Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: In this paper, the authors provided their theoretical understanding on the translation and reconstruction guarantees of the cycle-consistent GANs. They showed that the translators based on deep ReLU networks can prevent the input information from dissipating during generation cycles, which means we can lower bound the gap. On the other hand, there are positive results. The authors proved that the same translators not only achieve zero generation loss asymptotically but the generated distribution converges to the target distribution almost surely. The authors also showed the equivalence between L1 norm and 1-Wasserstein distance in the cyclic loss. 
strengths_and_weaknesses: This paper is very solid in the theory. The writing quality of this paper is high. And most importantly, the problem analyzed by the authors is very fundamental and important in the field of generative models. 

Although the literature review of this paper does not cover all the theoretical papers on GANs, it is not a big deal. The results are original and significant. 
questions: Is it possible to take the optimization trajectory or process into consideration, since we know that GAN is notorious for its hardness of training and instability of training. 
limitations: I think there is no potential negative societal impact of this work since it is completely theoretical.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: In this paper the authors analyze the properties of image to image translation networks that are based on cycle consistency. 
In particular, the authors show that when the translation networks are based on ReLU activations they behave as Information preserving transformations (IPTs). Next the authors also show that 1-Wasserstein distance and L1 cyclic distance are equivalent for Sobolev-smooth data and analyze the effects of ill-posedness on the regeneration. 
strengths_and_weaknesses: The setting is rigorously formulated and all the claims made in the manuscript appear to be shown in ample detail. 
questions: Additional discussion about other architectural choices for the translation networks should be helpful. 
Moreover, description of how these assumptions correspond to existing image-to-image translation methods should be very useful. 
limitations: The relevant assumptions about the network architectures and the data are stated clearly by the authors
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: In this paper, the authors propose a novel statistic framework to analyze the consistency of Cycle-GAN type neural networks, where translator and inverse-translator between two domains in the images spaces are trained. In particular, given ReLU DNN translators, the authors prove that they are information preserving based on delicate functional analysis tools. Moreover, the paper provides theoretical attests to previous empirical studies on the equivalence of L1 norm and 1-W distance in cyclic loss. It is also proved that using translation consistent translators, a cyclic-consistent network can be built.
strengths_and_weaknesses: The strengths of the paper include rigorous statistical analysis to studying the cyclic-consistency of cycle-GAN type networks: sharp non-asymptotic bounds on the translation error under mild conditions are rigorously proven. Also, for the first time, a deterministic bound on cumulative reconstruction error is given. This sheds new light on where the ill-posedness of cycle-GAN type networks stems from. The results are clearly stated and the motivations are explained well. Also,  commonly seen cycle-GAN type networks are reviewed and compared under the proposed framework, showing the great value in the application of the proposed analyzing framework. 

It might be more illustrative if numerical experiment results are given to support the theoretical results. But since the paper is fruitful theoretically, it is only a minor drawback.
questions: - in line 374, would you elaborate more on the self-attacking behavior?

limitations: The limitations of the paper are discussed in the last section.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: I do not think my expertise matches this paper so my summary of the paper and its contributions may not be accurate. This paper provides a series of math explanations and proof for the statistical translation and regeneration guarantees of cycle-consistent networks. The contribution of the paper is establishing an analytical way for not only cycle-consistent networks but also other deep generative models.
strengths_and_weaknesses: Strengths:
There are rare papers that provide the translation of the deep generative models based on rigorous mathematics. Therefore, I do think this paper has high originality and may have a good impact on the readers to conduct further research in the future.

Weaknesses:
Although math is a beautiful way to elaborate an algorithm, sometimes telling a story by images has more affinity to the readers and may help readers to know the whole picture faster.
questions: N/A.
limitations: N/A.
review_title: null

Meta-review: metareview: This paper analyzes the consistency of cycle-consistent GANs. The authors provide theoretical insights in how the information is preserved by ReLU DNN translators via functional analysis. They additionally show the equivalence of L1 and 1-W distance under the cyclic loss context. The highlight of this paper is the rigorous statistical analysis and fundamental theoretical insights in generative models. A minor point is that the reviewers suggest that more illustrative and intuitive presentation of the results will smooth the reading. In general it's a interesting paper and the AC recommends acceptance. 
confidence: Certain
Decision: Accept