Learning Distributions on Manifolds with Free-Form Flows

Peter Sorrenson, Felix Draxler, Armand Rousselot, Sander Hummerich, Ullrich Koethe

Keywords: generative model, riemannian geometry, riemannian manifolds, free-form flows, normalizing flows

TL;DR: We propose Manifold Free-form Flows, the first generative model for data on arbitrary manifolds that sample in a single function evaluation at high quality.

Abstract: 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.

Primary Area: Generative models

Submission Number: 6245