Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows

TL;DR

This paper presents a method to convert trained normalizing flows into more optimal Monge maps using optimal transport theory, improving efficiency while preserving model performance.

Contribution

It introduces a novel approach leveraging Brenier's theorem and geodesic constraints to enhance normalizing flows with minimal density change.

Findings

Reduces OT cost in existing models

Maintains original model performance

Produces smoother flows

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.