Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: The paper introduces kernel neural optimal transport. Prior work on neural optimal transport makes use of a weak cost. The paper proves that weak costs admit non-solutions in the sense that the resulting plans do not satisfy the marginal constraints. The authors also propose that the weak costs are related to optimization instability. As a solution, weak kernel costs are introduced. Extensive testing demonstrates the effectiveness of weak kernel costs including via unpaired image to image translation tasks. 
strength_and_weaknesses: Strengths: 
- Neural OT and related methods are of increasing interest across the community. 
- The paper is generally well written. The math is clear and appropriate. Proofs are thorough, complete, and in my checking correct.
- The theoretical results are accompanied by detailed simulations. 

Weaknesses:
- The claimed limitations of weak costs were not detailed as comprehensively as I would have liked. Specifically, how far off are the "fake" plans? Also, how bad is the instability? These seem like important points, given that they motivate the paper in the first place. 
- I found the exposition difficult. The introduction is quite short, has jargon that is not explained, and hence doesn't completely set up the problem. After reading the paper through, I appreciate the contribution but it think it would help to try to get more of the details earlier in the paper. 
- There are a large number of figures with image to image translation. I don't see what the unique contribution is for each. It seems like one would suffice in the main text. 

Other detailed comments
- I didn't really understand what figure 1 was supposed to show. How would I know if the result is "good"? 
- Please define strong and weak transport costs in the introduction. What is the significance of the difference aside from the fact that people use them? 
- Please define fake solutions. It would be helpful to be specific here so readers understand the gravity of the situation. 
- Please define a weak kernel cost when one uses it the first time. 
- It is not clear why imagine to image translation is a good way of testing performance. 
- "may be not an optimal stochastic map. In this paper, we show that for the γ-weak quadratic cost (3) this may be problematic: the arg infT sets might contain fake
solutions T" I don't quite understand whether this is the same point, i.e. it is problematic because it is not an optimal stochastic map, or whether there are other aspects that are problematic. If it is problematic because it is not an optimal stochastic map, and the fake solutions are of this type, it would help to use simpler language (fewer different names) and adopt a label more descriptive than "fake solutions". 
- I appreciated the inclusion of a derivation of (9) in the appendix. 
- "Should we really care about solutions being fake?" It would be great for the answer to this question to appear early in the paper. 
- It would be nice to have some sense of how bad fake solutions are. Are these truly pathological? (In the sense of being far from Q?) Also, can we demonstrate the stability issues more concretely? 
- I don't see the added value of having the example figures in the paper (beyond the first one). 
clarity,_quality,_novelty_and_reproducibility: The paper is generally clear, of high quality, and novel. 
summary_of_the_review: The paper introduces kernel NOT, which uses weak kernel costs, to address issues in earlier neural OT work. There are both theoretical and extensive simulation results. The simulations do not document the limitations of and solution to the prior work as clearly as I would have liked. Overall, this is a nice contribution.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: This paper proposes an extended method for neural optimal transport.
The authors first point out that the theoretical flaws of neural optimal transport with using the weak quadratic cost in terms of potentially learning face plans.
Then, they proposed a method of using kernel costs of neural optimal transport to overcome learning fake plans.
This paper also show the theoretical justification that the proposed method can solve the lack of the previous method.
The experiments are conducted on the image-to-image translation task, and demonstrate that the proposed method produces better results.

strength_and_weaknesses: Strength:
* The proposed method seems reasonable, and motivation is clear.
* This paper provides the theoretical justification of superiority of the proposed method.
* The figures help to intuitively understand the methods.


Weaknesses:
* The most of the experimental results shown in this paper are qualitative comparisons with small number of evaluation data. Therefore, it may exist the doubt of cherry picking.
* Similarly, it is hard to say that the proposed method really provides the better performance by solving the fake plan issue from the results shown in this paper. Therefore, I feel that the experimental results do not much support the main claim of this paper.

clarity,_quality,_novelty_and_reproducibility: I am not an expert in neural optimal transport. 
My correctness checks for all equations and notations are limited, and I may have missed something that needs to be corrected.
Therefore, I do not discuss this point in my review.

The motivation and problem that this paper address is clear and clearly understandable.
Moreover, to my knowledge, the proposed method seems novel and provides new findings to the community.

This paper promises to release all the source code and checkpoints to reproduce the experiments in this paper. 
Therefore, reproducibility should be no problem.
summary_of_the_review: I think this paper (potentially) has sufficient contributions to be accepted to the conference.
However, as I wrote above, I am not a specialist in neural optimal transport, so that it is hard for me to check the correctness of the theoretical justification of this paper.
My recommendation is based on the fact that this paper shows the correct justification.
review_title: null

final_score: rating: 5: marginally below the acceptance threshold
review_content: summary_of_the_paper: This paper introduced kernel weak quadratic costs to solve the problem that NOT with the weak quadratic cost might learn fake plans which are not optimal. 
And it conducts the theoretical and empirical analysis of the saddle point optimization problem of NOT algorithm for the weak quadratic cost. 
Finally, they analyze from theory and examples to show that NOT generates fake solutions which do not provide an OT plan.
strength_and_weaknesses: Strengths: This paper conducts the theoretical and empirical analysis of the saddle point optimization problem of NOT algorithm for the weak quadratic cost, this show it may appear fake solutions. 
This paper introduces kernel quadratic costs to solve above problem, and it proves the feasibility of the proposed algorithm via theoretical analysis and examples.

Weaknesses: The cost function with kernel is too complex in practice. Comparison with other neural network-based algorithms for solving OT mappings is lacking. 
This paper does not explain the advantages of the proposed algorithm compared with the classical algorithm, and there is no experimental support.
clarity,_quality,_novelty_and_reproducibility: The overall structure of this paper is clear, the shortcomings of NOT are analyzed theoretically,  and the KNOT algorithm is proposed, but the experimental comparison of the two algorithms is not given in the experiment, which lacks the support of the experiment. 
Only an improvement of kernel cost function is given, which lacks some novelty. Can not be reproduced.
summary_of_the_review: In this paper, a new neural optimal transport algorithm based on kernel cost function is proposed, and the deficiency of NOT is analyzed theoretically. 
However, the lack of experimental support, the key is the lack of experimental comparison of NOT, does not reflect the advantages of improving the cost function. 
Moreover, there is a lack of comparison of classical neural OT algorithms, such as ICNN.
review_title: null

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: Authors propose a novel technique to compute weak optimal transport plan between two distributions, given in possibly two different spaces. Based on an earlier work of Gozlan et al, the computation of OT plan is formulated as a certain minimax task. The major issue that the paper addresses is that some saddle points of the latter task can correspond to fake plans. Existence of fake plans is proven theoretically and demonstrated on toy experiments.
strength_and_weaknesses: First, it looks like existence of fake plans is a result of some kind of degeneracy in the problem formulation (like the multicollinearity problem in linear regression). Can it be tackled by some form of regularization? Interesting question was asked on page 6: "Should we really care about solutions being fake?" and noted that "the map T may fluctuate between fake solutions rather than converge." The question is: does it fluctuate between fake solutions, or, in fact, it fluctuates as some convex combination of stochastic OT plans? The Proposition 2 hints that it could be the second case. 
Another issue is that existence of fake plans are demonstrated on toy experiments. A natural question is: are fake plans common when the technique is applied to real-world data, e.g. to unpaired image-to-image translation task?

Minor comments:
Page 1: (a) Celeba (female) → shoes  ->   (a) Celeba (female) → anime  
Page 2:  fomulations ->  formulations  
Page 8: under perform -> underperform
clarity,_quality,_novelty_and_reproducibility: Exposition is clear and easy to read. To my knowledge, it is also original.
summary_of_the_review: Overall, a quantity and a quality of experiments convinces me in correctness of major points of the paper. Though, for me it is hard to judge experiments with unpaired image-to-image translation task. Some discussions of  unpaired image-to-image translation task, or a citation where one can learn details of that problem is needed.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: In this paper, the authors propose a kernel version of neural optimal transport method. The authors provide the theoretical and empirical analysis of the saddle point optimization problem of the neural OT algorithm for the weak quadratic cost. Overall the authors proposed an interesting OT method with theoretical guarantee, and many of reviewers puts positive reviews. Thus, I also vote for an acceptance.
justification_for_why_not_higher_score: This paper is an interesting formulation of OT problems. However, the contribution of this paper is relatively focused to small number of researchers. Thus, presenting it as a poster is reasonable.
justification_for_why_not_lower_score: This paper can be potentially rejected since many reviewers are not that excited about the result. However, the proposed method is solid and it is good to have the paper at the conference if there is enough space to present.
Decision: Accept: poster