We propose Manifold Free-Form Flows (M-FFF), a simple new generative model for data on manifolds. The existing approaches to learning a distribution on arbitrary manifolds are expensive at inference time, since sampling requires solving a differential equation. Our method overcomes this limitation by sampling in a single function evaluation. The key innovation is to optimize a neural network via maximum likelihood on the manifold, possible by adapting the free-form flow framework to Riemannian manifolds. M-FFF is straightforwardly adapted to any manifold with a known projection. It consistently matches or outperforms previous single-step methods specialized to specific manifolds, and is competitive with multi-step methods with typically two orders of magnitude faster inference speed. We make our code public at https://github.com/vislearn/FFF.

翻译：我们提出了一种简单的新型生成模型——流形自由形式流（M-FFF），用于处理流形上的数据。现有学习任意流形上分布的方法在推理阶段计算成本高昂，因为采样需要求解微分方程。我们的方法通过单次函数评估完成采样，克服了这一限制。其核心创新在于将自由形式流框架适配到黎曼流形上，从而能够通过最大似然法在流形上优化神经网络。对于任何已知投影的流形，M-FFF均可直接适配。该方法在特定流形专用单步方法中始终保持相当或更优的性能，并与多步方法具有竞争力，同时推理速度通常快两个数量级。代码已公开于 https://github.com/vislearn/FFF。