Venue: CLeaR 2022 Conference

Reviews:

final_score: rating: 6: Marginally above acceptance threshold
review_content: Summary: The paper proposes to perform clustering before causal structure learning, so as to identify structurally homogenous subsets of samples. To perform such clustering, they propose a distance-covariance based kernel. The main contribution is the proposal of the so-called dependence contribution kernel, so that the resulting kernel space is isometric to the space of causal ancestral graphs.
Main review: 1. An earlier reference for the distance correlation is "Feuerverger, A. (1993), “A consistent test for bivariate dependence,” International Statistical Review/Revue Internationale de Statistique, 61, 419–433."

2. Why do you choose distance covariance in particular, rather than alternative nonparametric metrics of dependence, such as the mutual information, or the ball covariance?

3. Causal structure learning is notoriously different with a large number of covariates. Performing causal structure learning within subgroups identified by the clustering algorithm appears to reduce the sample size even further. There seems to be a tradeoff between homogeneity within each cluster, and the number of observations left for structure learning within each cluster. So my worry is that maybe the best number of clusters for the purpose of clustering, which only takes homogeneity into account, may not yield the best result in terms of structure learning. 

As a related comment, in Section 1.1, the authors assume there are enough samples for statistical inference. This does not sound realistic, especially for structure learning.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: Summary: This paper propose the dependence contribution kernel to do causal clustering. DEPCON could identify samples with different latent causal structures in heterogeneous pooled data so that actual learning task could get a more accurate result in each cluster. The proposed method could be used in many problems such as invariant learning.
Main review: reasons to reject
    There are some notion errors in this paper.(e.g section2.1,paragraph2)
    The reason(advantages) for choosing distance covariance could be explained clearly.(There are still some other methods to measure nonlinear dependence)

reasons to accept
    In general, the presented idea seems to be novel and interesting. Identifying heterogeneous causal structures from pooled data is important and could be used in many other applications.
    The motivation of the work is well explained and the pros and cons of different methods are clearly discussed.
    The proof is sufficient.
    The experiment setting is clear and the proposed method performs really good.
review_title: null

final_score: rating: 9: Top 15% of accepted papers, strong accept
review_content: Summary: This paper proposed a novel kernel method to measure similarities between two causal structures. The validity and theoretical properties of this kernel is established, and it's use in a causal clustering setting is demonstrated through a simulated and real world dataset.
Main review: Originality: Yes, this paper proposed a novel kernel based method for measuring the similarity between two causal structures. To the best of my knowledge this work is novel and hasn't been solved by existing methods. 

Significance: The problem of getting data generated from heterogenous causal structures arises in various settings such as genomics, economics or any other system that can be thought of as comprising various sub modules. And so, the results are of importance to anyone interested in utilizing causal discovery methods, regardless of domain. This paper proposed a new way of looking at the problem of causal clustering using kernels.

Technical quality: Yes, the proposed method seems technically sound. The claims are substantiated by both empirical and theoretical results. To the best of my understanding, the proofs in this paper are correct. 
Question for authors: For equivalence classes defined in the traditional causal discovery way, how does this kernel differentiate between two causal graphs within the same equivalence class (maybe I missed this in my reading of the paper)? 

Question for authors (I understand this may be very complicated one): Do you have an statistical guarantees for a method to choose the number of causal clusters?

Clarity: Yes - this paper is clear, well written and well explained. The main theme of the paper is conveyed effectively, as well as a intuition behind the method is presented. Great paper overall.
review_title: null

Meta-review: This paper proposes to use a distance covariance-based kernel to measure the similarity between two nonlinear independence structures for nonlinear clustering. They prove that the corresponding feature map is a statistically consistent estimation of nonlinear independence structure. They further showed that the kernel space is isometric to the causal ancestral graphs. The reviewers appreciate these theoretical results.

Their method classifies observations based on their nonlinear independence structures, and the nonlinear independence structures correspond to unconditional equivalence classes over ancestral graphs. Still, I wonder if it does not necessarily mean they classify observations on their causal structures. Each equivalence class will include different causal structures. If this is so, the title, causal clustering, might sound exaggerated. I want the authors to clarify this point. I also want the authors to clearly state the limitations raised by the reviewers.
Decision: Accept (Poster)