Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion
  Processes
Jaehyeong Jo Sung Ju Hwang
Learning the distribution of data on Riemannian manifolds is crucial for
modeling data from non-Euclidean space, which is required by many applications
from 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
process on manifolds as a mixture of endpoint-conditioned diffusion processes
instead of relying on the denoising approach of previous diffusion models, for
which the generative process is characterized by its drift guiding toward the
most probable endpoint with respect to the geometry of the manifold. We further
propose a simple yet efficient training objective for learning the mixture
process, that is readily applicable to general manifolds. Our method
outperforms previous generative models on various manifolds while scaling to
high dimensions and requires a dramatically reduced number of in-training
simulation steps for general manifolds.