Landing with the Score: Riemannian Optimization through Denoising

We consider Riemannian optimization over manifolds given implicitly by a probability distribution supported on them. We show how the Stein score function used for denoising can be used as an approximation for certain manifold operations such as retraction and projection onto the tangent space. We then propose an optimizing gradient flow and give asymptotic error bounds on the norm of the Riemannian gradient at its accumulation points in terms of the diffusion temperature. Finally we provide numerical simulations of this flow for diffusion models trained on known manifolds, confirming the validity of our method