The Geometry of Phase Transitions in Diffusion Models: Tubular Neighbourhoods and Singularities

Manato Yaguchi, Kotaro Sakamoto, Ryosuke Sakamoto, Masato Tanabe, Masatomo Akagawa, Yusuke Hayashi, Masahiro Suzuki, Yutaka Matsuo

Keywords: diffusion models, geometry, tubular neighbourhoods

TL;DR: We investigate the relationship between the geometry of diffusion models and singularities of the generative dynamics.

Abstract: Diffusion models undergo phase transitions during the generative process where data features suddenly emerge in the final stages. The current study aims to elucidate this critical phenomenon from the geometrical perspective. We employ the concept of ``injectivity radius'', a quantity that characterises the structure of the data manifold. Through theoretical and empirical evidence, we demonstrate that phase transitions in the generative process of diffusion models are closely related to the injectivity radius. Our findings offer a novel perspective on phase transitions in diffusion models, with potential implications for improving performance and sampling efficiency.

Primary Area: generative models

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.

Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2025/AuthorGuide.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Submission Number: 8715