# Schr"odinger Bridge Over A Compact Connected Lie Group

arXiv: https://arxiv.org/abs/2603.14049 · HTML: https://mlsanity.com/paper/2603.14049/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 14 Mar 2026.

## Builds on (most cited first, 11 of 11)

- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [Entropic and displacement interpolation: a computational approach using the Hilbert metric](https://mlsanity.com/paper/1506.04255/) (2015)
- [Reflected Schr"odinger Bridge: Density Control with Path Constraints](https://mlsanity.com/paper/2003.13895/) (2020)
- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [Finite Horizon Density Steering for Multi-input State Feedback Linearizable Systems](https://mlsanity.com/paper/1909.12511/) (2019)
- [Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly](https://mlsanity.com/paper/2307.14442/) (2023)
- [Prediction and Optimal Feedback Steering of Probability Density Functions for Safe Automated Driving](https://mlsanity.com/paper/2009.09055/) (2020)
- [On the Hopf-Cole Transform for Control-affine Schr"odinger Bridge](https://mlsanity.com/paper/2503.17640/) (2025)
- [Optimal Transport on the Lie Group of Roto-translations](https://mlsanity.com/paper/2402.15322/) (2024)
- [Finite Horizon Density Control for Static State Feedback Linearizable Systems](https://mlsanity.com/paper/1904.02272/) (2019)

## Built on by (most cited first, 2 of 2)

- [Lifted Schr"odinger Bridges for Gaussian Mixture Endpoints: Projection Gaps and Path-Space Obstructions](https://mlsanity.com/paper/2605.24795/) (2026)
- [On the Equivalence of Stochastic Control and Path Space Formulations for Schr"odinger Bridges over Compact Connected Lie Groups](https://mlsanity.com/paper/2609.13758/) (2026)

## Abstract (arXiv)

This work studies the Schr"odinger bridge problem for the kinematic equation on a compact connected Lie group. The objective is to steer a controlled diffusion between given initial and terminal densities supported over the Lie group while minimizing the control effort. We develop a coordinate-free formulation of this stochastic optimal control problem that respects the underlying geometric structure of the Lie group, thereby avoiding limitations associated with local parameterizations or embeddings in Euclidean spaces. We establish the existence and uniqueness of solution to the corresponding Schr"odinger system. Our results are constructive in that they derive a geometric controller that optimally interpolates probability densities supported over the Lie group. To illustrate the results, we provide numerical examples on mathsfSO(2) and mathsfSO(3). The codes and animations are publicly available at https://github.com/gradslab/SbpLieGroups.git .