Scaling Riemannian Diffusion Models
Aaron Lou Minkai Xu Stefano Ermon
Riemannian diffusion models draw inspiration from standard Euclidean space
diffusion models to learn distributions on general manifolds. Unfortunately,
the additional geometric complexity renders the diffusion transition term
inexpressible in closed form, so prior methods resort to imprecise
approximations of the score matching training objective that degrade
performance and preclude applications in high dimensions. In this work, we
reexamine these approximations and propose several practical improvements. Our
key observation is that most relevant manifolds are symmetric spaces, which are
much more amenable to computation. By leveraging and combining various
ans\"{a}tze, we can quickly compute relevant quantities to high precision. On
low dimensional datasets, our correction produces a noticeable improvement,
allowing diffusion to compete with other methods. Additionally, we show that
our method enables us to scale to high dimensional tasks on nontrivial
manifolds. In particular, we model QCD densities on $SU(n)$ lattices and
contrastively learned embeddings on high dimensional hyperspheres.