A Likely Geometry of Generative Models

TMLR Paper8687 Authors

Abstract: The geometry of generative models serves as the basis for interpolation, model inspection, and more. Although certain generative models admit an implicit geometric structure, there is no broadly applicable framework that captures a principled notion of geometry across generative models without imposing restrictive assumptions on the model class or data dimensionality. In this paper, we show how to equip generative models with a general geometry compatible with different metrics and probability distributions to analyze generative models. Our method does not require additional training. We consider curves analogous to geodesics constrained to a suitable data distribution aimed at targeting high-density regions learned by generative models. We formulate this as a (pseudo-)metric and prove correspondence to a Newtonian system on a Riemannian manifold. We show that shortest paths can here be characterized by a system of ordinary differential equations, which, along the optimal path, locally correspond to geodesics under a suitable Riemannian metric. Numerically, we derive a novel algorithm to efficiently compute interpolation and generalized Fréchet means. Quantitatively, we show that curves using our metric traverse regions of higher likelihood areas than baselines across a range of models and datasets.

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

Assigned Action Editor: ~Richard_Nock1

Submission Number: 8687