Neural Ordinary Differential Equations on Manifolds

Abstract

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.

References in corpus (7)

• How to train your neural ODE: the world of Jacobian and kinetic regularization
• Normalizing Flows on Tori and Spheres
• Explorations in Homeomorphic Variational Auto-Encoding
• A Wrapped Normal Distribution on Hyperbolic Space for Gradient-Based Learning
• Normalizing Flows on Riemannian Manifolds
• Latent Variable Modelling with Hyperbolic Normalizing Flows
• Laplace Transformations of Submanifolds

Cited by in corpus (9)

• Riemannian Continuous Normalizing Flows
• Deep Efficient Continuous Manifold Learning for Time Series Modeling
• The Power Spherical distribution
• Introduction to Normalizing Flows for Lattice Field Theory
• Moser Flow: Divergence-based Generative Modeling on Manifolds
• Neural Spatio-Temporal Point Processes
• Optimal Potential Shaping on SE(3) via Neural ODEs on Lie Groups
• Learning ODEs via Diffeomorphisms for Fast and Robust Integration
• Equivariant Manifold Flows