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.

翻译：基于分数的基因变异模型(SGM)是一个强大的基因变异模型(SGM),具有惊人的经验性。基于分数的基因变异模型(SGM)由“SGM”阶段和基因变异模型组成,前者使用“SGM”阶段,逐渐在数据中增加高斯噪音,后者则使用“SGM”阶段,后者通过接近扩散时间反射来界定“Dodisois”过程。现有的SGMs假设数据在欧几里德空间上得到支持,即具有平坦几何学的多重数据。在许多领域,如机器人、地球科学或蛋白质建模领域,数据通常自然地通过流传到里曼多元体的分布和当前SGM技术来描述。我们在这里引入了“Riemannian多级基因变异模型(RSGM)”一类的基因变异模型(RSGM),将SGMs延伸到里曼多体。我们展示了我们对于多种元,特别是地球和气候科学球系数据的方法。