Neural Manifold Ordinary Differential Equations
This work addresses the challenge of developing flexible and generalizable deep generative models on non-Euclidean spaces for researchers and practitioners in machine learning, representing a novel method rather than an incremental step.
The paper tackles the problem of constructing normalizing flows on arbitrary manifolds by introducing Neural Manifold Ordinary Differential Equations, which generalize Neural ODEs to enable Manifold Continuous Normalizing Flows (MCNFs) that require only local geometry and use continuous change of variables, resulting in a marked improvement for density estimation and downstream tasks.
To better conform to data geometry, recent deep generative modelling techniques adapt Euclidean constructions to non-Euclidean spaces. In this paper, we study normalizing flows on manifolds. Previous work has developed flow models for specific cases; however, these advancements hand craft layers on a manifold-by-manifold basis, restricting generality and inducing cumbersome design constraints. We overcome these issues by introducing Neural Manifold Ordinary Differential Equations, a manifold generalization of Neural ODEs, which enables the construction of Manifold Continuous Normalizing Flows (MCNFs). MCNFs require only local geometry (therefore generalizing to arbitrary manifolds) and compute probabilities with continuous change of variables (allowing for a simple and expressive flow construction). We find that leveraging continuous manifold dynamics produces a marked improvement for both density estimation and downstream tasks.