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.

翻译：条件扩散过程的模拟是随机过程推断、数据插补、生成建模和几何统计学中的核心工具。尽管在欧氏空间中模拟扩散桥过程已具挑战性，但考虑黎曼流形上的扩散过程时，几何结构会引入更多复杂性。在更高泛化层次上，从黎曼几何推进到子黎曼几何不仅引入了亚椭圆性，还消除了为扩散过程评分寻找合适显式近似的可能性。我们通过展示如何改进机器学习领域的最新进展以实现在子黎曼流形上训练评分近似器，解决了这些挑战，并构建了子黎曼流形上的桥模拟方法。由于梯度依赖于水平分布，我们利用随机泰勒展开将常规去噪损失泛化为适用于非完整标架，并在海森堡群上显式地以及更一般地借助适应坐标展示了所得方案。我们通过数值实验验证了海森堡群上桥过程的样本及其在小时间区间内的浓度特性。