Learning Distributions on Manifolds with Free-form Flows
Peter Sorrenson Felix Draxler Armand Rousselot Sander Hummerich Ullrich Köthe
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.