Riemannian Diffusion Models

Chin-Wei Huang, Milad Aghajohari, Joey Bose, Prakash Panangaden, Aaron Courville

Keywords: diffusion models, density estimation, generative models, Riemannian manifolds, variational inference

TL;DR: We propose a continuous-time diffusion model for data represented on a Riemannian manifold.

Abstract: 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 for 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.