Riemannian Diffusion

Riemannian diffusion models extend the successful framework of diffusion models from Euclidean spaces to the more complex setting of Riemannian manifolds, which are curved spaces with a defined metric. Current research focuses on developing efficient algorithms for training these models, including adaptations of stochastic gradient descent and novel approaches to score matching, often leveraging the properties of symmetric spaces for computational tractability. These advancements enable the application of diffusion models to data residing on manifolds, improving generative modeling, inference, and optimization in various fields, such as robotics, brain-computer interfaces, and scientific data analysis where data inherently possesses non-Euclidean structure. The resulting improvements in accuracy and efficiency are significant for applications requiring the modeling of complex, non-linear data.