Learning Distributions on Manifolds with Free-Form Flows
This addresses the inference bottleneck for researchers and practitioners working with manifold data, offering a more efficient alternative to existing methods.
The paper tackles the problem of expensive inference in generative models for data on manifolds by proposing Manifold Free-Form Flows (M-FFF), which enables sampling in a single function evaluation, achieving two orders of magnitude faster speed than multi-step methods while matching or outperforming previous single-step methods in likelihoods.
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. It is typically two orders of magnitude faster than multi-step methods based on diffusion or flow matching, achieving better likelihoods in several experiments. We provide our code at https://github.com/vislearn/FFF.