# Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models

arXiv: https://arxiv.org/abs/2510.05509 · HTML: https://mlsanity.com/paper/2510.05509/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 94 of its 116 references that are in the map.
- First posted on arXiv 7 Oct 2025.

## Placed in

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

## Builds on (most cited first, 30 of 94)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [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)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [BLIP-2: Bootstrapping Language-Image Pre-training with Frozen Image Encoders and Large Language Models](https://mlsanity.com/paper/2301.12597/) (2023)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [BLIP: Bootstrapping Language-Image Pre-training for Unified Vision-Language Understanding and Generation](https://mlsanity.com/paper/2201.12086/) (2022)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Prompt-to-Prompt Image Editing with Cross Attention Control](https://mlsanity.com/paper/2208.01626/) (2022)
- [An Image is Worth One Word: Personalizing Text-to-Image Generation using Textual Inversion](https://mlsanity.com/paper/2208.01618/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Null-text Inversion for Editing Real Images using Guided Diffusion Models](https://mlsanity.com/paper/2211.09794/) (2022)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Latent Consistency Models: Synthesizing High-Resolution Images with Few-Step Inference](https://mlsanity.com/paper/2310.04378/) (2023)
- [Attend-and-Excite: Attention-Based Semantic Guidance for Text-to-Image Diffusion Models](https://mlsanity.com/paper/2301.13826/) (2023)
- [Interpreting the Latent Space of GANs for Semantic Face Editing](https://mlsanity.com/paper/1907.10786/) (2019)
- [GANSpace: Discovering Interpretable GAN Controls](https://mlsanity.com/paper/2004.02546/) (2020)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [Diffusion Autoencoders: Toward a Meaningful and Decodable Representation](https://mlsanity.com/paper/2111.15640/) (2021)

## Built on by (most cited first, 3 of 3)

- [A Likely Geometry of Generative Models](https://mlsanity.com/paper/2510.26266/) (2025)
- [MaDeL: Manifold-Decomposed Feature Losses for Generative Modeling](https://mlsanity.com/paper/2610.04419/) (2026)
- [Geometry-Aware Preference Optimization for Text-to-Image Diffusion Models](https://mlsanity.com/paper/2610.04980/) (2026)

## Abstract (arXiv)

Diffusion models are powerful deep generative models, but unlike classical models, they lack an explicit low-dimensional latent space that parameterizes the data manifold. This absence makes it difficult to perform manifold-aware operations, such as geometrically faithful interpolation or conditional guidance that respects the learned manifold. We propose a training-free Riemannian metric on the noise space, derived from the Jacobian of the score function. The key insight is that the spectral structure of this Jacobian separates tangent and normal directions of the data manifold; our metric leverages this separation to encourage paths to stay tangential to the manifold rather than drift toward high-density regions. To validate that our metric faithfully captures the manifold geometry, we examine it from two complementary angles. First, geodesics under our metric yield perceptually more natural interpolations than existing methods on synthetic, image, and video frame datasets. Second, the tangent-normal decomposition induced by our metric prevents classifier-free guidance from deviating off the manifold, improving generation quality while preserving text-image alignment.