  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.
