Riemannian Convex Potential Maps
This work addresses a crucial challenge in understanding non-Euclidean data in fields like physics and geology, offering a novel method that is incremental in improving representational and computational tradeoffs.
The paper tackles the problem of modeling distributions on Riemannian manifolds, such as spheres and tori, by proposing flows based on convex potentials from Riemannian optimal transport, which are universal and do not require domain-specific architectural knowledge, and demonstrates their effectiveness on synthetic and geological data with available source code.
Modeling distributions on Riemannian manifolds is a crucial component in understanding non-Euclidean data that arises, e.g., in physics and geology. The budding approaches in this space are limited by representational and computational tradeoffs. We propose and study a class of flows that uses convex potentials from Riemannian optimal transport. These are universal and can model distributions on any compact Riemannian manifold without requiring domain knowledge of the manifold to be integrated into the architecture. We demonstrate that these flows can model standard distributions on spheres, and tori, on synthetic and geological data. Our source code is freely available online at http://github.com/facebookresearch/rcpm