Riemannian Continuous Normalizing Flows

TL;DR

This paper introduces Riemannian continuous normalizing flows, enabling flexible probability modeling on manifolds like spheres and hyperbolic spaces, addressing limitations of traditional flat-geometry flows.

Contribution

It proposes a novel Riemannian flow framework using ODEs for probability measures on manifolds, improving modeling accuracy on curved spaces.

Findings

Significant performance improvements on synthetic data.

Enhanced modeling on real-world manifold data.

Outperforms standard and projected flows.

Abstract

Normalizing flows have shown great promise for modelling flexible probability distributions in a computationally tractable way. However, whilst data is often naturally described on Riemannian manifolds such as spheres, torii, and hyperbolic spaces, most normalizing flows implicitly assume a flat geometry, making them either misspecified or ill-suited in these situations. To overcome this problem, we introduce Riemannian continuous normalizing flows, a model which admits the parametrization of flexible probability measures on smooth manifolds by defining flows as the solution to ordinary differential equations. We show that this approach can lead to substantial improvements on both synthetic and real-world data when compared to standard flows or previously introduced projected flows.