Riemannian Convex Potential Maps

TL;DR

This paper introduces a universal class of Riemannian flows based on convex potentials from optimal transport, capable of modeling complex distributions on various manifolds without domain-specific adjustments.

Contribution

It proposes a novel Riemannian flow framework that is universal and does not require prior domain knowledge, advancing distribution modeling on manifolds.

Findings

Successfully models distributions on spheres and tori

Demonstrates effectiveness on synthetic and geological data

Source code is publicly available

Abstract

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