Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows

Guillaume Morel, Lucas Drumetz, Simon Benaïchouche, Nicolas Courty, François Rousseau

Abstract: Normalizing Flows (NF) are powerful likelihood-based generative models that are able to trade off between expressivity and tractability to model complex densities. A now well established research avenue leverages optimal transport (OT) and looks for Monge maps, i.e. models with minimal effort between the source and target distributions. This paper introduces a method based on Brenier's polar factorization theorem to transform any trained NF into a more OT-efficient version without changing the final density. We do so by learning a rearrangement of the source (Gaussian) distribution that minimizes the OT cost between the source and the final density. The Gaussian preserving transformation is implemented with the construction of high dimensional divergence free functions and the path leading to the estimated Monge map is further constrained to lie on a geodesic in the space of volume-preserving diffeomorphisms thanks to Euler's equations. The proposed method leads to smooth flows with reduced OT costs for several existing models without affecting the model performance.

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

Code: https://github.com/morel-g/GPFlow

Assigned Action Editor: ~George_Papamakarios1

License: Creative Commons Attribution 4.0 International (CC BY 4.0)

Submission Number: 816