  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.
