将归一化流转化为 Monge 映射：测地高斯保持流
Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows
AI总结：
本文提出基于 Brenier 极分解定理和欧拉方程的方法，将已训练的归一化流转化为沿保体积微分同胚测地线的 Monge 映射，在保持模型性能的同时降低最优传输代价。
摘要：
归一化流（Normalizing Flows, NF）是强大的基于似然的生成模型，能够在表达能力和可解性之间进行权衡，以建模复杂密度。一个现已成熟的研究方向利用最优传输（optimal transport, OT），寻找 Monge 映射，即在源分布和目标分布之间具有最小代价的模型。本文引入一种基于 Brenier 极分解定理的方法，在不改变最终密度的情况下，将任意已训练的 NF 转化为更具 OT 效率的版本。我们通过学习源（高斯）分布的重排来实现这一点，该重排最小化源分布与最终密度之间的 OT 代价。借助欧拉方程，我们进一步约束通往估计 Monge 映射的路径位于保体积微分同胚空间中的测地线上。所提出的方法在不影响模型性能的情况下，为多个现有模型带来了具有降低 OT 代价的平滑流。
英文摘要：
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.
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