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 https://github.com/vislearn/FFF.

翻译：许多真实世界的数据，特别是在自然科学和计算机视觉中，位于已知的黎曼流形上，例如球面、环面或旋转矩阵群。在此类流形上学习分布的主流方法需要求解微分方程来从模型中采样并评估密度，这导致采样时间因高次函数评估而显著延长。在本工作中，我们提出了一种替代方案，仅需单次函数评估后投影至流形。训练通过将近期提出的自由形式流形框架适配到黎曼流形来实现，核心思想是通过在切空间中计算迹来估计负对数似然的梯度。我们在多种流形上评估了该方法，发现与先前工作相比，推理速度显著提升且性能具有竞争力。我们已在 https://github.com/vislearn/FFF 公开代码。