Venue: ICLR 2022 Conference

Reviews:

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: The authors frame the bivariate causal discovery problem in terms of the analysis of a dynamical-system. They use results from the field of optimal transport to interpret additive noise models from this framework. They also develop a novel criterion and a causal discovery algorithm based thereupon, and compare their results with previous methods used in bivariate causal discovery.
main_review: I believe that the authors' attempt of reformulating bivariate causal discovery problem is a novel and inspiring endeavor. Given that the problem of causal discovery is a conceptual problem as much as it is a statistical problem, the introduction of such new perspectives are certainly helpful for the discussion in this field. The authors' presentation is concise and clear, and the paper is easy to follow (modulo the additional comments I provide below).

One potential avenue for improvement is the motivation of the present reframing. What shortcoming of the information geometric, information theoretic, probabilistic, or more recently, topological analyses of the problem of causal discovery motivated the authors to present an optimal transport-based perspective? What additional advantages do the authors' formulation (or its potential sequels) provide to our understanding and methodology regarding causality? I find their work interesting regardless, but the preceding point is crucial for any work that formulates a novel mathematical framework causal discovery.

The authors limit their presentation to bivariate causal discovery problem. I do not think this is necessarily a crippling limitation, given the novelty of their theoretical perspective. However, the authors still need to provide an outlook for other cases as the authors do not seem to position their work as a purely conceptual development. If practical implications are to be considered, at least initial discussions of the multivariate case, as well as other standard challenges such as latent confounding, selection bias etc. should be provided.

Another potential development of the current work is to improve its presentation. If accepted, I encourage the authors to use the extra space conferred by the camera-ready version to provide a better introduction to the field of optimal transport, its prevalent use cases, as well as various concepts that are used in the field. The authors could also consider moving the related work section to the end of the introduction to further strengthen their initial setup.
summary_of_the_review: The current paper proposes a novel and well-presented formulation of the causal discovery problem, and thus is of interest to the conference audience. Improved comparison with existing frameworks and expanded discussion regarding practical use cases are recommended to improve the paper.
review_title: null

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: The paper considers the problem of establishing causal direction in a bivariate system under the assumptions of additive noise (possibly post-nonlinear) and no unobserved confounding. It establishes a connection between functional causal models (FCM) and optimal transport problems in dynamical systems to derive a new divergence metric that can be efficiently computed as a conditional variance estimate, and shows this quantity is zero iff in the causal direction..Efficacy of the metric is evaluated on synthetic data as well as the real-world Tuebingen data stand found to outperform existing measures.
main_review: Good and very interesting paper on a well-known but challenging problem. In particular the derivation of the new measure through the connection with dynamical systems is both intriguing and inspiring. 

Perhaps main drawback is that the information density in the paper is quite high and technical, and assumes familiarity of the reader with literature from other fields that may not be well-known to the wider ICLR audience, making it hard to understand certain parts of the paper in isolation. Even with a background in physics I had think back hard to remember details of vector field divergences and incompressible flow mechanics. That said, the ideas in the paper are well motivated and supported by rigorous theoretical derivations, making it sufficiently worthwhile for an interested reader to spend the extra effort.
I briefly entertained the hope the approach might also apply in the presence of confounding, but as with similar approaches it does still require causal sufficiency and additive noise components. Performance shows improvement on existing methods, though still in the somewhat disappointing range of around 75% accuracy for a collection of 50-50 decisions. But that just shows the fundamental difficulty of the problem.

In short: good paper on challenging subject, bit technical but clear accept.

Few remarks/questions:
p1, bottom: ‘.. of which the modern form was initiated by ..’, ‘.. by augmenting a time-dimension …’, ‘ … in the context of the Jacobian problem ..’ => these references are unlikely to be known by the majority of readers, and it would help to give just a minor hint of what these might entail 
p2,top, point 2. ‘pressureless flow’ => do you mean the more standard notion of ‘incompressible flow’? or else explain the difference
p2,mid: ‘.. from p_0(x) to p_T(x) …’ => technically the random variables do not need to obtain their value after a constant time T, e.g. different coin throws end up heads or tails, but in the actual underlying dynamical system different throws can take different times to come to rest. Does this impact the interpretation in terms of the map M or is the actual value of T essentially irrelevant? If so then explain or perhaps remove it form the description.
p3,bot: ‘The sense of causality is enforced .. FCMs’ => this sentence is overly vague / ambiguous: please clarify
p4, final sentence above 3.2 .’Therefore optimality .. problem.’ => the implication of this sentence is unclear
p5, Prop2: can the authors comment/speculate on whether in the presence of unobserved confounders, in almost all cases we would expect to find D(v) =/= 0 in both the causal and anti-causal direction ?
idem: ‘ .. and the identifiability conditions of ANMs (Hoyer,2008) …’ => these conditions are crucial and should be explicitly stated in the current paper
p7, Ext. to PNL: it is not entirely clear how this invertible function should be chosen based on a given data set
p7, ‘Minimisation’ , ‘Note .. is not necessary to be the true distribution or even significantly different from the true one’ => this sentence is unclear

summary_of_the_review: Good and inspiring paper on challenging subject, bit technical but clear accept.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: The paper tackles the problem of causal discovery in the basic case where a pair of variables is considered.
In particular, it is concerned with Functional Causal Models and how to establlish the causal direction for a pair of continuous variables.
It motivates the contibution by poiting out thet the performance of availabel algorithms is sensitive to the model assumptions, which
makes it difficult for practitioners to use them in real settings.
Then, the main contibution is a novel dynamical-system view of Functional Causal Models which aims to identify the causal direction in the case of pairs of variables. The paper exploits connection between functional causal models and optimal transport. Then, the paper studies the problem of optimal transport by taking into account the constraint originating from the considered functional causal model.
main_review: As strenghts of the proposed method I see that is is nicely grounded on theorethical results, the paper is well structured, organized and written, and I personally enjoyed reading it.

Weaknesses, as weaknesses are cocerned I see the following main weak points:
1) it is not clear what happens, or at least I was not able to understand what happens in the case when the considered pair of variables are not related at all. Indeed, te proposed algorithm does not take into account this possibility. Lines 6-9 of Algorithm 1, which furthermore ignores that all we have is a sample and that which is which should be at least addressesd, what about the quantity you use are the same or nearly the same?
2) the decision to remove outliers is in itself something that practically could also be understood but from the theoretical point of view opens some doubts about the effectiveness of the proposed apporach. Indeed, standardization and subsequent removal of observations acconding to the 2 standard deviation limits brings bias in my opinion
3) numerical results seem to be in line with state of the art algorithms, however it seems that the proposed approach suffers from greater uncertainty about accuracy when compared to other literature approaches.

I also found some minor issues with typos.
summary_of_the_review: The paper tackles an interesting and well studied problem under a general setting, i.e. assuming uncounfondedness.
It is well written and proposes a nice method to discover the direction of causal relationship when considering pair of variables in isolation.
I have expected some more discussion about what to do in the case where the terms used in Algorithm 1 were not too different or extremely close. Furthermore, the results reported in Table 1 witness that the proposed methods achieves accuracy values similar to state of the art algorithms, while bringing more uncertainty about the true value of accuracy than other methods. Furthermore, for ANM the proposed method seems to be inferior to other state of the art algorithms.
review_title: null

Meta-review: This paper is a solid contribution to researchers in this field, as it provides a new idea for the basic problem of determining the direction of causality between two variables, using the functional causal model as a dynamical system and optimal transport.
Decision: Accept (Spotlight)