# Higher-Order Group Synchronization

arXiv: https://arxiv.org/abs/2505.21932 · HTML: https://mlsanity.com/paper/2505.21932/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 51 references that are in the map.
- First posted on arXiv 28 May 2025.

## Builds on (most cited first, 22 of 22)

- [SE-Sync: A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group](https://mlsanity.com/paper/1612.07386/) (2016)
- [Detector-Free Structure from Motion](https://mlsanity.com/paper/2306.15669/) (2023)
- [What are higher-order networks?](https://mlsanity.com/paper/2104.11329/) (2021)
- [Message Passing Least Squares Framework and its Application to Rotation Synchronization](https://mlsanity.com/paper/2007.13638/) (2020)
- [Robust Group Synchronization via Cycle-Edge Message Passing](https://mlsanity.com/paper/1912.11347/) (2019)
- [Message-passing algorithms for synchronization problems over compact groups](https://mlsanity.com/paper/1610.04583/) (2016)
- [Minimal Problems for the Calibrated Trifocal Variety](https://mlsanity.com/paper/1611.05947/) (2016)
- [GPSfM: Global Projective SFM Using Algebraic Constraints on Multi-View Fundamental Matrices](https://mlsanity.com/paper/1812.00426/) (2018)
- [A New Rank Constraint on Multi-view Fundamental Matrices, and its Application to Camera Location Recovery](https://mlsanity.com/paper/1702.03023/) (2017)
- [On the Instability of Relative Pose Estimation and RANSAC's Role](https://mlsanity.com/paper/2112.14651/) (2021)
- [Robust Group Synchronization via Quadratic Programming](https://mlsanity.com/paper/2206.08994/) (2022)
- [Non-unique games over compact groups and orientation estimation in cryo-EM](https://mlsanity.com/paper/1505.03840/) (2015)
- [The Geometry of Synchronization Problems and Learning Group Actions](https://mlsanity.com/paper/1610.09051/) (2016)
- [Estimation of Camera Locations in Highly Corrupted Scenarios: All About that Base, No Shape Trouble](https://mlsanity.com/paper/1804.02591/) (2018)
- [Scalable Cluster-Consistency Statistics for Robust Multi-Object Matching](https://mlsanity.com/paper/2201.04797/) (2022)
- [Synchronization over Z2 and community detection in multiplex signed networks with constraints](https://mlsanity.com/paper/1504.04061/) (2015)
- [Representation Theoretic Patterns in Multi-Frequency Class Averaging for Three-Dimensional Cryo-Electron Microscopy](https://mlsanity.com/paper/1906.01082/) (2019)
- [Efficient Detection of Long Consistent Cycles and its Application to Distributed Synchronization](https://mlsanity.com/paper/2407.04260/) (2024)
- [A Spectral Method for Joint Community Detection and Orthogonal Group Synchronization](https://mlsanity.com/paper/2112.13199/) (2021)
- [Tensor-Based Synchronization and the Low-Rankness of the Block Trifocal Tensor](https://mlsanity.com/paper/2409.09313/) (2024)
- [Algebraic Constraints and Algorithms for Common Lines in Cryo-EM](https://mlsanity.com/paper/2403.16879/) (2024)
- [Condition numbers in multiview geometry, instability in relative pose estimation, and RANSAC](https://mlsanity.com/paper/2310.02719/) (2023)

## Built on by (most cited first, 1 of 1)

- [QuadSync: Quadrifocal Tensor Synchronization via Tucker Decomposition](https://mlsanity.com/paper/2602.22639/) (2026)

## Abstract (arXiv)

Group synchronization is the problem of determining reliable global estimates from noisy local measurements on networks. The typical task for group synchronization is to assign elements of a group to the nodes of a graph in a way that respects group elements given on the edges which encode information about local pairwise relationships between the nodes. In this paper, we introduce a novel higher-order group synchronization problem which operates on a hypergraph and seeks to synchronize higher-order local measurements on the hyperedges to obtain global estimates on the nodes. Higher-order group synchronization is motivated by applications to computer vision and image processing, among other computational problems. First, we define the problem of higher-order group synchronization and discuss its mathematical foundations. Specifically, we give necessary and sufficient synchronizability conditions which establish the importance of cycle consistency in higher-order group synchronization. Then, we propose the first computational framework for general higher-order group synchronization; it acts globally and directly on higher-order measurements using a message passing algorithm. We discuss theoretical guarantees for our framework, including convergence analyses under outliers and noise. Finally, we show potential advantages of our method through numerical experiments. In particular, we show that in certain cases our higher-order method applied to rotational and angular synchronization outperforms standard pairwise synchronization methods and is more robust to outliers. We also show that our method has comparable performance on simulated cryo-electron microscopy (cryo-EM) data compared to a standard cryo-EM reconstruction package.