Unpaired image-to-image translation has attracted significant interest due to the invention of CycleGAN, a method which utilizes a combination of adversarial and cycle consistency losses to avoid the need for paired data. It is known that the CycleGAN problem might admit multiple solutions, and our goal in this paper is to analyze the space of exact solutions and to give perturbation bounds for approximate solutions. We show theoretically that the exact solution space is invariant with respect to automorphisms of the underlying probability spaces, and, furthermore, that the group of automorphisms acts freely and transitively on the space of exact solutions. We examine the case of zero 'pure' CycleGAN loss first in its generality, and, subsequently, expand our analysis to approximate solutions for 'extended' CycleGAN loss where identity loss term is included. In order to demonstrate that these results are applicable, we show that under mild conditions nontrivial smooth automorphisms exist. Furthermore, we provide empirical evidence that neural networks can learn these automorphisms with unexpected and unwanted results. We conclude that finding optimal solutions to the CycleGAN loss does not necessarily lead to the envisioned result in image-to-image translation tasks and that underlying hidden symmetries can render the result utterly useless.
This paper focuses on CycleGAN method to show theoretically when the exact solution space is invariant with respect to authomorphisms of the underlying probability spaces for unpaired image-to-image translation. 
- The paper provides interesting theoretical results on identifying conditions under which CycleGAN admits nontrivial symmetries and has a natural structure of a principal homogeneous space.  Proposition 2.1 provides interesting insights into the invariance of the kernel space. 
- Propositions 2.5 and 2.6 are interesting in that they show that the existence of authomorphisms can worsen the performance of CycleGAN, however, it is unclear that in practice, how could one verify the conditions efficiently before applying CycleGAN.
- The experimental results are interesting, however, they are very limited. Having a toy experiment is a good sanity check, but it would be more interesting to see the performance on a real-world applications, such as medical images or other use-cases brought in the introduction. Also, more discussion on the results provided in Fig 3, confusion matrices, would be very helpful. Are there any intuitions behind the large and low values in the table? It could be interesting to see what are the effects of other parameters such as alpha in producing the results. 
- Overall this paper presents interesting results regarding the theory of CycleGANs, however, the numerical results are very limited, and do not justify the motivations discussed in the introduction and the abstract. Moreover, although the paper introduces novel attempts and theoretically analyzing the CycleGAN, the scope of the work seems to be limited, and thus, it does not have a sufficient significance to be published in ICLR. I strongly suggest the authors to expand and provide more experimental evaluations.
** update:
Thanks for your comments! I found the additional experiment useful and better aligned with the purpose of the model. The discussion added clarified the confusion about the automorphism. That is why I decided to change my score.