Given an undirected measurement graph $\mathcal{H} = ([n], \mathcal{E})$, the classical angular synchronization problem consists of recovering unknown angles $\theta_1^*,\dots,\theta_n^*$ from a collection of noisy pairwise measurements of the form $(\theta_i^* - \theta_j^*) \mod 2\pi$, for all $\{i,j\} \in \mathcal{E}$. This problem arises in a variety of applications, including computer vision, time synchronization of distributed networks, and ranking from pairwise comparisons. In this paper, we consider a dynamic version of this problem where the angles, and also the measurement graphs evolve over $T$ time points. Assuming a smoothness condition on the evolution of the latent angles, we derive three algorithms for joint estimation of the angles over all time points. Moreover, for one of the algorithms, we establish non-asymptotic recovery guarantees for the mean-squared error (MSE) under different statistical models. In particular, we show that the MSE converges to zero as $T$ increases under milder conditions than in the static setting. This includes the setting where the measurement graphs are highly sparse and disconnected, and also when the measurement noise is large and can potentially increase with $T$. We complement our theoretical results with experiments on synthetic data.

翻译：给定一个无向测量图 $\mathcal{H} = ([n], \mathcal{E})$，经典的角度同步问题旨在从一组带噪声的成对测量中恢复未知角度 $\theta_1^*,\dots,\theta_n^*$，这些测量形式为 $(\theta_i^* - \theta_j^*) \mod 2\pi$，适用于所有 $\{i,j\} \in \mathcal{E}$。该问题出现在多种应用中，包括计算机视觉、分布式网络的时间同步以及基于成对比较的排序。本文研究了该问题的动态版本，其中角度以及测量图随 $T$ 个时间点演化。假设潜在角度的演化满足平滑性条件，我们推导了三种算法用于联合估计所有时间点的角度。此外，针对其中一种算法，我们在不同的统计模型下建立了均方误差（MSE）的非渐近恢复保证。特别地，我们证明了在比静态设定更宽松的条件下，随着 $T$ 的增加，MSE 收敛于零。这包括测量图高度稀疏且不连通的情况，以及测量噪声较大且可能随 $T$ 增大的情况。我们通过合成数据实验对理论结果进行了补充验证。