  Normalizing flows are a powerful technique for obtaining reparameterizable
samples from complex multimodal distributions. Unfortunately, current
approaches are only available for the most basic geometries and fall short when
the underlying space has a nontrivial topology, limiting their applicability
for most real-world data. Using fundamental ideas from differential geometry
and geometric control theory, we describe how the recently introduced Neural
ODEs and continuous normalizing flows can be extended to arbitrary smooth
manifolds. We propose a general methodology for parameterizing vector fields on
these spaces and demonstrate how gradient-based learning can be performed.
Additionally, we provide a scalable unbiased estimator for the divergence in
this generalized setting. Experiments on a diverse selection of spaces
empirically showcase the defined framework's ability to obtain
reparameterizable samples from complex distributions.
