Higher-Order Group Synchronization

2025-05-28

AI Summary

This paper addresses the higher-order group synchronization problem on hypergraphs: consistently recovering global group element estimates for nodes from higher-order local group measurements encoded on hyperedges. We formally define this problem and establish necessary and sufficient conditions for synchronizability based on hypergraph cycle consistency. We propose the first message-passing framework explicitly designed for higher-order measurements, integrating hypergraph modeling, group theory, algebraic topology, and robust optimization. Theoretical analysis guarantees convergence of the algorithm. Experiments demonstrate that our method significantly outperforms conventional pairwise synchronization approaches on rotation and angular synchronization tasks, exhibiting superior robustness to outliers. On simulated cryo-EM data, it achieves reconstruction accuracy comparable to state-of-the-art software packages. These results validate the practical utility of our approach in computer vision and structural biology.

Technology Category

• Computer Vision: Other Foundations of Computer Vision
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Data transparency and provenance

Abstract

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.

Problem

• Extends group synchronization to hypergraphs for global estimates
• Develops synchronizability conditions based on cycle consistency
• Proposes message-passing algorithm for robust higher-order synchronization

Innovation

• Higher-order synchronization on hypergraphs
• Message passing algorithm for global estimates
• Robust to outliers and noise