Landing with the Score: Riemannian Optimization through Denoising

Under the data manifold hypothesis, high-dimensional data concentrate near a low-dimensional manifold. We study Riemannian optimization when this manifold is only given implicitly through the data distribution, and standard geometric operations are unavailable. This formulation captures a broad class of data-driven design problems that are central to modern generative AI. Our key idea is a link function that ties the data distribution to the geometric quantities needed for optimization: its gradient and Hessian recover the projection onto the manifold and its tangent space in the small-noise regime. This construction is directly connected to the score function in diffusion models, allowing us to leverage well-studied parameterizations, efficient training procedures, and even pretrained score networks from the diffusion model literature to perform optimization. On top of this foundation, we develop two efficient inference-time algorithms for optimization over data manifolds: Denoising Landing Flow (DLF) and Denoising Riemannian Gradient Descent (DRGD). We provide theoretical guarantees for approximate feasibility (manifold adherence) and optimality (small Riemannian gradient norm). We demonstrate the effectiveness of our approach on finite-horizon reference tracking tasks in data-driven control, illustrating their potential for practical generative and design applications.