Neural Ordinary Differential Equations on Manifolds
This work addresses a limitation in machine learning for generating samples on complex geometries, though it appears incremental as it builds on existing neural ODE techniques.
The authors tackled the problem of extending normalizing flows to spaces with non-trivial topology by proposing neural ODEs on smooth manifolds, resulting in a general methodology for building such flows on these spaces.
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.