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。

子里曼布里奇抽样的评分匹配

Score matching for sub-Riemannian bridge sampling

The set $\Cal C(G)$ of closed subgroups of a locally compact group $G$ has a
natural topology which makes it a compact space. This topology has been defined
in various contexts by Vietoris, Chabauty, Fell, Thurston, Gromov, Grigorchuk,
and many others.
The purpose of the talk was to describe the space $\Cal C(G)$ first for a few
elementary examples, then for $G$ the complex plane, in which case $\Cal C(G)$
is a 4--sphere (a result of Hubbard and Pourezza), and finally for the
3--dimensional Heisenberg group $H$, in which case $\Cal C(H)$ is a
6--dimensional singular space recently investigated by Martin Bridson, Victor
Kleptsyn and the author \cite{BrHK}.
These are slightly expanded notes prepared for a talk given at several
places: the Kortrijk workshop on {\it Discrete Groups and Geometric Structures,
with Applications III,} May 26--30, 2008; the {\it Tripode 14,} \'Ecole Normale
Sup\'erieure de Lyon, June 13, 2008; and seminars at the EPFL, Lausanne, and in
the Universit\'e de Rennes 1. The notes do not contain any other result than
those in \cite{BrHK}, and are not intended for publication.

描述了具有正规拓扑的本地紧群 G 的封闭子群集合 C (G) 的性质，并给出了几个例子，包括复平面和 Heisenberg 群。