Caffarelli Regularity and Hierarchical Phase Boundaries in Diffusion Model Latent Space
Authors: Alexander Lobashev, Dmitry Guskov, Victor Borruat, Maria Larchenko
🎯 Key Contributions
- Reveals hierarchical phase boundaries in diffusion model latent spaces
- Connects Caffarelli regularity theory to diffusion model geometry
- Demonstrates fractal-like structure of phase transitions
- Provides theoretical framework linking optimal transport and diffusion dynamics
Abstract
This paper investigates the geometric structure of diffusion model latent spaces through the lens of optimal transport theory and Caffarelli regularity. We reveal hierarchical phase boundaries that organize the latent space into distinct regions with different geometric properties.
Our analysis shows that these phase boundaries exhibit fractal-like structures and correspond to qualitative changes in the generated content. We provide theoretical foundations connecting Monge-Ampère equations and diffusion model dynamics, demonstrating how Caffarelli regularity conditions relate to the smoothness and stability of latent space trajectories.
These findings offer new insights into the interpretability and control of diffusion models, with implications for guided generation, interpolation quality, and understanding emergent properties in generative AI systems.
📋 Citation
@article{lobashev2025caffarelli,
title={Caffarelli Regularity and Hierarchical Phase Boundaries in Diffusion Model Latent Space},
author={Lobashev, Alexander and Guskov, Dmitry and Borruat, Victor and Larchenko, Maria},
journal={arXiv preprint},
year={2025}
}