在平滑性约束下的动态角同步
Dynamic angular synchronization under smoothness constraints
AI总结：
本文提出在平滑性约束下动态角同步的算法，通过非渐近保证的均方误差收敛性，解决了时间演变中的角度估计问题。
摘要：
给定一个无向测量图H=([n],E)，经典的角同步问题是在一组噪声的成对测量值（θ_i* - θ_j*）mod 2π的基础上恢复未知的角度θ_1*,…,θ_n*。这个问题出现在各种应用中，包括计算机视觉、分布式网络的时间同步以及从成对比较中进行排名。在本文中，我们考虑了一个动态版本的问题，其中角度和测量图在T个时间点上演变。假设潜在角度的演变具有平滑性条件，我们推导出三种联合估计所有时间点上角度的算法。此外，对于其中一个算法，我们建立了在不同统计模型下均方误差（MSE）的非渐近恢复保证。特别是，我们证明当T增加时，MSE收敛到零，其条件比静态设置更宽松。这包括测量图高度稀疏且不连通的情况，以及当测量噪声较大且可能随T增加的情况。我们通过合成数据的实验补充了我们的理论结果。
英文摘要：
Given an undirected measurement graph $\mathcal{H} = ([n], \mathcal{E})$, the classical angular synchronization problem consists of recovering unknown angles $θ_1^*,\dots,θ_n^*$ from a collection of noisy pairwise measurements of the form $(θ_i^* - θ_j^*) \mod 2π$, 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.
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