Many real world data, particularly in the natural sciences and computer
vision, lie on known Riemannian manifolds such as spheres, tori or the group of
rotation matrices. The predominant approaches to learning a distribution on
such a manifold require solving a differential equation in order to sample from
the model and evaluate densities. The resulting sampling times are slowed down
by a high number of function evaluations. In this work, we propose an
alternative approach which only requires a single function evaluation followed
by a projection to the manifold. Training is achieved by an adaptation of the
recently proposed free-form flow framework to Riemannian manifolds. The central
idea is to estimate the gradient of the negative log-likelihood via a trace
evaluated in the tangent space. We evaluate our method on various manifolds,
and find significantly faster inference at competitive performance compared to
previous work. We make our code public at this https URL

我们提出了一种在 Riemannian 流形上进行分布学习的替代方法，该方法只需要一次函数评估，然后将结果投影到流形上。通过在切空间中评估的迹来估计负对数似然的梯度，我们在各种流形上评估了我们的方法，并发现相比之前的工作，推断速度显著提高且具有竞争性的性能。我们在该网址上公开了我们的代码。