Riemannian Diffusion Schr\"odinger Bridge

TL;DR

This paper introduces Riemannian Diffusion Schr"odinger Bridge, a novel method extending diffusion models to Riemannian manifolds, enabling improved data interpolation on non-Euclidean geometries with validation on synthetic and real-world datasets.

Contribution

It generalizes the Diffusion Schr"odinger Bridge to Riemannian manifolds and extends Riemannian score-based models beyond initial reversals, addressing sampling and interpolation challenges.

Findings

Effective on synthetic data

Successful application to Earth and climate data

Advances in non-Euclidean diffusion modeling

Abstract

Score-based generative models exhibit state of the art performance on density estimation and generative modeling tasks. These models typically assume that the data geometry is flat, yet recent extensions have been developed to synthesize data living on Riemannian manifolds. Existing methods to accelerate sampling of diffusion models are typically not applicable in the Riemannian setting and Riemannian score-based methods have not yet been adapted to the important task of interpolation of datasets. To overcome these issues, we introduce \emph{Riemannian Diffusion Schr\"odinger Bridge}. Our proposed method generalizes Diffusion Schr\"odinger Bridge introduced in \cite{debortoli2021neurips} to the non-Euclidean setting and extends Riemannian score-based models beyond the first time reversal. We validate our proposed method on synthetic data and real Earth and climate data.