  Diffusion models are recent state-of-the-art methods for image generation and
likelihood estimation. In this work, we generalize continuous-time diffusion
models to arbitrary Riemannian manifolds and derive a variational framework for
likelihood estimation. Computationally, we propose new methods for computing
the Riemannian divergence which is needed in the likelihood estimation.
Moreover, in generalizing the Euclidean case, we prove that maximizing this
variational lower-bound is equivalent to Riemannian score matching.
Empirically, we demonstrate the expressive power of Riemannian diffusion models
on a wide spectrum of smooth manifolds, such as spheres, tori, hyperboloids,
and orthogonal groups. Our proposed method achieves new state-of-the-art
likelihoods on all benchmarks.
