# Optimal Transport on the Lie Group of Roto-translations

arXiv: https://arxiv.org/abs/2402.15322 · HTML: https://mlsanity.com/paper/2402.15322/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 76 references that are in the map.
- First posted on arXiv 23 Feb 2024.

## Builds on (most cited first, 27 of 27)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Geometric Dataset Distances via Optimal Transport](https://mlsanity.com/paper/2002.02923/) (2020)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Roto-Translation Covariant Convolutional Networks for Medical Image Analysis](https://mlsanity.com/paper/1804.03393/) (2018)
- [A Wasserstein-type distance in the space of Gaussian Mixture Models](https://mlsanity.com/paper/1907.05254/) (2019)
- [Deep Shells: Unsupervised Shape Correspondence with Optimal Transport](https://mlsanity.com/paper/2010.15261/) (2020)
- [Accurate Point Cloud Registration with Robust Optimal Transport](https://mlsanity.com/paper/2111.00648/) (2021)
- [Scaling Algorithms for Unbalanced Transport Problems](https://mlsanity.com/paper/1607.05816/) (2016)
- [Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport](https://mlsanity.com/paper/1805.08331/) (2018)
- [PDE-based Group Equivariant Convolutional Neural Networks](https://mlsanity.com/paper/2001.09046/) (2020)
- [Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm](https://mlsanity.com/paper/2110.06798/) (2021)
- [Synchronizing Probability Measures on Rotations via Optimal Transport](https://mlsanity.com/paper/2004.00663/) (2020)
- [A Generative Model for Texture Synthesis based on Optimal Transport between Feature Distributions](https://mlsanity.com/paper/2007.03408/) (2020)
- [Quantitative Uniform Stability of the Iterative Proportional Fitting Procedure](https://mlsanity.com/paper/2108.08129/) (2021)
- [Parallelly Sliced Optimal Transport on Spheres and on the Rotation Group](https://mlsanity.com/paper/2401.16896/) (2024)
- [Analysis of (sub-)Riemannian PDE-G-CNNs](https://mlsanity.com/paper/2210.00935/) (2022)
- [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023)
- [Quantum Optimal Transport for Tensor Field Processing](https://mlsanity.com/paper/1612.08731/) (2016)
- [Highly corrupted image inpainting through hypoelliptic diffusion](https://mlsanity.com/paper/1502.07331/) (2015)
- [Geometrical Optical Illusion via Sub-Riemannian Geodesics in the Roto-Translation Group](https://mlsanity.com/paper/2002.08063/) (2020)
- [An algorithm for optimal transport between a simplex soup and a point cloud](https://mlsanity.com/paper/1707.01337/) (2017)
- [Entropic Wasserstein Gradient Flows](https://mlsanity.com/paper/1502.06216/) (2015)

## Built on by (most cited first, 4 of 4)

- [Schr"odinger Bridge Over A Compact Connected Lie Group](https://mlsanity.com/paper/2603.14049/) (2026)
- [Joint Manifold Learning and Optimal Transport for Dynamic Imaging](https://mlsanity.com/paper/2505.11913/) (2025)
- [Globally Solving Unbalanced Optimal Transport and Density Control for Gaussian Distributions](https://mlsanity.com/paper/2605.04246/) (2026)
- [Anatomy-Informed Neural Networks: Encoding Anatomic Priors in Loss and Architecture, with an SE(3) Formulation of Guidewire-Induced Aortoiliac Deformation](https://mlsanity.com/paper/2608.21332/) (2026)

## Abstract (arXiv)

The roto-translation group SE2 has been of active interest in image analysis due to methods that lift the image data to multi-orientation representations defined on this Lie group. This has led to impactful applications of crossing-preserving flows for image de-noising, geodesic tracking, and roto-translation equivariant deep learning. In this paper, we develop a computational framework for optimal transportation over Lie groups, with a special focus on SE2. We make several theoretical contributions (generalizable to matrix Lie groups) such as the non-optimality of group actions as transport maps, invariance and equivariance of optimal transport, and the quality of the entropic-regularized optimal transport plan using geodesic distance approximations. We develop a Sinkhorn like algorithm that can be efficiently implemented using fast and accurate distance approximations of the Lie group and GPU-friendly group convolutions. We report valuable advancements in the experiments on 1) image barycentric interpolation, 2) interpolation of planar orientation fields, and 3) Wasserstein gradient flows on SE2. We observe that our framework of lifting images to SE2 and optimal transport with left-invariant anisotropic metrics leads to equivariant transport along dominant contours and salient line structures in the image. This yields sharper and more meaningful interpolations compared to their counterparts on R^2