Higher-Order Group Synchronization

TL;DR

This paper introduces a higher-order group synchronization framework operating on hypergraphs, providing a novel approach that improves robustness and performance in applications like computer vision and cryo-EM.

Contribution

It defines the higher-order synchronization problem, establishes its mathematical foundations, and proposes the first computational message passing algorithm with theoretical guarantees.

Findings

Outperforms pairwise methods in rotational and angular synchronization.

More robust to outliers compared to standard methods.

Achieves comparable results to cryo-EM reconstruction packages.

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.