# Probability Density Geodesics in Image Diffusion Latent Space

arXiv: https://arxiv.org/abs/2504.06675 · HTML: https://mlsanity.com/paper/2504.06675/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 42 of its 58 references that are in the map.
- First posted on arXiv 9 Apr 2025.

## Placed in

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

## Builds on (most cited first, 30 of 42)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [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)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [BLIP: Bootstrapping Language-Image Pre-training for Unified Vision-Language Understanding and Generation](https://mlsanity.com/paper/2201.12086/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [DreamBooth: Fine Tuning Text-to-Image Diffusion Models for Subject-Driven Generation](https://mlsanity.com/paper/2208.12242/) (2022)
- [DreamFusion: Text-to-3D using 2D Diffusion](https://mlsanity.com/paper/2209.14988/) (2022)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [InstructPix2Pix: Learning to Follow Image Editing Instructions](https://mlsanity.com/paper/2211.09800/) (2022)
- [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)
- [CLEVR: A Diagnostic Dataset for Compositional Language and Elementary Visual Reasoning](https://mlsanity.com/paper/1612.06890/) (2016)
- [Imagic: Text-Based Real Image Editing with Diffusion Models](https://mlsanity.com/paper/2210.09276/) (2022)
- [Multi-Concept Customization of Text-to-Image Diffusion](https://mlsanity.com/paper/2212.04488/) (2022)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Plug-and-Play Diffusion Features for Text-Driven Image-to-Image Translation](https://mlsanity.com/paper/2211.12572/) (2022)
- [SVDiff: Compact Parameter Space for Diffusion Fine-Tuning](https://mlsanity.com/paper/2303.11305/) (2023)
- [P+: Extended Textual Conditioning in Text-to-Image Generation](https://mlsanity.com/paper/2303.09522/) (2023)
- [TOPIQ: A Top-down Approach from Semantics to Distortions for Image Quality Assessment](https://mlsanity.com/paper/2308.03060/) (2023)
- [Style Aligned Image Generation via Shared Attention](https://mlsanity.com/paper/2312.02133/) (2023)
- [Noise-Free Score Distillation](https://mlsanity.com/paper/2310.17590/) (2023)
- [DiffMorpher: Unleashing the Capability of Diffusion Models for Image Morphing](https://mlsanity.com/paper/2312.07409/) (2023)

## Built on by (most cited first, 9 of 9)

- [The Spacetime of Diffusion Models: An Information Geometry Perspective](https://mlsanity.com/paper/2505.17517/) (2025)
- [TP-Blend: Textual-Prompt Attention Pairing for Precise Object-Style Blending in Diffusion Models](https://mlsanity.com/paper/2601.08011/) (2026)
- [Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models](https://mlsanity.com/paper/2510.05509/) (2025)
- [Vibe Spaces for Creatively Connecting and Expressing Visual Concepts](https://mlsanity.com/paper/2512.14884/) (2025)
- [Geodesic Calculus on Implicitly Defined Latent Manifolds](https://mlsanity.com/paper/2510.09468/) (2025)
- [GeodesicNVS: Probability Density Geodesic Flow Matching for Novel View Synthesis](https://mlsanity.com/paper/2603.01010/) (2026)
- [Which Way from B to A: The role of embedding geometry in image interpolation for Stable Diffusion](https://mlsanity.com/paper/2511.12757/) (2025)
- [Energy Guided Geometric Flow Matching](https://mlsanity.com/paper/2509.25230/) (2025)
- [A Likely Geometry of Generative Models](https://mlsanity.com/paper/2510.26266/) (2025)

## Abstract (arXiv)

Diffusion models indirectly estimate the probability density over a data space, which can be used to study its structure. In this work, we show that geodesics can be computed in diffusion latent space, where the norm induced by the spatially-varying inner product is inversely proportional to the probability density. In this formulation, a path that traverses a high density (that is, probable) region of image latent space is shorter than the equivalent path through a low density region. We present algorithms for solving the associated initial and boundary value problems and show how to compute the probability density along the path and the geodesic distance between two points. Using these techniques, we analyze how closely video clips approximate geodesics in a pre-trained image diffusion space. Finally, we demonstrate how these techniques can be applied to training-free image sequence interpolation and extrapolation, given a pre-trained image diffusion model.