Diffusion Models for Inverse Problems on Riemannian Manifolds

Score-based diffusion models have become powerful learned priors for solving Bayesian inverse problems in Euclidean spaces, but existing diffusion-based posterior samplers do not extend to data that is intrinsically supported on a Riemannian manifold. We introduce Manifold DPnP, a generalization of the diffusion plug-and-play (DPnP) sampler to arbitrary complete Riemannian manifolds, by reformulating each sampling iteration as a manifold-valued SDE driven by Brownian heat flow and using the Bismut--Elworthy--Li formula to estimate the score of the likelihood under this flow. We further derive Manifold DPS, an extension of the DPS sampler obtained from a Riemannian Tweedie's formula based on Varadhan's short-time heat-kernel asymptotics. Experiments on the flat torus, on real earthquake locations on the sphere, and on a random-walk trajectory reconstruction problem show that both samplers substantially outperform a prior-free MALA baseline. Our results close a gap between Euclidean diffusion-based inverse problem solvers and intrinsic manifold-valued posterior sampling.