Understanding the Local Geometry of Generative Model Manifolds

• Main Authors: Humayun, Ahmed Imtiaz, Amara, Ibtihel, Schumann, Candice, Farnadi, Golnoosh, Rostamzadeh, Negar, Havaei, Mohammad
• Format: Journal Article
• Language: English
• Published: 15.08.2024
• Subjects: Computer Science - Computer Vision and Pattern Recognition; Computer Science - Learning
• DOI: 10.48550/arxiv.2408.08307

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.