Re: Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators (ID=1064). We thank the reviewers for reviewing our work. We will update the paper based on the suggestions. On what occasion would the diffeomorphic universality results be useful other than distribution approximation? Thank you for pointing out the missing references.

Additional Feedback: [POST REBUTTAL] --------- I thank the authors for the detailed response. I guess I see how one can parameterize some (Real NVP-style) linear couplings combined with permutation and sign flipping to represent any regular matrices (regular here denotes invertible I suppose?). Perhaps this could be explicitly constructed in the paper to complement the results. Does it also imply the the general linear group in the main result can be replaced with permutation group sign flipping? Furthermore, if someone is only concerned with diffeomorphisms with a jacobian having strictly positive eigenvalues, then can the sign flipping be dropped?

The paper received a positive feedback from the four reviewers. The reviewers have raised a few concerns, which have been addressed in the rebuttal. The area chair agrees with the reviewer's asssessment and follows their recommendation.

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.

Invertible neural networks based on coupling flows (CF-INNs) are neural network architectures with invertibility by design [1, 2]. Endowed with the analytic-form invertibility and the tractability of the Jacobian, CF-INNs have demonstrated their usefulness in various machine learning tasks such as generative modeling [3-7], probabilistic inference [8-10], solving inverse problems [11], and feature extraction and manipulation [4, 12-14]. The attractive properties of CF-INNs come at the cost of potential restrictions on the set of functions that they can approximate because they rely on carefully designed network layers. To circumvent the potential drawback, a variety of layer designs have been proposed to construct CF-INNs with high representation power, e.g., the affine coupling flow [3, 4, 15-17], the neural autoregressive flow [18-20], and the polynomial flow [21], each demonstrating enhanced empirical performance. Despite the diversity of layer designs [1, 2], the theoretical understanding of the representation power of CF-INNs has been limited. Indeed, the most basic property as a function approximator, namely the universal approximation property (or universality for short) [22], has not been elucidated for CF-INNs. The universality can be crucial when CF-INNs are used to learn an invertible transformation (e.g., feature extraction [12] or independent component analysis [14]) because, informally speaking, lack of universality implies that there exists an invertible transformation, even among well-behaved ones, that CF-INN can never approximate, and it would render the model class unreliable for the task of function approximation.