Dynamic angular synchronization under smoothness constraints
This addresses synchronization challenges in applications like computer vision and distributed networks, offering improved robustness in dynamic settings, though it is incremental as it extends static methods to time-varying scenarios.
The paper tackles the dynamic angular synchronization problem, where unknown angles and measurement graphs evolve over time, by developing three algorithms for joint estimation and proving that the mean-squared error converges to zero under milder conditions than static methods, including sparse graphs and high noise.
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.