# Solving Orthogonal Group Synchronization via Convex and Low-Rank Optimization: Tightness and Landscape Analysis

arXiv: https://arxiv.org/abs/2006.00902 · HTML: https://mlsanity.com/paper/2006.00902/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2020.

## Built on by (most cited first, 10 of 10)

- [Sharp Global Guarantees for Nonconvex Low-rank Recovery in the Noisy Overparameterized Regime](https://mlsanity.com/paper/2104.10790/) (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)
- [Dynamic Ranking and Translation Synchronization](https://mlsanity.com/paper/2207.01455/) (2022)
- [Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective](https://mlsanity.com/paper/2209.15130/) (2022)
- [SDP Achieves Exact Minimax Optimality in Phase Synchronization](https://mlsanity.com/paper/2101.02347/) (2021)
- [A Spectral Method for Joint Community Detection and Orthogonal Group Synchronization](https://mlsanity.com/paper/2112.13199/) (2021)
- [A Complete Loss Landscape Analysis of Regularized Deep Matrix Factorization](https://mlsanity.com/paper/2506.20344/) (2025)
- [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)

## Abstract (arXiv)

Group synchronization aims to recover the group elements from their noisy pairwise measurements. It has found many applications in community detection, clock synchronization, and joint alignment problem. This paper focuses on the orthogonal group synchronization which is often used in cryo-EM and computer vision. However, it is generally NP-hard to retrieve the group elements by finding the least squares estimator. In this work, we first study the semidefinite programming (SDP) relaxation of the orthogonal group synchronization and its tightness, i.e., the SDP estimator is exactly equal to the least squares estimator. Moreover, we investigate the performance of the Burer-Monteiro factorization in solving the SDP relaxation by analyzing its corresponding optimization landscape. We provide deterministic sufficient conditions which guarantee: (i) the tightness of SDP relaxation; (ii) optimization landscape arising from the Burer-Monteiro approach is benign, i.e., the global optimum is exactly the least squares estimator and no other spurious local optima exist. Our result provides a solid theoretical justification of why the Burer-Monteiro approach is remarkably efficient and effective in solving the large-scale SDPs arising from orthogonal group synchronization. We perform numerical experiments to complement our theoretical analysis, which gives insights into future research directions.