Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows

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

Keywords: Euler's equations, Geodesic, Optimal transport, Normalizing flows, Brenier's polar factorization theorem

TL;DR: We propose a method to reduce the optimal transport (OT) cost of any pre-trained NF model without changing the target density and constrain the learned flow to be geodesic in the sense of the Euler equations.

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. We further constrain the path leading to the estimated Monge map 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 cost for several existing models without affecting the model performance.