# Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry

Source: [Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry](<https://www.sota2.com/research/paper/68fa710a1e209131cea4ecf0>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-26T01:56:42.060Z

Authors: Yong-Hyun Park, Mingi Kwon, Jaewoong Choi, Junghyo Jo, Youngjung Uh

Published: 2023-07-24T15:06:42.000Z

## Abstract

Despite the success of diffusion models (DMs), we still lack a thorough understanding of their latent space. To understand the latent space $\\mathbf{x}\_t \\in \\mathcal{X}$, we analyze them from a geometrical perspective. Our approach involves deriving the local latent basis within $\\mathcal{X}$ by leveraging the pullback metric associated with their encoding feature maps. Remarkably, our discovered local latent basis enables image editing capabilities by moving $\\mathbf{x}\_t$, the latent space of DMs, along the basis vector at specific timesteps. We further analyze how the geometric structure of DMs evolves over diffusion timesteps and differs across different text conditions. This confirms the known phenomenon of coarse-to-fine generation, as well as reveals novel insights such as the discrepancy between $\\mathbf{x}\_t$ across timesteps, the effect of dataset complexity, and the time-varying influence of text prompts. To the best of our knowledge, this paper is the first to present image editing through $\\mathbf{x}$-space traversal, editing only once at specific timestep $t$ without any additional training, and providing thorough analyses of the latent structure of DMs. The code to reproduce our experiments can be found at https://github.com/enkeejunior1/Diffusion-Pullback.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

- [Image Editing on CelebA-HQ (test)](<https://www.sota2.com/research/sota/image-editing-on-celeba-hq-test.md>): Image Editing; CelebA-HQ (test); SSIM: 0.9.

- [Semantic Image Editing on CelebA-HQ (test)](<https://www.sota2.com/research/sota/semantic-image-editing-on-celeba-hq-test.md>): Semantic Image Editing; CelebA-HQ (test); Editing Time (s): 80.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2307.12868>)

[PDF](<https://arxiv.org/pdf/2307.12868.pdf>)