Riemannian Score-Based Generative Modelling
This work addresses a domain-specific problem for fields like robotics and geoscience where data naturally resides on curved manifolds, representing a novel method for a known bottleneck.
The paper tackles the problem of generating data on Riemannian manifolds, such as spherical data in climate science, by extending score-based generative models to non-Euclidean spaces, achieving competitive performance on various manifolds.
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.