Riemannian Continuous Normalizing Flows
This work addresses a domain-specific problem for researchers and practitioners in machine learning dealing with non-Euclidean data, such as in geometry or physics, and is incremental as it adapts continuous normalizing flows to Riemannian settings.
The paper tackles the problem of modeling probability distributions on Riemannian manifolds like spheres and hyperbolic spaces, where standard normalizing flows are misspecified due to flat geometry assumptions, by introducing Riemannian continuous normalizing flows, which show substantial improvements on synthetic and real-world data compared to existing methods.
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.