DeepAI AI Chat
Log In Sign Up

Neural Manifold Ordinary Differential Equations

by   Aaron Lou, et al.
cornell university

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.


page 8

page 20

page 21


Neural Ordinary Differential Equations on Manifolds

Normalizing flows are a powerful technique for obtaining reparameterizab...

Density estimation on low-dimensional manifolds: an inflation-deflation approach

Normalizing Flows (NFs) are universal density estimators based on Neuron...

Horizontal Flows and Manifold Stochastics in Geometric Deep Learning

We introduce two constructions in geometric deep learning for 1) transpo...

Riemannian Continuous Normalizing Flows

Normalizing flows have shown great promise for modelling flexible probab...

Thoughts on the Consistency between Ricci Flow and Neural Network Behavior

The Ricci flow is a partial differential equation for evolving the metri...

Neural Flows: Efficient Alternative to Neural ODEs

Neural ordinary differential equations describe how values change in tim...

Code Repositories


[NeurIPS 2020] Neural Manifold Ordinary Differential Equations (

view repo