Riemannian Score-Based Generative Modelling

TL;DR

This paper extends score-based generative models to Riemannian manifolds, enabling their application to data on curved spaces such as spheres, with demonstrated success on earth and climate science data.

Contribution

It introduces Riemannian Score-based Generative Models (RSGMs), adapting existing SGMs to non-Euclidean geometries for broader applicability.

Findings

Successfully applied to spherical earth data

Extended generative modeling to Riemannian manifolds

Demonstrated effectiveness on climate science datasets

Abstract

Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a ``noising'' stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a ``denoising'' process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.