Dynamic angular synchronization under smoothness constraints

2024-06-06

arXiv.org

AI Summary

This work addresses the dynamic angular synchronization problem: jointly estimating time-varying angular variables across $T$ time steps from sparse, time-varying, and noisy pairwise angular difference measurements, subject to graph signal smoothness constraints. We establish the first non-asymptotic mean squared error (MSE) bound for this problem, proving that the MSE vanishes as $T$ increases—even under graph sparsity, disconnection, and growing noise—with a convergence rate strictly faster than that of the static counterpart. We propose three joint estimation algorithms integrating graph signal modeling, temporal optimization, spectral methods, and convex relaxation, each accompanied by rigorous theoretical guarantees. Synthetic experiments demonstrate the robustness and superiority of our methods under low graph connectivity and high measurement noise.

Technology Category

• Intelligent Robots: State Estimation
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Given an undirected measurement graph $mathcal{H} = ([n], mathcal{E})$, the classical angular synchronization problem consists of recovering unknown angles $ heta_1^*,dots, heta_n^*$ from a collection of noisy pairwise measurements of the form $( heta_i^* - heta_j^*) mod 2pi$, 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.

Problem

• Estimating dynamic angles over time from noisy pairwise measurements
• Ensuring smooth evolution of angles across multiple time points
• Handling sparse, disconnected graphs and increasing noise levels

Innovation

• Dynamic angular synchronization over evolving graphs
• Smoothness-constrained joint angle estimation
• MSE convergence under sparse noisy conditions