Dynamic angular synchronization under smoothness constraints

TL;DR

This work addresses dynamic angular synchronization, where both latent angles and measurement graphs evolve over time under a smoothness constraint. It develops three algorithms—GTRS-DynSync, LTRS-GS-DynSync, and GMD-LTRS-DynSync—to jointly estimate angles across all time points, accompanied by non-asymptotic MSE guarantees for at least one method under AGN and Outliers noise models. Theoretical results show that, under mild smoothness, the MSE can vanish as $T\to\infty$, even with highly sparse or disconnected graphs and with noise that may grow with $T$, and are complemented by experiments on synthetic data. This dynamic framework extends static angular synchronization and ranking literature, offering practical methods for time-evolving synchronization problems with broad potential applications. The findings highlight robust performance of the matrix-denoising plus local-spectral approach, and suggest avenues for extending to other compact groups and dependent-time measurements.

Abstract

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.

Figures (9)

• Figure 1: RMSE (top) and Data Fidelity (bottom) for the Outliers model for a single MC run ($n = 30, T = 20, S_T = 1/T$, $\eta = p = 0.2$). Notice a sharp change in slope of the data-fidelity for LTRS-GS-DynSync and GMD-LTRS-DynSync at $\beta = 1$, which is also where the RMSE is minimized for these two methods. For GTRS-DynSync (with $\lambda_{\text{scale}} = 10$) the data-fidelity peaks at $\beta = 1$ which is where the RMSE is minimized.
• Figure 2: RMSE (top) and Data Fidelity (bottom) for the AGN model for a single MC run ($n = 30, T = 20, S_T = 1/T$, $\sigma = 3$). Notice a sharp change in slope of the data-fidelity for LTRS-GS-DynSync and GMD-LTRS-DynSync at $\beta = 1$, which is also where the RMSE is minimized for these two methods. For GTRS-DynSync (with $\lambda_{\text{scale}} = 10$) the data-fidelity peaks at $\beta = 1$, but the RMSE is minimized for $\beta = 20$.
• Figure 3: RMSE versus $T$ for AGN model ($n=30$, $\sigma = 3$) with $S_T \in \left\{{1/T,1,T^{1/4}}\right\}$, results averaged over $20$ MC runs. We choose $\lambda_{\text{scale}} = 10$ for $\texttt{GTRS-DynSync}$. Results on left show average RMSE, the right panel shows average RMSE $\pm$ standard deviation.
• Figure 4: RMSE versus $T$ for Outliers model ($n=30$, $\eta = p = 0.2$) with $S_T \in \left\{{1/T, 1, T^{1/4}}\right\}$, results averaged over $20$ MC runs. We choose $\lambda_{\text{scale}} = 10$ for $\texttt{GTRS-DynSync}$. Results on left show average RMSE, the right panel shows average RMSE $\pm$ standard deviation.
• Figure 5: RMSE versus $\gamma (= \sigma)$ for AGN model ($n=30$, $T = 20$) with $S_T \in \left\{{1/T, 1, T^{1/4}}\right\}$, results averaged over $20$ MC runs. We choose $\lambda_{\text{scale}} = 10$ for $\texttt{GTRS-DynSync}$. Results on left show average RMSE, the right panel shows average RMSE $\pm$ standard deviation.

Theorems & Definitions (24)

• Remark 1: Static anchored synchronization
• Remark 2: Complexity of \ref{['eq:opt_denoising']}
• Remark 3: Spectral local synchronization
• Remark 4: Denoising smooth signals on a graph
• Remark 5: Comparison with AKT_dynamicRankRSync
• Lemma 1: Bias bound
• proof
• Lemma 2: Variance bound
• proof
• Theorem 1: Main result: $\overline{G}^*$ recovery under AGN model