Universality of parametric Coupling Flows over parametric diffeomorphisms

Invertible neural networks (INNs) such as coupling flows are firstly introduced as a class of generative models with a tractable likelihood [11, 25, 40], and have shown their usefulness and powerfulness in various machine learning tasks such as inverse problems [2], probabilistic inference [29] and feature extraction [22] in recent years. With plenty of successful applications of INNs, one would wonder if such a type of models have the universal expressiveness. As most generative models mainly concern about the transform between distributions, existing works such as [19, 23] focused on the expressiveness from the distribution perspective. However, the expressiveness from the distribution perspective does not result in the expressiveness from the mapping perspective, as there are a large (or even infinite) number of diffeomorphisms mapping the given source µ to the given target ν. In many applications, knowing the distributional universality is not yet enough, one may be interested in knowing if the optimal transport [41], which finds emerging applications in many fields, e.g., machine learning [32], wireless communication [30] and economics [15], can be approximated by invertible neural networks.