The Geometry of Diffusion Models: Tubular Neighbourhoods and Singularities

Diffusion generative models have been a leading approach for generating high-dimensional data. The current research aims to investigate the relation between the dynamics of diffusion models and the tubular neighbourhoods of a data manifold. We propose an algorithm to estimate the injectivity radius, the supremum of radii of tubular neighbourhoods. Our research relates geometric objects such as curvatures of data manifolds and dimensions of ambient spaces, to singularities of the generative dynamics such as emergent critical phenomena or spontaneous symmetry breaking.