Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators
This provides theoretical reliability for CF-INNs in applications like image synthesis, though it is incremental as it builds on existing coupling flow methods.
The paper tackled the problem of whether coupling-based invertible neural networks (CF-INNs) can universally approximate invertible functions, showing that they are universal if layers include affine coupling and invertible linear functions, which resolves an unsolved problem about normalizing flow models.
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.