# Synchronizing Probability Measures on Rotations via Optimal Transport

arXiv: https://arxiv.org/abs/2004.00663 · HTML: https://mlsanity.com/paper/2004.00663/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Apr 2020.

## Built on by (most cited first, 19 of 19)

- [Weakly Supervised Learning of Rigid 3D Scene Flow](https://mlsanity.com/paper/2102.08945/) (2021)
- [MultiBodySync: Multi-Body Segmentation and Motion Estimation via 3D Scan Synchronization](https://mlsanity.com/paper/2101.06605/) (2021)
- [KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support](https://mlsanity.com/paper/2106.08929/) (2021)
- [Quantum Permutation Synchronization](https://mlsanity.com/paper/2101.07755/) (2021)
- [Multiway Non-rigid Point Cloud Registration via Learned Functional Map Synchronization](https://mlsanity.com/paper/2111.12878/) (2021)
- [NRDF: Neural Riemannian Distance Fields for Learning Articulated Pose Priors](https://mlsanity.com/paper/2403.03122/) (2024)
- [Quantum Motion Segmentation](https://mlsanity.com/paper/2203.13185/) (2022)
- [Deep Permutation Equivariant Structure from Motion](https://mlsanity.com/paper/2104.06703/) (2021)
- [Robust Group Synchronization via Quadratic Programming](https://mlsanity.com/paper/2206.08994/) (2022)
- [Chordal Averaging on Flag Manifolds and Its Applications](https://mlsanity.com/paper/2303.13501/) (2023)
- [Can Modifying Data Address Graph Domain Adaptation?](https://mlsanity.com/paper/2407.19311/) (2024)
- [Q-FW: A Hybrid Classical-Quantum Frank-Wolfe for Quadratic Binary Optimization](https://mlsanity.com/paper/2203.12633/) (2022)
- [Rotation Synchronization via Deep Matrix Factorization](https://mlsanity.com/paper/2305.05268/) (2023)
- [Self-Supervised Correspondence Estimation via Multiview Registration](https://mlsanity.com/paper/2212.03236/) (2022)
- [Optimal Transport on the Lie Group of Roto-translations](https://mlsanity.com/paper/2402.15322/) (2024)
- [Certifiably Optimal Anisotropic Rotation Averaging](https://mlsanity.com/paper/2503.07353/) (2025)
- [A Taxonomy of Structure from Motion Methods](https://mlsanity.com/paper/2505.15814/) (2025)
- [Riemannian Functional Map Synchronization for Probabilistic Partial Correspondence in Shape Networks](https://mlsanity.com/paper/2111.14762/) (2021)
- [Application and Evaluation of the Common Circles Method](https://mlsanity.com/paper/2602.17353/) (2026)

## Abstract (arXiv)

We introduce a new paradigm, textitmeasure synchronization, for synchronizing graphs with measure-valued edges. We formulate this problem as maximization of the cycle-consistency in the space of probability measures over relative rotations. In particular, we aim at estimating marginal distributions of absolute orientations by synchronizing the textitconditional ones, which are defined on the Riemannian manifold of quaternions. Such graph optimization on distributions-on-manifolds enables a natural treatment of multimodal hypotheses, ambiguities and uncertainties arising in many computer vision applications such as SLAM, SfM, and object pose estimation. We first formally define the problem as a generalization of the classical rotation graph synchronization, where in our case the vertices denote probability measures over rotations. We then measure the quality of the synchronization by using Sinkhorn divergences, which reduces to other popular metrics such as Wasserstein distance or the maximum mean discrepancy as limit cases. We propose a nonparametric Riemannian particle optimization approach to solve the problem. Even though the problem is non-convex, by drawing a connection to the recently proposed sparse optimization methods, we show that the proposed algorithm converges to the global optimum in a special case of the problem under certain conditions. Our qualitative and quantitative experiments show the validity of our approach and we bring in new perspectives to the study of synchronization.