Riemannian Score-Based Generative Modelling

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 <PRE_TAG>manifolds</POST_TAG> and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (R<PRE_TAG>SGMs</POST_TAG>), a class of generative models extending SGMs to Riemannian <PRE_TAG>manifolds</POST_TAG>. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.