Papers › Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry

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

24 Jul 2023NeurIPS 2023 11arXiv:2307.12868archive 2025-07-28

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

Despite the success of diffusion models (DMs), we still lack a thorough understanding of their latent space. To understand the latent space 𝐱ₜ ∈𝒳, we analyze them from a geometrical perspective. Our approach involves deriving the local latent basis within 𝒳 by leveraging the pullback metric associated with their encoding feature maps. Remarkably, our discovered local latent basis enables image editing capabilities by moving 𝐱ₜ, 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 𝐱ₜ 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 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.

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