# What's Inside Your Diffusion Model? A Score-Based Riemannian Metric to Explore the Data Manifold

arXiv: https://arxiv.org/abs/2505.11128 · HTML: https://mlsanity.com/paper/2505.11128/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 33 of its 60 references that are in the map.
- First posted on arXiv 16 May 2025.

## Placed in

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

## Builds on (most cited first, 30 of 33)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Adding Conditional Control to Text-to-Image Diffusion Models](https://mlsanity.com/paper/2302.05543/) (2023)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Null-text Inversion for Editing Real Images using Guided Diffusion Models](https://mlsanity.com/paper/2211.09794/) (2022)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2307.12868/) (2023)
- [Score-Based Generative Models Detect Manifolds](https://mlsanity.com/paper/2206.01018/) (2022)
- [Robust and interpretable blind image denoising via bias-free convolutional neural networks](https://mlsanity.com/paper/1906.05478/) (2019)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [DiffMorpher: Unleashing the Capability of Diffusion Models for Image Morphing](https://mlsanity.com/paper/2312.07409/) (2023)
- [Classification and Geometry of General Perceptual Manifolds](https://mlsanity.com/paper/1710.06487/) (2017)
- [Norm-guided latent space exploration for text-to-image generation](https://mlsanity.com/paper/2306.08687/) (2023)
- [The Geometry of Deep Generative Image Models and its Applications](https://mlsanity.com/paper/2101.06006/) (2021)
- [Optimization Techniques on Riemannian Manifolds](https://mlsanity.com/paper/1407.5965/) (2014)
- [Riemannian Laplace approximations for Bayesian neural networks](https://mlsanity.com/paper/2306.07158/) (2023)
- [NoiseDiffusion: Correcting Noise for Image Interpolation with Diffusion Models beyond Spherical Linear Interpolation](https://mlsanity.com/paper/2403.08840/) (2024)
- [Lagrangian Manifold Monte Carlo on Monge Patches](https://mlsanity.com/paper/2202.00755/) (2022)
- [The local low-dimensionality of natural images](https://mlsanity.com/paper/1412.6626/) (2014)
- [Scalable Stochastic Gradient Riemannian Langevin Dynamics in Non-Diagonal Metrics](https://mlsanity.com/paper/2303.05101/) (2023)

## Built on by (most cited first, 6 of 6)

- [Learning a distance measure from the information-estimation geometry of data](https://mlsanity.com/paper/2510.02514/) (2025)
- [Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models](https://mlsanity.com/paper/2510.05509/) (2025)
- [Riemannian Metric Matching for Scalable Geometric Modeling of Distributions](https://mlsanity.com/paper/2606.14334/) (2026)
- [RemEdit: Efficient Diffusion Editing with Riemannian Geometry](https://mlsanity.com/paper/2601.17927/) (2026)
- [Exploring the Intrinsic Geometry of Diffusion Models with Constrained Inverse Kinematics](https://mlsanity.com/paper/2606.26408/) (2026)
- [One Inverse Step is a Convex Program: Bayes-Limit Calibration of Diffusion Inversion](https://mlsanity.com/paper/2608.23094/) (2026)

## Abstract (arXiv)

Recent advances in diffusion models have demonstrated their remarkable ability to capture complex image distributions, but the geometric properties of the learned data manifold remain poorly understood. We address this gap by introducing a score-based Riemannian metric that leverages the Stein score function from diffusion models to characterize the intrinsic geometry of the data manifold without requiring explicit parameterization. Our approach defines a metric tensor in the ambient space that stretches distances perpendicular to the manifold while preserving them along tangential directions, effectively creating a geometry where geodesics naturally follow the manifold's contours. We develop efficient algorithms for computing these geodesics and demonstrate their utility for both interpolation between data points and extrapolation beyond the observed data distribution. Through experiments on synthetic data with known geometry, Rotated MNIST, and complex natural images via Stable Diffusion, we show that our score-based geodesics capture meaningful transformations that respect the underlying data distribution. Our method consistently outperforms baseline approaches on perceptual metrics (LPIPS) and distribution-level metrics (FID, KID), producing smoother, more realistic image transitions. These results reveal the implicit geometric structure learned by diffusion models and provide a principled way to navigate the manifold of natural images through the lens of Riemannian geometry.