# Landing with the Score: Riemannian Optimization through Denoising

arXiv: https://arxiv.org/abs/2509.23357 · HTML: https://mlsanity.com/paper/2509.23357/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 29 of its 73 references that are in the map.
- First posted on arXiv 27 Sep 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 29 of 29)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [Diffusion models as plug-and-play priors](https://mlsanity.com/paper/2206.09012/) (2022)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorial](https://mlsanity.com/paper/2402.07487/) (2024)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024)
- [A Brief Introduction to Manifold Optimization](https://mlsanity.com/paper/1906.05450/) (2019)
- [Fast and accurate optimization on the orthogonal manifold without retraction](https://mlsanity.com/paper/2102.07432/) (2021)
- [Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion](https://mlsanity.com/paper/2410.05898/) (2024)
- [Hessian Riemannian gradient flows in convex programming](https://mlsanity.com/paper/1811.10331/) (2018)
- [Complexity of Proximal augmented Lagrangian for nonconvex optimization with nonlinear equality constraints](https://mlsanity.com/paper/1908.00131/) (2019)
- [Orthogonal Directions Constrained Gradient Method: from non-linear equality constraints to Stiefel manifold](https://mlsanity.com/paper/2303.09261/) (2023)
- [Port-Hamiltonian Neural ODE Networks on Lie Groups For Robot Dynamics Learning and Control](https://mlsanity.com/paper/2401.09520/) (2024)
- [Local Manifold Approximation and Projection for Manifold-Aware Diffusion Planning](https://mlsanity.com/paper/2506.00867/) (2025)
- [Learning Manifold Implicitly via Explicit Heat-Kernel Learning](https://mlsanity.com/paper/2010.01761/) (2020)
- [MAP Estimation with Denoisers: Convergence Rates and Guarantees](https://mlsanity.com/paper/2507.15397/) (2025)
- [Finite-Horizon LQR Control of Quadrotors on SE2(3)](https://mlsanity.com/paper/2105.13935/) (2021)
- [Adjoint-Based Aerodynamic Shape Optimization with a Manifold Constraint Learned by Diffusion Models](https://mlsanity.com/paper/2507.23443/) (2025)

## Built on by (most cited first, 8 of 8)

- [Manifold-Aligned Guided Integrated Gradients for Reliable Feature Attribution](https://mlsanity.com/paper/2605.02167/) (2026)
- [Riemannian Metric Matching for Scalable Geometric Modeling of Distributions](https://mlsanity.com/paper/2606.14334/) (2026)
- [Diffusion Processes on Implicit Manifolds](https://mlsanity.com/paper/2604.07213/) (2026)
- [Recovering Weighted Tangent Geometry from a Single-Scale Score Field](https://mlsanity.com/paper/2608.22334/) (2026)
- [Safety-Critical Contextual Control via Online Riemannian Optimization with World Models](https://mlsanity.com/paper/2604.19639/) (2026)
- [Denoising as Projection: Constrained Optimization with Gradient-Guided Diffusion](https://mlsanity.com/paper/2608.29507/) (2026)
- [Matched Queries for Curvature and Density at Branching Junctions](https://mlsanity.com/paper/2609.01319/) (2026)
- [Intrinsic Geometry Matching: learning observation-equivariant process geometry](https://mlsanity.com/paper/2610.16995/) (2026)

## Abstract (arXiv)

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 perform optimization. Building on this foundation, we propose two efficient inference-time algorithms -- Denoising Landing Flow (DLF) and Denoising Riemannian Gradient Descent (DRGD) -- and provide theoretical guarantees for both feasibility (approximate manifold adherence) and optimality (small Riemannian gradient norm). Finally, we demonstrate the effectiveness of our approach on finite-horizon reference tracking tasks in data-driven control, highlighting its potential for practical generative and design applications.