Kati8/ICLR_2023_2025 huggingface dataset:
authors: Yingyu Lin
authors: Yuxing Huang
authors: Wenqin Liu
authors: Haoran Deng
authors: Ignavier Ng
authors: Kun Zhang
authors: Mingming Gong
authors: Yian Ma
authors: Biwei Huang
reviews: 
Comments: In this paper, the author proposed a novel causal learning algorithm for heteroscedastic symmetric noise models (HSNMs), where the noise term of a variable $X_i$ can be parameterized by $\sigma(pa(X_i))N_{X_i}$ with symmetric noise $N_{X_i}$. Specifically, the author showed that the skewness of the score function (SkewScore) in the $X_i$-coordinate is zero if and only if $X_i$ is a leaf node under certain assumptions. Therefore, the topological order can be identified by iteratively removing the variables with the least SkewScore. Conditional independence tests are then applied to determine the edges. Its validity for the HSNM with latent confounder is also discussed. Extensive experiments showed that the proposed method is effective on HSNMs and is robust to assumption violations.
Strengths: 1. The method is novel and can be potentially applied to causal discovery with latent confounders.
2. The theoretical justification of Assumption 1 seems sound, and the experiments showed that the proposed method works particularly well on sparse HSNMs.
3. The author extensively evaluated the method when the assumptions are violated, e.g. non-symmetric noise distribution, and latent confounder in the multivariate setting, and proved its effectiveness and robustness.
Weaknesses: 1. The author mainly used ER1 to evaluate the methods on HSNMs without latent confounders, which is relatively sparse.
2. I think the justification of Equation (6) was a bit missing. Proposition 5 justified that Assumption 1 is mild, and therefore Equation (5) in the bivariate setting holds. However, there is no such explanation for Equation (6). Can you provide some concrete examples where Equation (6) is reduced to some known results, or simplified by specific function and noise classes?
Recommendation: 6
Reviewer confidence: 3
Soundness/correctness: 3
Clarity: 4
reviews: 
Comments: ==Post-rebuttal update==

The authors' replies to my questions show their further insights into the problem and how the work could be deepened.

I also had a rough scan of other reviews and their respective replies, and I hold my original score.

This is an exceptionally well-written paper with valuable insights and high potential impact. Of course, no papers are perfect, but the current rating best describes my evaluation of the paper, i.e., it "should be highlighted at the conference".

==End update==

This paper proposes a method to identify causal directions in heteroscedastic symmetric noise models (HSNMs) using the skewness of the score function, which is the gradient of the log density. The skewness considered here is analogous to standard skewness but is distinct, offering computational advantages. Importantly, the paper proves that the score's skewness is zero only in the causal direction. This idea is later generalized to the multivariate case. The proposed method also shows robustness in the presence of hidden confounders, and a theoretical analysis is provided for an "additive" confounder in the bivariate case. The method demonstrates strong empirical performance.
Strengths: The paper provides a neat and effective solution to a critical problem. The connection between score skewness and causal direction is both theoretically grounded and practically effective.

The extensions to multivariate cases and the potential applicability under hidden confounding are valuable contributions.

The theoretical analysis appears rigorous (though I haven’t verified every detail).

The paper is clearly written, with helpful elements like the two properties of the score function, the motivation of score skewness by analogy to standard skewness, and well-designed figures.
Weaknesses: The generality of Assumption 1, which defines the applicable HSNMs, could be explored further, although the paper presents sufficient results for a conference submission.

The method's applicability under hidden confounding might be limited to special confounding structures.
Recommendation: 10
Reviewer confidence: 4
Soundness/correctness: 3
Clarity: 4
reviews: 
Comments: The paper presents a novel approach to uncovering causal relationships in data with heteroscedastic noise, which challenges traditional causal discovery models. The authors propose the use of a skewness-based criterion derived from the score of data distributions to identify causal directions, establishing that this measurement is zero in the causal direction but non-zero in the anti-causal direction.
Strengths: 1. The introduction of the SkewScore, which is leveraged to identify causal structures without requiring the extraction of exogenous noise.
2. A theoretical extension of the criterion to multivariate cases, along with empirical validations showing its robustness, particularly in scenarios involving latent confounders.
Weaknesses: See questions.
Recommendation: 5
Reviewer confidence: 2
Soundness/correctness: 3
Clarity: 3
reviews: 
Comments: This work proposes a causal discovery method for a class of restricted structural causal models with heteroscedastic symmetric noise. The identification of the causal graph is established using a score that quantifies the skewness of the score function. A causal discovery algorithm is then proposed based on the skewness score.
Strengths: The notation is well-organized and the paper is easy to follow overall.
Weaknesses: 1. Fact 2 is incorrect. Let X follow univariate exponential distribution with parameter $\lambda$, then $E[\frac{d \text{log} p(X)} { dx}] = -\lambda$. Its proof is also incorrect. 

2. The proof of Theorem 1 heavily relies on the assumption of symmetric noise. Without this symmetry, skewness no longer provides useful implications for the causal direction, as the proof of Theorem 1 breaks down. 

3. Definition 1 needs detailed discussions.  As stated, "this measure captures the asymmetry...," it would be helpful to demonstrate that it is zero when the distribution is symmetric.
Recommendation: 6
Reviewer confidence: 4
Soundness/correctness: 2
Clarity: 3
reviews: 
Comments: This paper provides a novel causal discovery algorithm that leverages skewness to determine causal direction when heteroscedastic noise (HN) is present.
Strengths: 1. An example in Section 5 provides a clear, practical illustration of the algorithm, connecting theoretical concepts with application. 

2. This paper provides a well-structured and self-contained summary of prior works in causal discovery algorithms based on functional causal models, allowing readers to clearly follow the evolution for learning causal graphs. As a reviewer, this summary enables me to track the advancements in the field and understand the recent progress. 

3. The core idea behind the proposed method is simple and intuitive, based on the asymmetry of score skewness to identify causal directions. This approach is conceptually accessible and computationally efficient solutions. 

4. The experiments provide strong empirical evidences of the paper’s contributions, with a wide range of scenarios that demonstrate the robustness of the proposed framework in heteroscedastic settings.
Weaknesses: 1. The paper provides correctness guarantees only for the two-variable case, where skewness in the score function can reliably distinguish causal directions. However, for settings with more than two variables, there is no formal guarantee of correctness for Algorithm 1. 

2. Current empirical results seem insufficient in showcasing its performance in high-dimensional scenarios. In other words, it remains unclear whether SkewScore scales effectively as the number of variables grows significantly. 

3. While the authors claim robustness in the presence of latent confounders, this robustness appears to be limited to simple bivariate cases. In multivariate settings, SkewScore can still produce incorrect causal graphs when latent confounders are present. For instance, in the causal structure $X_1 \rightarrow X_2 \rightarrow X_3$ with a latent confounder $L$ influencing both $X_1$ and $X_3$ ($X_1 \leftarrow L \rightarrow X_3$), the method would infer the structure $X_1 \rightarrow X_2 \rightarrow X_3$ along with a spurious direct link $X_1 \rightarrow X_3$. This incorrect edge arises because the latent confounder $L$ induces a conditional dependence between $X_1$ and $X_3$, which SkewScore interprets as a direct causal link. As a result, while the method can handle some latent confounding, it does not offer sufficient mechanisms to detect and mitigate confounding in more complex structures, limiting its application in practical scenarios where latent variables are common.
Recommendation: 6
Reviewer confidence: 4
Soundness/correctness: 4
Clarity: 3