The authors focus on the challenge of causal discovery in the presence of unmeasured confounders. This is an important topic within the causal inference literature (and in fact many causal discovery algorithms often assume *no* unobserved confounders, which may often be unrealistic). Whilst some methods have been proposed, they often rely on assumptions such as pure 1-factor models with at least children per latent variable. The authors propose a two-stage method where they first use theorem 2 to find clusters based on whether triplets of variables satisfy a triad constraint. One important computational/algorithmic benefit of this first stage is that only independence testing (as opposed to conditional independence) testing is required. Given the clusters, the authors then focus on recovering the causal ordering over latent variables. The experiments are well executed. My only concern is the absence of traditional methods such as LiNGAM (even though it is misspecified). I would also have liked to see the performance of a very naive method which replaced the cluster finding using the triad constraints with simple clustering methods (eg k means clustering). This would help highlight which stage of the proposed method was really doing the heavy lifting (I would guess it is the first stage). Overall the paper is original and clearly written. There are some minor concerns regarding the experiments (some basic/misspecified baselines as discussed above would have been helpful). # minor comments/typos: - just before section 3: "equation 1)"

originality: the idea is very interesting, even though with heavy assumptions. Authors did explain the impact of each assumption, but it is still a very limited setting. quality: 1. the technical results are sound, but authors should state full assumptions for each theoretical results (such as Proposition 1). 2. One can view the work is closely related to some hierarchical tree/latent tree learning algorithm. It seems that the major different the latent variables can have arbitrary relationships. Author should explain in more details that how does the proposed algorithm compare with many latent tree algorithms? In Experiments, authors should also compare with these algorithms. 3. the consistency result of the algorithm is missing: is it sound or complete? 4. Does the method find an equivalent class of the graphs or the true graph? 5. what is a reason to choose noise term so small, with fifth power? It seems the algorithm could suffer from high noise? Clarity: the paper is well written, although one would wished that the authors should rely less on the supplementary materials and provide more intuition/explanation on proofs of the theorems. The examples are good. Significance: the idea is worth to pursue further and will have potential big impact. ===== I have read the authors' response. It would be interesting to see how the latent tree methods perform on the real dataset, since Figure 5 is basically a latent tree.

Update after author response: - Based on author response and some reflection on the problem itself (see below), I have increased my score to a 7. - Latent variable modeling is a challenging problem and any insights into additional constraints beyond standard conditional independence are valuable. The triad constraints mentioned here, while limited in scope, due to the parametric and structural assumptions posed in this paper, may be an interesting gateway to more generalized constraints in much the same way that tetrad constraints motivated this paper. - The notion of pseudo-residuals is also a concept worthy of further investigation, given its history in providing breakthroughs in other aspects of graphical modeling such as the Residual Iterative Conditional Fitting algorithm proposed in https://www.stat.washington.edu/~md5/Papers/2004uai.pdf. - If the paper gets accepted, I would ask that the authors change some of the language in the paper. Hyperbole such as "This goes far beyond the Tetrad constraints" can be off-putting and while Triad constraints are an improvement over Tetrad constraints, I am not sure they go "far beyond", or it should be left to the reader to decide if they do. ++++++++++++++++++++++++++++++++++++++++++++++ - The literature review in the introduction is very thorough! - "Overall, learning the structure of latent variables is still a challenging problem; for instance, none of the above methods is able to recover the causal structure as shown in Figure 1." Is the method proposed here, strictly more general than all of the other methods previously proposed. That is, are there a class of graphs that the present work would not be able to recover but previous methods would? My feeling is that it is strictly more general than those that use Tetrad constraints but it's unclear to me if it is more general than ICA-type methods like Hoyer et al (I don't think this is the case). - "It first finds pure clusters (clusters of variables having only one common latent variable and no observed parent) from observed data in phase I" -- this part of the methodology that requires a single latent common cause and no observed parents seems restrictive to me. - In definition 1 part 1), could you clarify what you mean by there is no direct causal relation between observed variables. Does this mean absence of a directed path meaning one cannot be an ancestor of the other or just absence of a directed edge meaning one cannot be a parent of the other. I interpret the current definition to mean the latter but if it is the former, this rules out important causal graphs such as the front-door graph A->M->Y, A<->Y (when viewing the latent projection) so being a little more explicit in the definition may be important. The former constraint however, is similar to the ancestrality property in ancestral graphs where the presence of both A <-> B and a directed path from A to B or vice versa is disallowed, and could be justified as such. - In definition 1 part 1), if my interpretation above for this part of the definition is correct i.e. there is no directed edge between the two observed variables, this is graphically equivalent to the absence of "bow-arcs" A->B, A<->B in the latent projection. This may affect the generalizability of the method beyond the current restrictions imposed on the latent structure (because latent projections define a class of infinite latent variable DAGs). The bow-free property combined with 2) and 3) are not sufficient for identifiability of all the parameters in linear SEMs with correlated errors. These graphs must lack C-trees or convergent arborescences in order to be everywhere identifiable. See [1]https://projecteuclid.org/download/pdfview_1/euclid.aos/1299680957. An example (I think) of a graph that fits Definition 1 but does not meet the criteria for identifiability is the following: A->B->C->D, A<->C, A<->D, B<->D. Since some of these parameters will correspond to coefficients or covariances involved in the computation of residuals, it seems like this would pose a challenge to generalizing this method further. - In definition 1 part 3) could it not be relaxed to at most one noise term is Gaussian? This would be similar to the assumption in other papers on causal discovery using additive noise models. Or do all noise terms have to be non-Gaussian for the DS-theorem? - In theorem 1: could you be precise, does "directed connected mean" existence of a directed path? Usually directed or bidirected connected implies the presence of a path but I think here it means L_a -> L_b or L_b -> L_a - The exposition in section 4.1 is nice and does a good job explaining the algorithm. - In section 4.1 -- regarding the replacement of the latent variable with an observed, does the correctness of this step rely on the linearity of the model i.e., collapsing directed paths is equivalent to multiplication/addition of coefficients as in circuit diagrams/path analysis (it kind of looks like that from some of the analysis in the supplement)? It might be useful for exposition to mention that if it is true. - In table 1, a comparison against an ICA-type method would be nice and would probably also the answer the question of how a triad based method compares with an ICA-type one. Minor comments: - Example 1 there is a typo -- vaiolated -> violated.