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

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.

Certifications: Featured Certification

Submission Length: Regular submission (no more than 12 pages of main content)

Changes Since Last Submission: - Fixed an incorrect OpenReview link.

Code: https://github.com/yagumana/lateinit

Assigned Action Editor: ~Arash_Mehrjou1

Submission Number: 4775