  Normalizing flows are a powerful technique for obtaining reparameterizable
samples from complex multimodal distributions. Unfortunately current approaches
fall short when the underlying space has a non trivial topology, and are only
available for the most basic geometries. Recently normalizing flows in
Euclidean space based on Neural ODEs show great promise, yet suffer the same
limitations. Using ideas from differential geometry and geometric control
theory, we describe how neural ODEs can be extended to smooth manifolds. We
show how vector fields provide a general framework for parameterizing a
flexible class of invertible mapping on these spaces and we illustrate how
gradient based learning can be performed. As a result we define a general
methodology for building normalizing flows on manifolds.
