# Bridge Simulation and Metric Estimation on Lie Groups and Homogeneous Spaces

arXiv: https://arxiv.org/abs/2112.00866 · HTML: https://mlsanity.com/paper/2112.00866/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Dec 2021.

## Built on by (most cited first, 1 of 1)

- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)

## Abstract (arXiv)

We present schemes for simulating Brownian bridges on complete and connected Lie groups and homogeneous spaces. We use this to construct an estimation scheme for recovering an unknown left- or right-invariant Riemannian metric on the Lie group from samples. We subsequently show how pushing forward the distributions generated by Brownian motions on the group results in distributions on homogeneous spaces that exhibit non-trivial covariance structure. The pushforward measure gives rise to new parametric families of distributions on commonly occurring spaces such as spheres and symmetric positive tensors. We extend the estimation scheme to fit these distributions to homogeneous space-valued data. We demonstrate both the simulation schemes and estimation procedures on Lie groups and homogenous spaces, including SPD(3) = GL_+(3)/SO(3) and mathbb S2 = SO(3)/SO(2).