Paper ID: 2408.08307

Understanding the Local Geometry of Generative Model Manifolds

Ahmed Imtiaz Humayun, Ibtihel Amara, Candice Schumann, Golnoosh Farnadi, Negar Rostamzadeh, Mohammad Havaei

Deep generative models learn continuous representations of complex data manifolds using a finite number of samples during training. For a pre-trained generative model, the common way to evaluate the quality of the manifold representation learned, is by computing global metrics like Fr\'echet Inception Distance using a large number of generated and real samples. However, generative model performance is not uniform across the learned manifold, e.g., for \textit{foundation models} like Stable Diffusion generation performance can vary significantly based on the conditioning or initial noise vector being denoised. In this paper we study the relationship between the \textit{local geometry of the learned manifold} and downstream generation. Based on the theory of continuous piecewise-linear (CPWL) generators, we use three geometric descriptors - scaling ($\psi$), rank ($\nu$), and complexity ($\delta$) - to characterize a pre-trained generative model manifold locally. We provide quantitative and qualitative evidence showing that for a given latent, the local descriptors are correlated with generation aesthetics, artifacts, uncertainty, and even memorization. Finally we demonstrate that training a \textit{reward model} on the local geometry can allow controlling the likelihood of a generated sample under the learned distribution.

Submitted: Aug 15, 2024