From: ICLR.cc/2023/Conference/Paper4045/Authors
title: Submission Withdrawn by the Authors

From: ICLR.cc/2023/Conference/Paper4045/Authors
title: Response to reviewer cT2j
comment: Thank you for the comments. They help to make our work stronger and more complete.

### Presentation

* The new form of SEM in Eq. (8) introduces the new noise variables $\mathbf{N}_s$. Other, than that it is equivalent to the classical linear SEM. In particular, when $\mathbf{N}_s = \textbf{0}$: $$\mathbf{X} = \mathbf{XA} + \left(\mathbf{C + N_f}\right)\Leftrightarrow \mathbf{X} = \left(\mathbf{C + N_f}\right)\left(\mathbf{I + \overline{A}}\right)$$  So, in fact our reformulation doesn't try to model new settings that cannot be modeled by the linear SEM when $\mathbf{N}_s = \textbf{0}$. With the new formulation we try to give motivation for *a new way of interpreting the data generation process*. Defining the spectral components is equivalent to introducing some hidden variables (the cause of the data) which are mutually independent. So our objective is equivalent to trying to minimize the number of these hidden variables, as you correctly pointed out. The spectrum matrix only gives a connection with Fourier analysis. It is interesting that the sparsity assumption makes sense in our scenario as this is typically also the case for other types of Fourier domains.

* We left the empirical solution of the optimization problem (11) without explanation. Our implementation uses the optimization procedure of NOTEARS and substitutes the mean-square loss with the L1 norm of the approximated spectrum. We strongly believe that there exist optimization methods that could run more efficiently for our setting. We are currently considering this direction for the extension of our work.

### Empirical Analysis

* We provide the computation of the varsortability for the various experimental settings (see updated appendix). We notice that in general we have high varsortability. However, our measurements are in general lower than the measurements reported in ([1], Appendix G.1) for linear SEMs.  In some cases is around $5\%$ lower than those of linear SEMs, which is a sign that in our experimental setting it is not trivial to learn the DAG. This can also be deduced from the performance of sortnregress, which has decent performance but is not achieving the best results. It is also worth mentioning that the fixed sparsity support case seems to be the most interesting, since all methods fail in this case and varsortability is very low. Therefore, there is at least one scenario where our model successfully proposes a hard problem for DAG learning.

* We added comparison with GOLEM and sortnregress (see updated table in the paper). This doesn't affect our conclusions. GOLEM is in most cases more successful than NOTEARS, but overall slightly worse than Möbius. Regarding the trivial variance-sorting approach, called "sortnregress," we deduce that it is to some extent successful, but with worse performance. Of course this relates to the varsortability measure.

* Please see general comments regarding the disagreement with the Sachs et al. dataset.

### Minor comments
Thanks for the minor comments, we fixed them.

[1] Reisach, Alexander, Christof Seiler, and Sebastian Weichwald. "Beware of the simulated dag! causal discovery benchmarks may be easy to game." Advances in Neural Information Processing Systems 34 (2021): 27772-27784.

From: ICLR.cc/2023/Conference/Paper4045/Authors
title: Response to reviewer R97q
comment: Thank you for pointing out the strengths and the novelty of our work. Your comments regarding the theoretical aspects of this work are very interesting and make our work more complete.

* Our data generating process reformulates the linear SEM: $$\mathbf{X} = \mathbf{XA} + \left(\mathbf{C + N_f}\right)\Leftrightarrow \mathbf{X} = \left(\mathbf{C + N_f}\right)\left(\mathbf{I + \overline{A}}\right)$$
The way we generate the independent causes is as follows. Each data point $c$ in the spectrum is set to $0$ with probability $(1-p)$, otherwise set uniformly at random to a value $[0,1]$. Then at each such point we add the corresponding noise  $n_f$. Consequently, each node the variable $c+n_f$ has the same variance. Therefore, this case is contained in the scenario of equal noise variances for linear SEMs and is thus identifiable. The fact that our method achieves very high accuracy on discovering the true DAG, empirically confirms this.

* The scenario where we have fixed sparsity support is not generated with equal variances and thus cannot be justified to be identifiable in the same way. We are currently working towards analyzing the theory for the identifiability of our model. As a preliminary result we may say that when the support of the sparsity is concentrated on the sink nodes then our model is identifiable due to the conditional variance criterion [1].

* We multiply by the inverse of $\left(\mathbf{C + N_f}\right)^T\left(\mathbf{C + N_f}\right)$ which exists since $\mathbf{C + N_f}$ is assumed to have full rank. Therefore, indeed there is no assumption on the variances of the noise. Intuitively with continuous noise distribution we will very likely get full rank. This lemma works as deterministic identifiability criterion that eliminates the case of two different DAGs being able to generate the same data through the same causes.

* The $L^1$ norm has been widely used as a regularizer for promoting sparsity. Here, we use it as the main term in our optimization to promote sparsity in the approximated spectrum, such that assumption (9) holds. Empirically this seems to work very well in our experiments. We are currently analyzing why the global minimizer of the objective gives the true DAG.

* We didn't consider interventions/intepretability of our model, as we mainly focused on DAG learning rather than the more general problem of learning causality. However, this seems as an interesting extension for our work which we may consider in the future.

* Please see the general comments about sparsity in the spectrum.


[1] Park, Gunwoong. "Identifiability of Additive Noise Models Using Conditional Variances." J. Mach. Learn. Res. 21.75 (2020): 1-34.

From: ICLR.cc/2023/Conference/Paper4045/Authors
title: Response to reviewer uHH8
comment: Thank you for pointing out our effort for writing our paper in a clear way. Regarding the weaknesses:

* The theoretical results may indeed come from trivial linear algebra manipulations. However the interpretation of the resulting equation $\mathbf{X} = \left(\mathbf{C}+\mathbf{N}_f\right)\left(\mathbf{I} + \overline{\mathbf{A}}\right) + \mathbf{N}_s$ is novel and non-trivial. We support it by giving a concrete example (river network) of a causal setting where in the spectral values only a few causes appear.

* We added an additional experiment with scale-free (Albert-Barabasi) graphs. We get similar conclusions also in that setting.

* We chose default hyperparameters for the baseline methods. The reason is that none of the previous methods has hyperparameters relevant to the sparsity in the spectrum.

* We also notice a possible confusion regarding the sparsity of the graph (edges/vertices ratio) and the sparsity of the spectrum. In our experiment we mainly change the sparsity in the spectrum and only in one experiment we deal with the graph sparsity (average degree $d=3$ instead of default $2$). In that case we modified the NOTEARS threshold so that it proposes more edges. For an $L^1$-norm regularizer equal to $10^{-3}$ instead of $5\cdot10^{-3}$ we get no significant improvement in TPR (from $0.89$ to $0.90$) and the SHD gets worse, from $11.3$ to $17.7$. Thus, there is no significant change in our experiment.

* The default setting is the ideal experimental scenario to promote sparsity in the spectrum. In other scenarios we try to approximately violate this assumption by altering any parameter. If the sparsity disappears then our solution has no meaning anymore. For this reason high noise deviation is not desired as it will violate the sparsity. We could have higher noise deviation if the spectrum was assumed to be in a range larger than $[0,1]$.

* There can also be a possible confusion here regarding the std of our noise and the std of the noise in the linear SEM model. For example consider our equation $$\mathbf{X} = \left(\mathbf{C}+\mathbf{N}_f\right)\left(\mathbf{I} + \overline{\mathbf{A}}\right)$$
Each data point $c$ in the spectrum is set to $0$ with probability $(1-p)$, otherwise it is set uniformly at random to a value $[0,1]$. Then at each such point we add the corresponding noise $n_f$. So this means the deviation of $c+n_f$ is larger than the noise $n_f$ itself (which has std at most $0.1$).
If we wanted to compare this to the std computed for linear SEMs we would have to write $$\mathbf{X} = \mathbf{X}\mathbf{A} + \mathbf{C}+\mathbf{N}_f$$
so the standard deviation of $c+n_f$ is exactly the std we would have for the noise in the linear SEM scenario.

* We added a comparison with GOLEM. This doesn't affect our conclusions. GOLEM is in most cases more successful than NOTEARS, but overall still slightly worse than Möbius.

* Please see our general comments regarding the implementation details for solving problem (11).

* We added the table with the SHD in the appendix.

From: ICLR.cc/2023/Conference/Paper4045/Authors
title: Response to reviewer o8xF
comment: In the paper we made an effort to make the content of this work understandable for a large audience. We give an additional explanation of our contributions:

* We express linear SEMs in a new form (Eq. 8) via the causal Fourier transform. This introduces the new variables of the spectrum that give a new interpretation to the data generating process. In the river network example the (spectrum) causes are the pollution from each node, whereas the observation is the accumulated pollution from upstream nodes.

* We illustrate that Fourier-sparsity has a meaning for our example as it makes sense to assume that only a few nodes cause pollution at each time.

* Connecting these two fields of DAG Learning and causal Fourier analysis is novel and gives new insights to generate other types of synthetic DAG datasets.

* We furthermore propose a solution to the stated problem which improves over the performance of current state-of-the-art methods for DAG learning from linear SEMs.

Our paper is different from (Seifert et al. 2022b) as our goal is to solve the problem of learning DAGs. In particular, we only leverage the theory presented in (Seifert et al. 2022b) to address the DAG learning problem from a novel angle.

From: ICLR.cc/2023/Conference/Paper4045/Authors
title: General comment
comment: We thank all the reviewers for their reviewing effort and the very useful comments on how our paper can be improved. We revised it based on your comments to improve it.

We want to emphasize the following changes.

**Theoretical Analysis**

* A discussion regarding identifiability and varsortability was added in the supplementary material.

**Motivating sparsity in the spectrum**

* We strongly believe that it makes sense to assume that the data are generated only by a few causes (in our model). This belief is based on realistic scenarios like the river network explained in the text, the Fourier-sparse infection signal in [1] and furthermore it is possible to interpret the educational model in [2] (from the NeurIPS 2022 competition) as a signal generated by a few causes.


**Experiments**

* We added experiments with scale-free graphs in addition to Erdös-Renyi.

* We added comparison to GOLEM and sortnregress. This doesn't affect our conclusions.

* We computed mean varsortability for all different experimental scenarios. This clarifies that our generated datasets are not trivial and have slightly less varsortability than classical SEMs. In particular we have much less varsortability in the case of fixed sparsity support. 

* We fixed the experiment with the Sachs dataset (see updated table in the paper). Now the results are much closer to the ones reported in the corresponding papers. The disagreement was due to the use of [Causal Discovery Toolbox](https://fentechsolutions.github.io/CausalDiscoveryToolbox/html/tutorial_1.html?highlight=sachs), which turns out to have a different ground truth matrix with $18$ edges instead of the conventional one with $17$ edges. Furthermore, people didn't always used the entire dataset to predict the underlying DAG, but they might use only the first $853$ data points instead of the full $7466$. So this time we only used $853$ data points as well.  In particular NOTEARS and DAG-GNN use $7466$ samples and the DAG has $20$ edges, GOLEM and [3] use $853$ data points and the ground truth has $17$ edges. In our new implementation we use the ground truth DAG with $17$ edges and the first $853$ data points from the dataset. We take our data from the supplementary material of the original paper from (Sachs et al. 2005). In any case we did not want to just copy the reported results from the reference of each baseline, as it is always better to have every method executed in the same environment. Of course, there are still slight differences in the reported performances which could be due to the different choice of hyperparameters. In fact, we could not find any public implementation for evaluation on the Sachs et al. dataset. In any case, this experiment is only used as a proof-of-concept that our method could be useful for real datasets, too.

* We left the empirical solution of the optimization problem (11) without explanation, because it was not the main point of our paper. Our implementation uses the optimization procedure of NOTEARS and substitutes the mean-square loss with the L1 norm of the approximated spectrum. It is possible that there exist optimization methods that could run more efficiently for our setting. We are currently considering this direction for extending our work.

[1] Bastian Seifert, Chris Wendler, and Markus Püschel. Causal Fourier Analysis on Directed Acyclic Graphs and Posets
. arXiv, 2022b.

[2] Gong, Wenbo, et al. "Instructions and Guide: Causal Insights for Learning Paths in Education." arXiv preprint arXiv:2208.12610 (2022).

[3] Reisach, Alexander, Christof Seiler, and Sebastian Weichwald. "Beware of the simulated dag! Causal Discovery benchmarks may be easy to game." Advances in Neural Information Processing Systems 34 (2021): 27772-27784.

From: ICLR.cc/2023/Conference/Paper4045/Reviewer_o8xF
confidence: 2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: This paper studied the problem of learning directed acyclic graphs (DAGs) from data. The authors proposed to use L1-norm minimization on the approximated spectrum to learn DAGs from Fourier-sparse data. The numerical experiments demonstrate the effectiveness of the proposed method.


strength_and_weaknesses: Strength:

The paper presents a new perspective for learning DAGs based on causal Fourier analysis. The authors further proposed to minimize the L1-norm of the spectrum, which improves the performance of graph learning as shown in experiments.

Weakness:

The paper is not well-written. The authors should make clear the contributions of this paper, and the difference with the reference (Seifert et al., 2022b). Compared to the prior work, the contributions and novelty of this paper seem not enough to be accepted for publication.


clarity,_quality,_novelty_and_reproducibility: The paper is not clearly written. The authors should make clear the contributions and novelty of this paper compared to the prior work. The practical significance is unknown.
summary_of_the_review: This paper considered the problem of learning directed acyclic graphs (DAGs) from data. Overall the paper does not read well. The authors should make clear the contributions and novelty of this paper compared to the prior work. This paper basically builds upon the prior work (Seifert et al., 2022b), and the novelty and technical contributions of this paper are not clarified well.
correctness: 3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.
technical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
empirical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
flag_for_ethics_review: NO.
recommendation: 5: marginally below the acceptance threshold

From: ICLR.cc/2023/Conference/Paper4045/Reviewer_uHH8
confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
summary_of_the_paper: The authors study the problem of learning DAGs from observational data. An assumption based on Fourier-sparsity is introduced for linear SEMs for which an L1 penalty of the loss is proposed to solve the sparse problem. Some experiments are provided to evaluate the performance of the proposed method.
strength_and_weaknesses: It has been difficult to find the strengths of this work. The technical contributions are poor and the empirical evaluations are not extensive at all. If something, the paper presents the core ideas in a clear way so that's a plus.

On the other hand, I could find many weaknesses in the paper. 
* Regarding the technical results, they are either trivial or known results. 

* Regarding the experiments, there is very little to no detail on the hyperparameters of the baselines, and only Erdos-Renyi graphs were considered. I emphasize the hyperparameters in this case because this is a different data generation process than prior work. To give a concrete example, did the authors care to change the l1 coefficient of NOTEARS for the very sparse regimes? The default setting is also strange, to say the least, the variances are very small which is likely unrealistic to assume in practice. Moreover, the weight bounds are also small, I wonder if the authors also cared to play with the weight threshold of NOTEARS for these different regimes since it has been recently brought to attention that the existing acyclicity functions can suffer from numerical problems for small weight values (see [1]). Also, the authors should consider testing the GOLEM algorithm from [2] which tends to work better than NOTEARS for linear SEMs.

* There is no explanation at all as to how the optimization problem was solved. (1) Did the authors use the augmented Lagrangian or quadratic penalty method or anything else? (2) Did the authors use a first-order sub-gradient method to deal with the L1 norm on the loss or perhaps a quasi-Newtown method with box constraints? One can only guess, but this lack of detail for an ICLR paper is rather disappointing.

* Table 1 should report SHD and not simply the TPR. 

[1] Bello et al. (2022), DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization.
[2] Ng et al. (2020), On the Role of Sparsity and DAG Constraints for Learning Linear DAGs.
clarity,_quality,_novelty_and_reproducibility: The paper is mostly clear. I would suggest perhaps cutting some text that is common knowledge to save space and perhaps adding more discussion on the sparsity assumption, optimization procedure, and a clear explanation of the benefits of this type of modeling.
As per quality and novelty, I believe the work does not meet the bar for an ICLR paper in its current form. 
summary_of_the_review: The main points of my review are in the above sections.
correctness: 4: All of the claims and statements are well-supported and correct.
technical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
empirical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
flag_for_ethics_review: NO.
recommendation: 3: reject, not good enough

From: ICLR.cc/2023/Conference/Paper4045/Reviewer_R97q
confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: This paper presents a connection between DAG learning and causal Fourier analysis. The authors show that linear SEMs can be characterized in the frequency domain, and further propose a data generating procedure that is a generalized version of the linear SEMs based on Fourier sparsity. The authors further provide a continuous DAG learning method based on Fourier sparsity, and validate it through experiment results.
strength_and_weaknesses: Strength
1. The connection between DAG learning and Fourier analysis/sparsity is interesting and, to my knowledge, novel, which may bring further insights to the DAG learning community.
2. The proposed method achieves promising empirical results.

Weakness
1. The formulation of NOTEARS with least squares loss is appropriate for linear SEM with equal error variances that is shown to be identifiable (Loh and Buhlmann, 2014). That is, the global minimizer corresponds to the true DAG under some assumptions. However, it does not seem straightforward to me whether the new data generating process considered in Eq. (8) is identifiable, and whether global minimizer of Eq. (11) corresponds to the true DAG of the new data generating process. Therefore, it would make the paper stronger if the authors could provide more analysis/discussion on it.
2. In Lemma 3, how do we go from the second equality to the third one? This result, if true, may be surprising because it does not involve any restriction on the variances of the noises. I.e., it is generally known that linear SEM where the noises have unequal variances is not identifiable, and further restriction on the noise variances is needed. Not to mention Eq. (8) is more general than the linear SEM.
3. Section 4 mentions that the first L1 term in Eq. (11) is sparse when Assumption (9) holds and $N_S(I+\bar{A})^{-1}$ has entries of low magnitude. It would be more convincing if the authors could formally show that minimizing the first L1 term (or even with L0 norm) in Eq. (11) will encourage "Fourier sparsity", which is the key motivation of the proposed method.
4. A key benefit of causal graphical model/SEM is that it provides a framework to reason about interventions/counterfactuals, and is also highly interpretable. However, this is not the case for the data generating process proposed in Eq. (8). Can the authors provide more explanation on it?
5. The key motivation of using Eq. (8) and fourier sparsity may seem unclear. Apart from the river network example in Section 3, can the authors provide more real-world examples?
clarity,_quality,_novelty_and_reproducibility: This is a well written paper and the formulation is novel. The experiments should be reproducible because the code is included as supplementary material. 
summary_of_the_review: It is unclear whether the proposed data generating process is identifiable, and whether the global minimizer of the proposed formulation corresponds to the true DAG.
correctness: 2: Several of the paper’s claims are incorrect or not well-supported.
technical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
empirical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
flag_for_ethics_review: NO.
recommendation: 5: marginally below the acceptance threshold

From: ICLR.cc/2023/Conference/Paper4045/Reviewer_cT2j
confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
summary_of_the_paper: The paper leverages the recent work of [Seifert et al, 2022ab] and proposes a new way of modelling/generating data from a DAG, which boils down to extending the standard linear Structural Equation Models so that not only the roots of the graph can inject new mass in the flow (can be "causes" using the paper's terminology). More formally, the standard linear SEMs consider that the spectrum of the causes of the graph is non-null only on its roots, while the proposed extension allows it to be non-null on intermediate nodes as well.
This new modelling gives rise to a new loss function for learning the DAG's adjacency matrix from data, which fosters solutions with small numbers of causes. 

The resulting method is then empirically compared with methods of the literature that handles data generated from linear SEMs, on two settings: on data generated from the new Fourier-sparse SEM and one standard real dataset whose underlying SEM is unknown.
strength_and_weaknesses: ## Strengths
The paper is generally well-written: the contributions are clearly stated, an example is provided to guide the reader in understanding the novelty of the formulation, the derivations are reported in great level of detail. To the best of my knowledge the proposed loss for learning DAGs is novel.

## Weaknesses
### Presentation
1. The paper does not provide a clear motivation for the new structural equation model. The example of Section 2.2 helps the reader understand the different components of the model. However from the paper, it is not clear which settings can be modelled by the new SEM that could not be modelled by the standard linear SEM. Couldn't the spectrum be integrated in the noise variables $N$ or modelled as additional (and potentially hidden) variables $X$? What is the interest of introducing a spectrum matrix? Interpreting instead the intermediate causes as hidden variables might actually support the use of the proposed sparsity-inducing loss in the partially observable settings, where we want to model the data supposing that we have as few hidden variables as possible.

2. It is never mentioned how Problem 11 is solved in practice, apart from saying that it leverages continuous optimization. In particular, how is the acyclicity constraint enforced in practice? Is it relaxed to a penalty term? 

### Empirical analysis
1. The empirical evaluation of the method should be extended to include a study of the triviality of the generated problems and to include two essential baselines. 

- The cited work of [Reisach et al. 2021] showed that most synthetic datasets used in the literature, and in particular those generated by linear SEMs, can be trivially solved by ordering the variables by increasing marginal variance (and then applying standard sparse linear regression), a property that they named "varsortability". It would be insightful to study the varsortability of the datasets used in Section 6.1. Firstly this would allow to validate the empirical evaluation, as it would not be insightful if carried out on non-trivial problems. Secondly this would allow to check whether the generalized SEM is able to generate non-trivial problems, and in that case, it would add value to the paper as the new SEM could be used as testbed for future work. The community is currently lacking synthetic non-trivial datasets, and contributing in this direction would be highly significant.

- The simple baseline proposed in the same paper [Reisach et al. 2021] should be included in the comparison to check that the new method performs at least better than that.
The second missing baseline is Golem, the cited work of Ng et al., which is the state-of-the-art on data generated by a linear SEM and in practice performs better than NoTEARS (which is the second best method in Figure 3 after Möbius).

2. The results on Sachs reported in Table 2 are weirdly worse than usually reported for the selected baselines (see for instance the cited [Yu et al. 2021], [3]). In the paper no difference in the experimental setup is highlighted that would justify this discrepancy.

### Minors
- page 1: The end of the first paragraph suggests that by making assumptions on the data generation process the problem of learning the DAG from data is solvable. This phrasing is not correct as even when making assumptions on the underlying SEM it is not always feasible to reconstruct the DAG, because (i) the search space is too large or (ii) the data distribution can be modelled by multiple DAGs.

- page 1, "learning a DAG from data is still very challenging problem" - > "learning a DAG from data is still a very challenging problem"

- page 2, "which could be length of shortest path" - > "which could be the shortest path's length"

- page 3, "a pollutant at i reaches j" - > "a pollutant at i that reaches j"

- Equation 9, "$||C||_0$ is small" should be written formally

- page 6, "criterions" - > "criteria"

- page 6, "the mean-squared loss is the optimal loss function". What does it mean optimal in this context? The mean-squared loss can be obtained by maximizing the log likelihood and making assumptions about the noise distribution, but other formulations could be considered instead, such as the one derived in Golem [Ng et al. 2020].

- page 6, "applicable to DAGs since the the"

- page 6, "with the computational advance of deep learning" - > "advances"

- page 6, Permutation-based methods date way back the two cited works. See for instance [1], [2].

[1] Marc Teyssier, Daphne Koller: Ordering-Based Search: A Simple and Effective Algorithm for Learning Bayesian Networks. UAI 2005
[2] Nir Friedman, Daphne Koller: Being Bayesian About Network Structure. A Bayesian Approach to Structure Discovery in Bayesian Networks. Mach. Learn. 2003
[3] Paul Rolland, Volkan Cevher, Matthäus Kleindessner, Chris Russell, Dominik Janzing, Bernhard Schölkopf, Francesco Locatello:
Score Matching Enables Causal Discovery of Nonlinear Additive Noise Models. ICML 2022
clarity,_quality,_novelty_and_reproducibility: **clarity** The paper is generally very clear, but lacks a convincing motivation for modelling the data generation process with the new SEM and does not describe how the proposed problem is solved in practice (see Weaknesses). 

**quality** The formulation is derived in great level of detail, but the empirical analysis is not sufficient to validate the method and support the conclusions (see Weaknesses). Few statements need to be rephrased and some seminal works should be cited (see Minors).

**novelty** I am not very familiar with the work of [Seifert et al, 2022ab] on which this paper is based. As far as I understand, the new SEM was proposed before but the sparsity-inducing loss is novel.

**reproducibility** The implementation was submitted as supplementary material. The paper does not mention how Problem 11 is solved in practice (can be easily fixed) and reports results on Sachs that are very different from those reported in the literature.
summary_of_the_review: Overall the paper is well written and clear, but it is hard to judge the significance of the new formulation and loss in the context of DAG learning, and the significance of the empirical results.

1. The paper should provide a convincing motivation for using the generalized linear SEM, e.g. by describing settings that cannot be modelled by adding additional variables to the standard linear SEM or by changing the noise model. This point can be easily addressed in a revision.

2. The empirical analysis cannot be validated as it misses necessary baselines and a study of the triviality of the new setting. Also the surprising results reported for known baselines on Sachs hint to possible issues in the implementation or experimental setup. This point will be hard to address in a revision.

correctness: 2: Several of the paper’s claims are incorrect or not well-supported.
technical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
empirical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
flag_for_ethics_review: NO.
recommendation: 3: reject, not good enough