[2509.23357] Landing with the Score: Riemannian Optimization through Denoising

Computer Science > Machine Learning arXiv:2509.23357 (cs) [Submitted on 27 Sep 2025 (v1), last revised 28 Feb 2026 (this version, v2)] Title:Landing with the Score: Riemannian Optimization through Denoising Authors:Andrey Kharitenko, Zebang Shen, Riccardo de Santi, Niao He, Florian Doerfler View a PDF of the paper titled Landing with the Score: Riemannian Optimization through Denoising, by Andrey Kharitenko and 4 other authors View PDF HTML (experimental) Abstract:Under the data manifold hypothesis, high-dimensional data are concentrated near a low-dimensional manifold. We study the problem of Riemannian optimization over such manifolds when they are given only implicitly through the data distribution, and the standard manifold operations required by classical algorithms are unavailable. This formulation captures a broad class of data-driven design problems that are central to modern generative AI. Our key idea is to introduce a link function that connects the data distribution to the geometric operations needed for optimization. We show that this function enables the recovery of essential manifold operations, such as retraction and Riemannian gradient computation. Moreover, we establish a direct connection between our construction and the score function in diffusion models of the data distribution. This connection allows us to leverage well-studied parameterizations, efficient training procedures, and even pretrained score networks from the diffusion model literature to p...