Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators

TL;DR

This paper proves that coupling-based invertible neural networks are universal diffeomorphism approximators, demonstrating their broad representational capacity for invertible functions and distributions, which was previously unresolved.

Contribution

It establishes a criterion for universality of CF-INNs and resolves the open question of their ability to approximate all invertible functions and distributions.

Findings

CF-INNs are universal if they include affine coupling and invertible linear functions.

Normalizing flow models with affine coupling are universal distributional approximators.

A general theorem on the equivalence of universality for certain diffeomorphism classes.

Abstract

Invertible neural networks based on coupling flows (CF-INNs) have various machine learning applications such as image synthesis and representation learning. However, their desirable characteristics such as analytic invertibility come at the cost of restricting the functional forms. This poses a question on their representation power: are CF-INNs universal approximators for invertible functions? Without a universality, there could be a well-behaved invertible transformation that the CF-INN can never approximate, hence it would render the model class unreliable. We answer this question by showing a convenient criterion: a CF-INN is universal if its layers contain affine coupling and invertible linear functions as special cases. As its corollary, we can affirmatively resolve a previously unsolved problem: whether normalizing flow models based on affine coupling can be universal distributional approximators. In the course of proving the universality, we prove a general theorem to show the equivalence of the universality for certain diffeomorphism classes, a theoretical insight that is of interest by itself.