Unsupervised Domain Translation (UDT) consists in finding meaningful correspondences between two domains, without access to explicit pairings between them. Following the seminal work of CycleGAN, many variants and extensions of this model have been applied successfully to a wide range of applications. However, these methods remain poorly understood, and lack convincing theoretical guarantees. In this work, we define UDT in a rigorous, non-ambiguous manner, explore the implicit biases present in the approach and demonstrate the limits of theses approaches. Specifically, we show that mappings produced by these methods are biased towards low energy transformations, leading us to cast UDT into an Optimal Transport (OT) framework by making this implicit bias explicit. This not only allows us to provide theoretical guarantees for existing methods, but also to solve UDT problems where previous methods fail. Finally, making the link between the dynamic formulation of OT and CycleGAN, we propose a simple approach to solve UDT, and illustrate its properties in two distinct settings.
Given pairs of elements from two different domains, domain translation consists in learning a mapping from one domain to another, linking these paired elements together. If we consider a photograph of a given scene, and an artistic painting of the same scene, we may want to learn the map associating a photograph to paintings describing the same scene, and conversely, for paintings to photographs. A wide range of problems can be formulated as translation, including image-toimage (Isola et al. (2016)), or video-to-video (Wang et al. (2018), image captioning ), natural language translation (Bahdanau et al. (2015)), etc... However, obtaining paired examples can be difficult thus motivating the unpaired setting where only samples from both domains are available which allows us to tackle a wider range of problems. A seminal work in this direction has been the CycleGAN model proposed in Zhu et al. (2017a), which has led to extensions for many applications and has given impressive results.
The starting point of this work is to understand and study this successful approach as there remains little theoretical understanding of why these models work. ; Yang et al. (2018) have observed that the approach first introduced in Zhu et al. (2017a) is ill-posed: in most cases, any pairing between samples of both domains -many of which are unwanted pairings -satisfies their objective function. This is in contradiction with the empirical results of the model and shows that there must be an implicit bias towards well-behaved mappings.  made a first step in this direction by demonstrating that desirable mappings are obtained by networks of minimal complexity, relating this notion of complexity to the number of hidden layers of the neural network implementing the mapping. They then show that the number of mappings satisfying the objective is expected to be small 1 . This is indeed a first step but some ambiguity remains: For example, when translating photographs to paintings, instead of preserving the content of the input, we may want to preserve the same color palette. Here, both tasks are in contradiction and no mapping can solve both problems: What task will be solved by the CycleGAN model? What is the right notion of complexity to explain this convergence? Can we steer the model toward a certain task?
In this work, we aim to answer these questions and further bridge the gap between practice and theory, by first rigorously and unambiguously defining the UDT problem. Optimal Transport theory then appears as a natural tool to regularize UDT models in this context and its dynamical formulation allows us to build a model which overcomes the shortcomings of CycleGAN-like models.
Our main contributions are the following:
• Highlighting the need for a more rigorous treatment of the problem of UDT, we reformulate it in a rigorous, non-ambiguous and generic way, showing the theoretical limits of the CycleGAN approach.
• We provide an explanation for the success of CycleGAN, bridging the gap between theory and empirical findings and shedding light on the implicit biases in those models.
• More specifically, this allows us to link UDT and Optimal Transport. We show that any UDT problem can be cast into the Optimal Transport framework and prove that any UDT Task can be solved by selecting the appropriate transportation cost.
• Exploiting the connection between residual networks and dynamical systems, we present a general and flexible approach to solve UDT, based on the Dynamical formulation of Optimal Transport. This allows us to train a steerable one-sided mapping, and obtain the inverse mapping for free after training, along with smooth interpolations between both domains.
• We empirically highlight the implicit bias and limitations of the CycleGAN approach, and demonstrate the potential of the proposed approach in several different scenarios.
The paper proposes an analysis of CycleGAN methods. Observing that the class of solutions (mappings) covered by those methods is very large, they propose to restrict the set of solutions by looking at low energy mappings (energy being defined wrt. a dedicated cost). A natural formulation of the associated problem is found in optimal transport (OT) theory. They examine the underlying problem in its dynamical formulation, for which a direct connection can be made with ResNet architecture that are commonly used in cycleGANs. They illustrate these results on simple examples, involving pairing swapped digits from MNIST and celebA male to female examples. As a matter of facts, results presented with the OT formulation are more constant. The main proposition of the paper is that the task at hand can be efficiently coded through the distance (cost) function of OT.
Overall the paper is well written and the proposition is reasonable. Yet some parts seem unnecessary long to me, or bring little information, notably the formalisation of 2.1 and 2.2. The fact that cycleGANs are severely ill-posed problems is well known from the computer vision community. Variants that can include a few paired samples can be found (not exhaustive): 
Tripathy, S., Kannala, J., & Rahtu, E. (2018, December). Learning image-to-image translation using paired and unpaired training samples. In Asian Conference on Computer Vision (pp. 51-66). Springer, Cham.
Oe that try to regularize the associated flow:
DLOW: Domain Flow for Adaptation and Generalization Rui Gong, Wen Li, Yuhua Chen, Luc Van Gool; The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2019, pp. 2477-2486
In this spirit, I wonder if a comparison with only the vanilla cycleGAN is sufficient to really assess the interest of using the OT formulation of the problem. Notably, in the example of digit swaps, a cost is learnt by finding a representation of the digits that eliminates the importance of position in the representation. Training such a classifier assumes having some labelled data, that could theoretically be paired, and thus making amenable variants of cycleGans that use a few paired samples. In this sense, I think that the paper fails in giving convincing arguments that advocate the use of OT here. As the dynamical formulation is known and already used to learn mappings (  see Trigila, G., & Tabak, E. G. (2016). Data‐driven optimal transport. Communications on Pure and Applied Mathematics, 69(4), 613-648. For instance). Also variants of OT that estimate a Monge mapping could have been included (e.g. V. Seguy, B. B. Damodaran, R. Flamary, N. Courty, A. Rolet, M. Blondel, Large-Scale Optimal Transport and Mapping Estimation, International Conference on Learning Representations (ICLR), 2018.)
As a summary:
Pros:
A nice interpretation of CycleGAN with OT
The paper is fairly well written
Cons: 
Overall the quantity of novelties is, in the eyes of the reviewer, somehow limited. At least the contributions should be clarified;
The experimental section is not convincing in explaining why the OT formulation is better than variants of cycleGAN or also other schemes for computing OT than the dynamical formulation
Minor remark:
A reference to Benamou, Brenier 2000 could have been given regarding section 3.2 and the dynamical formulation of OT.