# Score matching for sub-Riemannian bridge sampling

arXiv: https://arxiv.org/abs/2404.15258 · HTML: https://mlsanity.com/paper/2404.15258/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 64 references that are in the map.
- First posted on arXiv 23 Apr 2024.

## Builds on (most cited first, 10 of 10)

- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Simulating Diffusion Bridges with Score Matching](https://mlsanity.com/paper/2111.07243/) (2021)
- [Diffusion bridges for stochastic Hamiltonian systems and shape evolutions](https://mlsanity.com/paper/2002.00885/) (2020)
- [Simulation of multivariate diffusion bridge](https://mlsanity.com/paper/1405.7728/) (2014)
- [A piecewise deterministic Monte Carlo method for diffusion bridges](https://mlsanity.com/paper/2001.05889/) (2020)
- [Bridge Simulation and Metric Estimation on Landmark Manifolds](https://mlsanity.com/paper/1705.10943/) (2017)
- [Inference for partially observed Riemannian Ornstein-Uhlenbeck diffusions of covariance matrices](https://mlsanity.com/paper/2104.03193/) (2021)

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

- [Neural Guided Diffusion Bridges](https://mlsanity.com/paper/2502.11909/) (2025)
- [Score Matching Diffusion Based Feedback Control and Planning of Nonlinear Systems](https://mlsanity.com/paper/2504.09836/) (2025)
- [Sampling conditioned diffusions via Pathspace Projected Monte Carlo](https://mlsanity.com/paper/2506.15743/) (2025)
- [Learning the Geometry of Data: A Mathematical Review of Shape Space Analysis](https://mlsanity.com/paper/2606.17022/) (2026)

## Abstract (arXiv)

Simulation of conditioned diffusion processes is an essential tool in inference for stochastic processes, data imputation, generative modelling, and geometric statistics. Whilst simulating diffusion bridge processes is already difficult on Euclidean spaces, when considering diffusion processes on Riemannian manifolds the geometry brings in further complications. In even higher generality, advancing from Riemannian to sub-Riemannian geometries introduces hypoellipticity, and the possibility of finding appropriate explicit approximations for the score of the diffusion process is removed. We handle these challenges and construct a method for bridge simulation on sub-Riemannian manifolds by demonstrating how recent progress in machine learning can be modified to allow for training of score approximators on sub-Riemannian manifolds. Since gradients dependent on the horizontal distribution, we generalise the usual notion of denoising loss to work with non-holonomic frames using a stochastic Taylor expansion, and we demonstrate the resulting scheme both explicitly on the Heisenberg group and more generally using adapted coordinates. We perform numerical experiments exemplifying samples from the bridge process on the Heisenberg group and the concentration of this process for small time.