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Normalizing Flows (NF) are powerful likelihood-based generative models thatare able to trade off between expressivity and tractability to model complexdensities. A now well established research avenue leverages optimal transport(OT) and looks for Monge maps, i.e. models with minimal effort between thesource and target distributions. This paper introduces a method based onBrenier's polar factorization theorem to transform any trained NF into a moreOT-efficient version without changing the final density. We do so by learning arearrangement of the source (Gaussian) distribution that minimizes the OT costbetween the source and the final density. We further constrain the path leadingto the estimated Monge map to lie on a geodesic in the space ofvolume-preserving diffeomorphisms thanks to Euler's equations. The proposedmethod leads to smooth flows with reduced OT cost for several existing modelswithout affecting the model performance.

Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows