On The Local Geometry of Deep Generative Manifolds

In this paper, we study theoretically inspired local geometric descriptors of the data manifolds approximated by pre-trained generative models. The descriptors – local scaling (ψ), local rank (ν), and local complexity (δ) — characterize the uncertainty, dimensionality, and smoothness on the learned manifold, using only the network weights and architecture. We investigate and emphasize their critical role in understanding generative models. Our analysis reveals that the local geometry is intricately linked to the quality and diversity of generated outputs. Additionally, we see that the geometric properties are distinct for out-of-distribution (OOD) inputs as well as for prompts memorized by Stable Diffusion, showing the possible application of our proposed descriptors for downstream detection and assessment of pre-trained generative models.