Caffarelli Regularity and Hierarchical Phase Boundaries in Diffusion Models

Alexander Lobashev, Dmitry Guskov, Victor Kawasaki-Borruat, Maria Larchenko

Keywords: diffusion models, optimal transport, phase transitions, Caffarelli regularity

Abstract: Recent studies have shown phase-transition-like behavior in diffusion models, where a small perturbation of the initial Gaussian noise sample can cause an abrupt change in the generated image. The underlying mechanism of these transitions, however, remains theoretically underexplored. In this work, we investigate this phenomenon through the lens of the pullback metric on the latent space induced by the perceptual similarity between generated images. We observe a hierarchical emergence of phase boundaries: coarse boundaries appear in the early denoising steps, while finer boundaries progressively emerge within these regions as the denoising process advances. Moreover, we observe that diffusion distillation shifts boundary formation towards earlier denoising steps and reduces final complexity by decreasing the number of sharp boundaries. To provide a theoretical foundation, we follow the JKO scheme and approximate the reverse diffusion dynamics by a discrete-time sequence of quadratic-cost optimal transport maps between successive noisy marginals. We show that mode splitting forces the diffusion generative map to develop large Lipschitz constant. Using Caffarelli’s regularity theory, we argue that these high-Lipschitz regions form contiguous sets, driven by the disjoint support of the real data distribution and giving rise to phase boundaries. We further note that the proposed theoretical framework does not depend on models' design, but describes the general properties of unimodal-to-multimodal diffusion mappings. This leads to an important practical implication: non-Lipschitzness of generative mapping is necessary for good mode coverage.

Primary Area: generative models

Submission Number: 24670