Neural Manifold Ordinary Differential Equations

arXiv2006.10254MaRDI QIDQ6343178

Isay Katsman, Qingxuan Jiang, Derek Lim, S.-N. Lim, Christopher de Sa, Leo Huang, Aaron Lou

Publication date: 17 June 2020

Abstract: 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.

Has companion code repository: https://github.com/CUAI/Riemannian-Residual-Neural-Networks

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