# The Geometry of Synchronization Problems and Learning Group Actions

arXiv: https://arxiv.org/abs/1610.09051 · HTML: https://mlsanity.com/paper/1610.09051/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Oct 2016.

## Built on by (most cited first, 7 of 7)

- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Bundle Networks: Fiber Bundles, Local Trivializations, and a Generative Approach to Exploring Many-to-one Maps](https://mlsanity.com/paper/2110.06983/) (2021)
- [Fiber Bundle Morphisms as a Framework for Modeling Many-to-Many Maps](https://mlsanity.com/paper/2203.08189/) (2022)
- [Quiver Laplacians and Feature Selection](https://mlsanity.com/paper/2404.06993/) (2024)
- [Learning from Frustration: Torsor CNNs on Graphs](https://mlsanity.com/paper/2510.23288/) (2025)
- [Higher-Order Group Synchronization](https://mlsanity.com/paper/2505.21932/) (2025)
- [A Probabilistic Approach to Pose Synchronization for Multi-Reference Alignment with Applications to MIMO Wireless Communication Systems](https://mlsanity.com/paper/2511.03280/) (2025)

## Abstract (arXiv)

We develop a geometric framework that characterizes the synchronization problem --- the problem of consistently registering or aligning a collection of objects. The theory we formulate characterizes the cohomological nature of synchronization based on the classical theory of fibre bundles. We first establish the correspondence between synchronization problems in a topological group G over a connected graph Gamma and the moduli space of flat principal G-bundles over Gamma, and develop a discrete analogy of the renowned theorem of classifying flat principal bundles with fix base and structural group using the representation variety. In particular, we show that prescribing an edge potential on a graph is equivalent to specifying an equivalence class of flat principal bundles, of which the triviality of holonomy dictates the synchronizability of the edge potential. We then develop a twisted cohomology theory for associated vector bundles of the flat principal bundle arising from an edge potential, which is a discrete version of the twisted cohomology in differential geometry. This theory realizes the obstruction to synchronizability as a cohomology group of the twisted de Rham cochain complex. We then build a discrete twisted Hodge theory --- a fibre bundle analog of the discrete Hodge theory on graphs --- which geometrically realizes the graph connection Laplacian as a Hodge Laplacian of degree zero. Motivated by our geometric framework, we study the problem of learning group actions --- partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations. A dual interpretation is to learn finitely generated subgroups of an ambient transformation group from noisy observed group elements. A synchronization-based algorithm is also provided, and we demonstrate its efficacy using simulations and real data.