  Learning the distribution of data on Riemannian manifolds is crucial for
modeling data from non-Euclidean space, which is required by many applications
in diverse scientific fields. Yet, existing generative models on manifolds
suffer from expensive divergence computation or rely on approximations of heat
kernel. These limitations restrict their applicability to simple geometries and
hinder scalability to high dimensions. In this work, we introduce the
Riemannian Diffusion Mixture, a principled framework for building a generative
diffusion process on manifolds. Instead of following the denoising approach of
previous diffusion models, we construct a diffusion process using a mixture of
bridge processes derived on general manifolds without requiring heat kernel
estimations. We develop a geometric understanding of the mixture process,
deriving the drift as a weighted mean of tangent directions to the data points
that guides the process toward the data distribution. We further propose a
scalable training objective for learning the mixture process that readily
applies to general manifolds. Our method achieves superior performance on
diverse manifolds with dramatically reduced number of in-training simulation
steps for general manifolds.
