁棒性。