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.

在本研究中，我们处理了模拟潜在困难条件下的扩散过程，并构建了一种在子黎曼流形上进行桥梁模拟的方法，通过展示如何将机器学习的最新进展改进为在子黎曼流形上训练评分近似器。我们使用随机泰勒扩展将通常的去噪损失概念推广为与非完全陈闭性框架一起工作，并通过在海森伯格群上和使用适应坐标的一般方式演示了所得到的方案。我们进行了数值实验，演示了来自海森伯格群桥梁过程的样本以及对于短时间的该过程的集中度。