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Invertible neural networks based on coupling flows (CF-INNs) have variousmachine learning applications such as image synthesis and representationlearning. However, their desirable characteristics such as analyticinvertibility come at the cost of restricting the functional forms. This posesa question on their representation power: are CF-INNs universal approximatorsfor invertible functions? Without a universality, there could be a well-behavedinvertible transformation that the CF-INN can never approximate, hence it wouldrender the model class unreliable. We answer this question by showing aconvenient criterion: a CF-INN is universal if its layers contain affinecoupling and invertible linear functions as special cases. As its corollary, wecan affirmatively resolve a previously unsolved problem: whether normalizingflow models based on affine coupling can be universal distributionalapproximators. In the course of proving the universality, we prove a generaltheorem to show the equivalence of the universality for certain diffeomorphismclasses, a theoretical insight that is of interest by itself.

Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators