We consider Riemannian optimization over manifolds given implicitly by a probability distribution supported on them. We show how the Stein score function, estimated by diffusion models, can be used as an approximation for certain manifold operations such as the closest-point and tangent space projections. We then propose a landing gradient flow and give asymptotic error bounds on the norm of the Riemannian gradient at its accumulation points in terms of the diffusion temperature. Additionally we obtain average gradient norm estimates for the classical Riemannian gradient descent algorithm and provide a modification that allows to perform one gradient step using a single Jacobian-vector product. Finally, we provide numerical experiments using the score function obtained from diffusion models trained on known synthetic manifolds and apply our method to finite-horizon reference tracking in the domain of data-driven control.

@inproceedings{kharitenko2025landing,
	title={Landing with the Score: Riemannian Optimization through Denoising},
	author={Kharitenko, Andrey and Shen, Zebang and De Santi, Riccardo and He, Niao and Doerfler, Florian},
	booktitle={International Conference on Learning Representations (ICLR)},
	year={2026},
	month={April},
	pdf={https://arxiv.org/abs/2509.23357},
}