Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows
This is an incremental improvement for researchers in generative modeling, focusing on enhancing existing normalizing flows with optimal transport principles.
The paper tackled the problem of making normalizing flows more optimal transport-efficient by transforming them into Monge maps, resulting in smooth flows with reduced OT cost without affecting model performance.
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.