# Dynamic angular synchronization under smoothness constraints

arXiv: https://arxiv.org/abs/2406.04071 · HTML: https://mlsanity.com/paper/2406.04071/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 48 references that are in the map.
- First posted on arXiv 6 Jun 2024.

## Builds on (most cited first, 20 of 20)

- [A Survey of Structure from Motion](https://mlsanity.com/paper/1701.08493/) (2017)
- [Near-optimal bounds for phase synchronization](https://mlsanity.com/paper/1703.06605/) (2017)
- [Nonconvex phase synchronization](https://mlsanity.com/paper/1601.06114/) (2016)
- [Tightness of the maximum likelihood semidefinite relaxation for angular synchronization](https://mlsanity.com/paper/1411.3272/) (2014)
- [Notes on computational-to-statistical gaps: predictions using statistical physics](https://mlsanity.com/paper/1803.11132/) (2018)
- [Total Variation Classes Beyond 1d: Minimax Rates, and the Limitations of Linear Smoothers](https://mlsanity.com/paper/1605.08400/) (2016)
- [Message-passing algorithms for synchronization problems over compact groups](https://mlsanity.com/paper/1610.04583/) (2016)
- [On the Estimation Performance and Convergence Rate of the Generalized Power Method for Phase Synchronization](https://mlsanity.com/paper/1603.00211/) (2016)
- [The Projected Power Method: An Efficient Algorithm for Joint Alignment from Pairwise Differences](https://mlsanity.com/paper/1609.05820/) (2016)
- [A Unified Approach to Synchronization Problems over Subgroups of the Orthogonal Group](https://mlsanity.com/paper/2009.07514/) (2020)
- [Ranking and synchronization from pairwise measurements via SVD](https://mlsanity.com/paper/1906.02746/) (2019)
- [Nonparametric Estimation in the Dynamic Bradley-Terry Model](https://mlsanity.com/paper/2003.00083/) (2020)
- [Non-Convex Exact Community Recovery in Stochastic Block Model](https://mlsanity.com/paper/2006.15843/) (2020)
- [Dynamic Ranking with the BTL Model: A Nearest Neighbor based Rank Centrality Method](https://mlsanity.com/paper/2109.13743/) (2021)
- [Synchronization over Z2 and community detection in multiplex signed networks with constraints](https://mlsanity.com/paper/1504.04061/) (2015)
- [Optimal Orthogonal Group Synchronization and Rotation Group Synchronization](https://mlsanity.com/paper/2109.13491/) (2021)
- [Seeded graph matching for the correlated Gaussian Wigner model via the projected power method](https://mlsanity.com/paper/2204.04099/) (2022)
- [Dynamic Ranking and Translation Synchronization](https://mlsanity.com/paper/2207.01455/) (2022)
- [Robust Angular Synchronization via Directed Graph Neural Networks](https://mlsanity.com/paper/2310.05842/) (2023)
- [Generalized Orthogonal Procrustes Problem under Arbitrary Adversaries](https://mlsanity.com/paper/2106.15493/) (2021)

## Abstract (arXiv)

Given an undirected measurement graph H = ([n], E), the classical angular synchronization problem consists of recovering unknown angles θ1^*,dots,θn^* from a collection of noisy pairwise measurements of the form (θi^* - θj^*) mod 2π, for all i,j ∈ 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.