# A Unified Approach to Synchronization Problems over Subgroups of the Orthogonal Group

arXiv: https://arxiv.org/abs/2009.07514 · HTML: https://mlsanity.com/paper/2009.07514/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Sep 2020.

## Built on by (most cited first, 15 of 15)

- [Joint Community Detection and Rotational Synchronization via Semidefinite Programming](https://mlsanity.com/paper/2105.06031/) (2021)
- [Optimal Orthogonal Group Synchronization and Rotation Group Synchronization](https://mlsanity.com/paper/2109.13491/) (2021)
- [Orthogonal Group Synchronization with Incomplete Measurements: Error Bounds and Linear Convergence of the Generalized Power Method](https://mlsanity.com/paper/2112.06556/) (2021)
- [ReSync: Riemannian Subgradient-based Robust Rotation Synchronization](https://mlsanity.com/paper/2305.15136/) (2023)
- [Non-Convex Joint Community Detection and Group Synchronization via Generalized Power Method](https://mlsanity.com/paper/2112.14204/) (2021)
- [Robust Angular Synchronization via Directed Graph Neural Networks](https://mlsanity.com/paper/2310.05842/) (2023)
- [NS-RGS: Newton-Schulz based Riemannian gradient method for orthogonal group synchronization](https://mlsanity.com/paper/2604.07372/) (2026)
- [Generalized Orthogonal Procrustes Problem under Arbitrary Adversaries](https://mlsanity.com/paper/2106.15493/) (2021)
- [Decentralized Weakly Convex Optimization Over the Stiefel Manifold](https://mlsanity.com/paper/2303.17779/) (2023)
- [Dual Quaternion SE(3) Synchronization with Recovery Guarantees](https://mlsanity.com/paper/2602.00324/) (2026)
- [Primal-Dual Methods for Nonsmooth Nonconvex Optimization with Orthogonality Constraints](https://mlsanity.com/paper/2604.04130/) (2026)
- [Dynamic angular synchronization under smoothness constraints](https://mlsanity.com/paper/2406.04071/) (2024)
- [ℓ1-norm rank-one symmetric matrix factorization has no spurious second-order stationary points](https://mlsanity.com/paper/2410.05025/) (2024)
- [TriP: A Triangle Puzzle Approach to Robust Translation Averaging](https://mlsanity.com/paper/2605.07143/) (2026)
- [Recovery Theory for Projected Power Iterations in Permutation Synchronization](https://mlsanity.com/paper/2609.09502/) (2026)

## Abstract (arXiv)

The problem of synchronization over a group G aims to estimate a collection of group elements G^*1, dots, G^*n ∈ G based on noisy observations of a subset of all pairwise ratios of the form G^*i G^*j-1. Such a problem has gained much attention recently and finds many applications across a wide range of scientific and engineering areas. In this paper, we consider the class of synchronization problems in which the group is a closed subgroup of the orthogonal group. This class covers many group synchronization problems that arise in practice. Our contribution is fivefold. First, we propose a unified approach for solving this class of group synchronization problems, which consists of a suitable initialization step and an iterative refinement step based on the generalized power method, and show that it enjoys a strong theoretical guarantee on the estimation error under certain assumptions on the group, measurement graph, noise, and initialization. Second, we formulate two geometric conditions that are required by our approach and show that they hold for various practically relevant subgroups of the orthogonal group. The conditions are closely related to the error-bound geometry of the subgroup -- an important notion in optimization. Third, we verify the assumptions on the measurement graph and noise for standard random graph and random matrix models. Fourth, based on the classic notion of metric entropy, we develop and analyze a novel spectral-type estimator. Finally, we show via extensive numerical experiments that our proposed non-convex approach outperforms existing approaches in terms of computational speed, scalability, and/or estimation error.