Diffusion models struggle when the data manifold bends

Diffusion models must learn both the support of the data distribution and how probability is distributed along that support. We show that these two tasks can differ substantially in difficulty. In particular, models can accurately land on the data manifold while systematically under-sampling regions of high curvature. We provide a theoretical explanation showing that curvature induces rapid variation in the score field, increasing the approximation burden on finite-capacity networks. Synthetic experiments confirm this separation and show that larger models and more sampling steps reduce the resulting bias. To study the effect in natural images, we introduce a scalable denoiser-based curvature estimator and apply it to Stable Diffusion 3.5. We find that generated images also under-represent high-curvature regions relative to real data, with classifier-free guidance partially mitigating the effect. These results identify curvature as a key source of distributional error in diffusion models.