Thank you for your review of our work, we briefly highlight the key strengths listed before addressing the weaknesses.

Strengths

Novel identifiability results for hypergraphical models.

Novel algorithm for learning hyper DAGs.

Weaknesses

Some terminology is not explained and may take some effort even for a knowledgeable reader to understand how they generalize to the hypergraphical setting.

We apologize that there is a lot of terminology which felt difficult to understand. We chose to introduce the definitions of body and multicollider during Theorem 1 because of their extremely close correspondence with skeleton and collider. We had hoped this would give easier intuition for readers who were already knowledgeable about these existing definitions. Based on the feedback from multiple reviewers, we understand this was not the case and we have added to Table 1 a hyperlink to a new section of the Appendix which more formally states these definitions.

To reiterate, the body of the HDAG takes the directed hypergraph and replaces every directed hyperedge with its undirected version. This is written as $H’ = \\{ S \cup \\{ j \\} : (S,j) \in H\\}$. A multicollider of an HDAG is any set of $n$ parents and one child. A multicollider is unshielded if the body of the hypergraph does not also contain the hyperedge with all $n$ parents. The intuition for this is that the only time all $n$ parents interact with one another is in creation of the normalizing constant $Z$ in equation 8. This is a special interaction which disappears under marginalization of the child, explaining why we cannot condition on colliders or descendants of colliders in classical DAG theory. (Please also note our correction to your question about Equation 8 below.) The immoralized hypergraph is the undirected hypergraph containing the body of the HDAG along with all of the $Z$ terms from Equation 8, i.e. the hyperedges created by simultaneously marrying all of the parents. It is hoped that how these terms correspond with existing terms is now more clear and we would be very happy to explain any further confusions regarding this.

As the authors argue in their abstract about "abundance of real-world processes exhibiting higher-order mechanisms", I would have hoped for more comprehensive experiments or some other kind of evidence about the significance of the work.

It is unfortunate that you felt misled about the nature of the experiments. This sentence is taken by the authors to mean that, despite the abundance of works showing that real-world processes do exhibit higher-order mechanisms [A, B, C], there remains to be sufficient treatment of how to learn such mechanisms in a causal framework. In our results, we find that the situation is even more dire than one might expect. Even in the most common setting (ER graphs), with the simplest adjustment to include higher-order mechanisms, most existing algorithms not only fail to learn the higher-order causal structure (Table 4), but also fail to learn the typical causal structure in the presence of these higher-order mechanisms (Tables 2 and 3).

These negative results point to the idea that maybe the reason causal structure learning cannot be applied to real-world systems is because existing algorithms are not able to handle higher-order phenomena. Our theoretical results then confirm that higher-order structure can be learned directly from the data, implying that we need to do work on improving the algorithms. Our proposed method HCAM is a first attempt in this direction. Our work can be seen as a similar result to [D], which provides negative results to emphasize that current synthetic data doesn’t represent fully realistic settings.

In this sense, even if one only tentatively believes in the abundance of higher-order mechanisms, what we feel has been clearly demonstrated by the current experiments is that existing methods will not be able to capture them in the situations where they do arise.

Experiments are only performed on sparse Erdős–Rényi graphs over 30 vertices, making them somewhat limited in my opinion.

Four of the six previous works we compare against also focus on ER graphs with a relatively small number of vertices. We agree that this comes with limitations, but it is also the fairest way to compare against existing algorithmic proposals.

Our larger concern is that even with very simple multilinear terms, existing algorithms fail on this limited setting. Additionally, we feel the limitation to 10K samples may also prove to be a key reason why algorithms fail in this setting. We are currently running experiments in this setting for HCAM (other algorithms do not scale well to this regime, which is why we did not originally compare in this setting in the first place).

[A] “Networks beyond pairwise interactions: Structure and dynamics”. Federico Battiston et al. 2020.

[B] “Higher-order genetic interactions and their contribution to complex traits”. Matthew Taylor et al. 2014.

[C] “Dynamics on higher-order networks: a review”. Soumen Majhi et al. 2022.

[D] “Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Game”. Alexander G. Reisach et al. 2021.

Minor Points

Def. 1, 3, 5: I'd add some literature reference here.

We have added references to “Probabilistic Graphical Models: Principles and Techniques” (defn 4.11/4.12 and defn 3.4) for the first two definitions and “Causal Discovery with Continuous Additive Noise Models” (equation 6) for the third definition.

Eq. (6), (7): inconsistent use of italics in Pa.

Italics removed from Eq (6).

Eq. (8): should Pa be HypPa?

No, it should not. Although the functional structure of how the child is generated according to different sets of hyperparents will vary with the chosen structure, the normalizing constant will in general depend on all possible parents in order to form a valid conditional probability.

Line 299: add citation to "Following previous works".

We have added a citation to CAM, RESIT, SCORE, and BOSS which all use ER graphs.

1.

Thank you for your detailed critique of our work. We will address each of your many points in what we believe to be descending order of importance. Reiterated questions are later removed due to space.

W1: Related literature on directed acyclic hypergraphs is not discussed, e.g. [1,2,3],

Thank you for these references with which you are familiar, we will add them to our work. We note that [2] is a pretty unrelated framework, creating a hypergraph over individual subjects for ITE rather than over variables. [3] is more related, but in their Figure 1, it is clear to see that they focus entirely on adding hypergraph structure to the latent marginalization. Under the causally sufficient setting we study, their work would reduce to a DAG. An adequate extension to include marginalizing and conditioning over latent variables for HDAGs would likely encompass both works (discussed further in the next paragraphs).

The last work, [1], is much more nuanced in its differences. In this work, they are specifically focused on merging the directed and undirected graphs under the LWF interpretation of the chain graph. This is different from our framework which focuses entirely on extending the DAG framework. We will call ours HDAG and theirs DAH throughout our response to avoid further confusion.

Using their language, we can say the “shadow” of a DAH is always a chain graph whereas the shadow of an HDAG is always a DAG. This is exactly the correspondence depicted in our Figure 1 where the shadow is the same for 1a, 1b, and 1c. If our understanding is correct, their Equation (7) on page 12 is an extension of our Definition 4. Their work does not handle continuous variables or additive noise models. Despite this generalization to chain-graph-like structures, they do not provide the appropriate generalizations of Markov characterization or identifiability results of the DAH. In particular, they only show each DAH respects the Markov properties of its LWF chain graph shadow. It is imagined the identifiability results herein could be generalized to also handle the DAH case. Nonetheless, given that the three major interpretations of chain graphs (including marginalization) have remained separated for decades, it seems unlikely this would overlap with the interpretation from the above paragraph. Lastly, it is noted that no practical experiments or algorithms are available in relation to the DAH.

R4: Directed acyclic hypergraphs are abbreviated in other works by DAH

This turns out to be quite convenient, because as discussed they use a different definition than ours. As mentioned above, we will continue to use the two different abbreviations DAH and HDAG to make it clear which work we are talking about throughout our reply.

Q2: What is the difference between a DAH HDAG and a DAG that introduces a latent variable for each hypergraph?

R3: The term hyper-Markov equivalence class slightly abuses the definition of the Markov Property, introducing a latent variable for each hyperedge may elegantly resolve the issue

Unfortunately, no, the naive addition of latent variables is not sufficient for our setting. Imagine adding a LV for each hyperedge. At the node of the LV, you could have a variable representing the correct functional term; however, you would need to have a special node to make sure that no additive noise occurs. Next, you would need to aggregate all of the functional terms into the next observed node and use additive noise. However, using the typical ANM structure would destroy the additive structure when combining the LVs; therefore, we would need to use a special edge implying CAM-style aggregation. Instead of adding a special node type and a special edge type, it is much more natural to introduce the hypergraph structure. The situation is similar for the classical setting.

Accordingly, we hope that it is now clear how our studied regime differentiates itself from the existing works using hypergraph structure and how our setup and theoretical results are a unique contribution to the causality literature compared with those related works. Moreover, our setting is not trivially the same as other well-known settings like considering the marginalizing and conditioning of latent variables. If there are any remaining points of confusion here, please let us know immediately so that we may attempt to address them. Otherwise, we will continue to the next section to discuss the listed strengths and technical misunderstandings of our work.

2.

Strengths

The submitted work addresses causal structure learning with directed acyclic hypergraphs under causal sufficiency

Theoretical proof of the identifiability of the causal hypergraph corresponding to an ANM

In addition to this, we have the identifiability result in the classical setting as well, giving identifiability at least up to the MEC and in some cases even more than this.

Technical Points

Q1, Q8

We have added another section to the appendix extending Table 1 to make sure the definitions of all the introduced terms are clear. See response to reviewer "nSx3".

R1

You are correct about this, but unfortunately much in the same sense that writing log p(x) = \xi(x) for some function \xi(x) is even more general than all four of these equations. In this case, we would just be doing probabilistic modeling instead of causal structure learning.

It is understood that you already know the usefulness of causal structure for understanding intervention effects. There are many reasons to also learn the mechanism structure, not least of which is that they are known to exist in a variety of physical phenomena [A, B, C]. Contrast this with our work's experimental results showing that existing algorithms cannot learn these mechanisms (even failing to learn the causal structure in this case!). If this more specific structure is identifiable directly from the data, then why shouldn’t we incorporate it? If this structure is also learnable, then why can’t current algorithms succeed in these simple cases?

L1

Indeed, as stated we make the assumption of causal sufficiency throughout the work, applying to all three subsections. Extending beyond this setting would likely be a great undertaking.

W6

We believe that this must be a misunderstanding of some kind. The ideas of Markov equivalence are from our second setting like in Section 2.2, whereas the proposed algorithm is applied to the third setting from Section 2.3.

Lastly, we will get into the minutiae of the concerns regarding the experimental setup and the writing.

3.

Experiment and Dataset Concerns

W2, R10, L3, W4, L2

We group these concerns because they are all focused on the same topic. We first emphasize that sparse ER graphs are an extremely common setting, employed by 4 of the 6 baseline methods which we compare against. Focusing on this setting does not significantly limit the findings or invalidate the use of the baseline.

It is also reminded that we provide experimental evidence in terms of a negative result, demonstrating that most existing algorithms cannot handle 2D or 3D data even for simple ANMs with Gaussian noise. Due to this, the simplicity of the multi-linear model is actually an advantage to the empirical evidence, rather than a limitation.

W3, Q4, R9

Data generation details are in W2 of “35ca”. Higher-order SHD is computed as the SHD of the bodies, where every algorithm except CAM is assumed to include all interactions like in Figure 1f to enable a comparison.

W8

Although we point to several works supporting our stated hypothesis that the largest reason is the small sample size relative to the exponentially large modeling space of probability densities, we agree that an experiment specifically verifying this would enhance the empirical contribution. A key challenge is that existing methods do not scale to 100K+ samples. We are running experiments now (at least for HCAM) on these larger datasets.

W9

This is another point which could enhance the empirical understanding of the HCAM algorithm. Is it suggested that we ablate parts of the algorithm with oracle knowledge? Since no other existing algorithms can handle HO structure, it is not straightforward to choose which aspects of the algorithm can be ablated. This is especially so because most of the work in the CAM and HCAM algorithms are done in step 2.

Writing and Presentation Concerns

W5

We hope with the adjustments to the writing made in accordance with the technical misunderstandings discussed in the previous part of our response, as well as the other points below, the clarity has been greatly enhanced.

R2

We apologize for the confusion, we have added the nonparametric SEMs to the bottom three panels of Figure 1. We agree that this will help to make Figure 1 more easily understandable.

The ANM does not explicitly specify the interactions in a graphical model. The point we are making here is that it does so implicitly. As mentioned above, the CAM or ANM assumptions lift to different HDAGs (a to d or c to f; our generalized version in the middle), but the shadow of all three HDAGs is the same DAG.

R8

We have added visible nodes and therefore colored in the ellipses. This visually helps a significant amount. We have also added a third picture to the second block.

R11

We have added algorithm pseudocode to the appendix section with the algorithm details.

R12, R13

We have bolded according to the logic “at least as good as the baseline which assumes independence”. We have added a sentence to make this clear and adjusted the current bolding slightly. BOSS’s good performance on 3D is not statistically significant, but we have added a sentence addressing this.

Typos

Everything resolved except for line 170 which is written as intended. We have added the equation $\theta_{ij}(x_i,x_j) + \theta_{ik}(x_i,x_k) + \theta_{jk}(x_j,x_k)$ to make the meaning more obvious.

(3)
W2, R10: To start with, 4 out of 6 does not make an “extremely common setting” (no proper reasoning).

Focusing on this setting does not significantly limit the findings or invalidate the use of the baseline.

Without doubts, the experimental evidence is limited to the test setting, namely sparse ER graphs. The advantage of sparsity favoring methods (e.g. zero baseline) is an artefact of this setting.

L2, W4: For the 2D setting, experimental evidence that the proposed method HCAM is competitive with some other baselines is only provided for the multilinear DGM with additive Gaussian noise, hence, it is certainly a limitation of the positive result. For the 3D setting, the authors transparently report a negative result of HCAM, since it performs exactly as the zero baseline. Although the other algorithms do not do better, it remains (also) a limitation for HCAM. The fact that both GES as well as HCAM reduces to the zero baseline should be discussed by the authors (only implicitly hinted by the very same numbers). In conclusion, both limitations do exist.

W3, Q4, Q5, R9: The generation of the graphs needs some further details to avoid ambiguity:

• For nodes of a sampled ER-graph with more than two or three variables: Are there exclusively multiple hyperedges sampled (overlapping hyperparental sets)? Let’s assume $X_1, X_2, X_3$ (random order) are the parents of some node in a sampled ER graph and the 2D setting is considered. Are the set of hyper-parents for the respective hyper-edges then $(X_1, X_2)$ and $(X_2, X_3)$ as in Figure 1e)? Is this meant by a cyclic fashion for a given random ordering?
• For the 3D model, are there any additional 2D hyperedges (when only two parents are present)? Conventional edges only exist when only a single parent is present? For sampling of the parameter, please clarify the (non-standard) generation:
• Why do you sample from the log uniform distribution and not the (standard) uniform distribution?
• Why do you rescale the coefficients by the square root of both expected moments?
• Why do you additionally divide by the total number of parents?

Q7: As stated, the higher-order SHD is computed on the hypergraph bodies (hyper-edges as well as undirected standard edges).I noted that the result of both SHD are exactly the same for CAM in the 1D setting. Does this imply that the standard SHD computes only the distances between skeletons? In addition, why is for 1D the higher-order SHD for HCAM smaller than the standard SHD?

W8: A very simple setting with a single graph of four nodes and a single hyperedge (three parents and their child) and parameters values close to the mean of the their distribution may be provide some insights (although clear simplification).

R12: The stated logic is not accurate, since the baseline itself is sometimes also bolded. In addition, the uncertainty over multiple runs (standard error of the mean) should be considered as well. Here, the validity of std over 3 runs have to be discussed.

R13:

BOSS’s good performance on 3D is not statistically significant, but we have added a sentence addressing this.

Firstly, an added sentence should be stated in the rebuttal as a direct quote. Secondly, the author’s notion of “statistical significance” needs some further explanation, in particular in the case that only three sampled graphs and a single set of parameters were tested.

General note:
Due to the changed rebuttal policy that does not allow to update the pdf, I would welcome a summary of added paragraphs or sections (both main paper and appendix) as a response to all reviewers. The added results are already sufficiently documented.

Polite reminder:
From the review there is a question (Q3) as well as a weaknesses (W7) that were not addressed by the authors in their rebuttal, but they may still want to do so. By contrast, Q6 has been addressed, although not clearly marked.

Thank you for the final clarification that the reported mean values and standard deviations in Table 2, 3 and 4 are only taken over 3 datasets. The answer also clarifies that for the same graph, not multiple parameter values were sampled.

As a consequence, the interpretation of the standard deviations using $N_\text{runs}=3$ and resulting claims have to be lowered by the authors accordingly. (Only) for the three same tested graphs in each setting, they allow for a direct comparison between the algorithms. Hence, the presented empirical evidence for each setting is very limited and certainly to weak to support general statements about the performance of the algorithms on the DGP.
This explains why I highly recommended to state the tested adjacency matrices and parameters.

Strengths

This paper introduces novel causal structure learning framework by incorporating higher order interactions.

There are also many useful theoretical results discussed in the paper which can be useful for future work in this direction.

Writing Weaknesses

W1: It might be trivial, but a lot of the technical results lack references. For example, the definitions related to the pairwise models.

Thank you for bringing this to our attention, please see the “minor points” section of our response to Reviewer “nSx3” where we make point-by-point improvements to the writing based on their feedback about this writing issue.

W2: The data used in examples or the equations used to generate them should be available and more explicit than what is currently in Section 5.

We have added more of these details to the appendix. The data is generated as follows. We first sample the ER graph and then sample a random ordering of nodes (it does not matter which order we do this in). In the 1D setting our HDAG is already complete. In the 2D and 3D setting, we choose hyperedges by taking a random ordering of parents and selecting pairs or triples in a cyclic fashion. This is the simplest HDAG which obeys the same DAG structure. Now we sample the parameters. For the beta coefficients, we sample from [0.5, 2.0] using a log uniform distribution. We sample gaussian standard deviations from the same distribution. We then rescale the coefficients by dividing by the square root of the expected Gaussian moments, 3.0 and 15.0, respectively, and the divide by the total number of parents as well. For the 1D case we follow the literature standard of sampling from a Gaussian process prior (randomly initialized sklearn.gaussian_process.GaussianProcessRegressor). Code will be released.

Question – Extendable to Time Series?

Q2: The data in the experiments are i.i.d. Would a similar greedy algorithm work for time-series data (let's assume stationarity)?

There are some existing works showing the identifiability of causal structure in time series settings [E, F]. Given that our theoretical results show that HDAGs are always identifiable when the DAGs are identifiable, we expect the same would hold for any TS generalization. If your question is about the greedy algorithm, already for our 3D experiments the algorithm is struggling, which would only become more challenging in TS. This might helps explains why TS researchers already make low-dimensional assumptions (e.g. channel independence) in practice.

Question – Other Algebraic Structures?

Q3: There are other higher order structures, such as directed simplicial complexes and cell complexes. Can these structures be exploited to infer causality from data similar to the hyper DAG?

Yes, if by complexes you are referring specifically to abstract simplicial complexes, then the structure defined by an ASC is exactly the same as the structure defined by the undirected hypergraph. In some sense, it would be more accurate to introduce our framework using ASCs because we already assume the hierarchy property or “closure under subsets” property of the hypergraph edges. However, we felt it would be much more natural to extend the existing graphical language which has been built up over decades.

Q1: The current definition of HyperDAG only has one child. It is clear that a more general form will be to have multiple child. Would this be useful in terms of causal structure learning? How will this affect the technical results obtained in the paper?

No, lifting the restriction to use hyperedges which have a single child (single arrowhead) does not have any use in this framework, as mentioned next to its definition. Because of the ordering on the variables coming from the DAG framework, we will only generate one variable at a time. Accordingly, if one tries to have a hyperedge with k children, it will be equivalent to splitting the hyperedge into k hyperedges with one child, only making the results of the paper less clear.

One could imagine multiple arrowheads being useful in an extension to latent variables such as in mixed graphs where bidirected edges are often used to represent confounding. However, the complexity of such an extension skyrockets. It took a decade of research in the 90s to go from DAGs to MAGs and exactly classifying the equivalence of different mixed graphs remains an open problem and an active area of research. For why latents are hard even in our CAM setting, see [G].

All that is to say, (a) for our paper’s setting without latents, it is not useful, and (b) for a setting with latents, it could be much more complex than “HDAGs with multiple children”.

[E] “Search for Additive Nonlinear Time Series Causal Models”. Tianjiao Chu and Clark Glymour. 2008.

[F] “Causal Inference on Time Series using Restricted Structural Equation Models” Jonas Peters et al. 2013.

[G] “Causal additive models with unobserved variables” Takashi Nicholas Maeda, Shohei Shimizu. UAI 2021.

Strengths

Using Hypergraphs for representing higher-order interactions is useful and can be used in various fields when such type of interactions are as important as knowing cause-effect relationships.

Extended identifiability results for hypergraphs are interesting and might be valuable for further researches.

Weaknesses

The main concern is about the usefulness of the hypergraph and higher-order interactions. It is still not so clear in what type of applications knowing (10) is better than (9).

Knowing (10) is true is always strictly better than knowing (9) is true, containing strictly more information about the structure. It is in this sense that we say the HDAG structure strictly extends the DAG structure without making any parametric SEM assumptions, like Eqn (1). Whenever the DAG is identifiable, we show that without any additional assumptions, the HDAG is also identifiable. Therefore, any time you can know (9), you can also know (10). This is the importance of our identifiability results both in classical DAGs and in ANMs.

All of these rewritings of $x_1 = x_2 + x_3 + x_4 + e_1$ have the exact same HDAG structure according to equation (10). Namely $x_1 = f(x_2) + f(x_3) + f(x_4) + e_1$. A slightly more interesting example could be $x_1 = log( x_2 \cdot x_3 \cdot x_4) + e_1$ which does not have the HDAG structure $x_1 = f(x_2,x_3,x_4) + e_1$, but instead we can see that $x_1 = log(x_2) + log(x_3) + log(x_4) + e_1$, so we have the same $x_1 = f(x_2) + f(x_3) + f(x_4) + e_1$ structure as before.

Eq. (4) is not clear. If \xi_S(x_s) is the log probability of p(x_s), then (4) is not correct. According to the Hammersley-clifford theorem, the joint probability can be represented using the cliques but the terms are not necessary the probabilities.

You are correct that Equation (4) is essentially the Hammersley–Clifford theorem and that in general the $\xi_S(x_S)$ is not equal to $\log(p(x_S))$. The same is true of Equation (5) below it. We apologize if this is confusing due to us defining $\xi_X$ and introducing the clique representation using $\xi_S$ in the same line.

Regarding the proposed greedy method, does its complexity grow with the order of interactions? This affects its scalability. Moreover, the order of interactions should be a hyper parameters, how is that selected?

You are correct, the complexity grows with the order of interactions. In particular, we take the maximal possible degree of an interaction as the hyperparameter (set equal to three throughout our experiments). This means the greedy, layerwise approach taken by SIAN is able to stop early if the real data has no interactions, but cannot add any interactions of degree 4 or higher. Setting the maximal degree to something larger like 5 would slow down phase 1 of the algorithm and also create a larger candidate set for phase 2.

These questions of how to learn higher-order interactions while balancing scalability are not addressed anywhere in current literature and we believe this contributes to most algorithms’ failure in Tables 2, 3, and 4. In the real world, the true degree will depend on the actual causal mechanisms behind the data generation. We believe this tradeoff is an important part of causal learning which is not shown in current literature, even though higher-order interactions are already known to exist from physics [A, B, C].

We hope that most of these concerns can be alleviated through better writing, and also through better positioning of how our claims fit into the current literature on causal structure learning. To enable a better comparison, we include a list of the major changes to improve the writing and it is hoped that this will allow for the possibility to reconsider the value of the proposed work.

Major Writing Changes

• Added further motivation on why higher-order is known to be important for real-world systems.
  • The paragraph at line 40 now reads: “In this work, we revisit the additive structural assumption of CAM to also incorporate higher-order interactions, extending the causal model to explicitly consider higher-order causal mechanisms. Higher-order mechanisms are known to exist in a variety of real-world processes [Battiston et al., 2020], and are critical for modeling a number of different scientific phenomena including dynamical systems [Majhi et al., 2022], bioscience [Taylor and Ehrenreich, 2015, Gaudelet et al., 2018], neurobiology [Amari et al., 2003, Petri et al., 2014], and social networks [Freeman and White, 1993, de Arruda et al., 2020]. Nevertheless, previous approaches to causality fail to adequately represent the higher-order mechanisms which could be at play in real-world data. Most algorithms take an all-or-nothing approach, either (a) directly following CAM-like assumptions of no interactions or (b) modeling all possible interactions between the parents of a child node.”
• We add to the end of contribution 2 at line 56, “Moreover, experiments on simple synthetic data demonstrate that most algorithms struggle to capture higher-order interactions, calling into question the practical learnability of real-world systems with higher-order interactions.”
• We move the paragraph at line 79 to just above Section 2.1
• Added references to existing definitions (defn 1, 3, 5) and existing experiment settings (line 299)
• Definition, clarity
  • Line 118 now reads: “It is straightforward to see that this again strictly generalizes the structures which are representable by the typical DAG. Importantly, this includes structures which are not easily represented through existing latent variable approaches. It can also be seen from equation 8 that we may restrict our attention to hypergraphs which are hierarchical, meaning $S \\in \\text{HypPa}\_{\\cal{H}}(j)$ and $T \\subseteq S$ implies that $T \\in \\text{HypPa}\_{\\cal{H}}(j)$ (much like a simplicial complex).”
• Results, additional sentences
  • “We bold those algorithms which beat the zero baseline. In the case of no statistically significant outperformance, we bold algorithms which perform no worse than the baseline.”
  • “In several cases, BOSS on 2D, and BOSS, GES, HCAM on 3D, we find that the performance of the baseline is matched by directly predicting the zero baseline (partially or completely).”
  • “In Figure 3, we see how the number of provided samples directly affects the ability to learn higher-order mechanism structure. Even for this small hypergraph, learning with 100K samples is not sufficient for perfect recovery 100% of the time.”

Major Presentation Changes

• Added a definitions section to the appendix
• Improved Figure 1 with depicted SEMs (and updated caption)
• Improved Figure 2 visually (and updated caption)
• Added all data generation details to appendix
• Added an experiment explicitly verifying that dataset size and scaling is the key issue
• Added algorithm pseudocode to the appendix

[H] “Higher-order molecular organization as a source of biological function” Gaudelet et al. 2018. [I] “Synchronous firing and higher-order interactions in neuron pool” Amarai et al. 2003. [J] “Homological scaffolds of brain functional networks” Petri et al. 2014. [K] “Using Galois Lattices to Represent Network Data” Freeman and White. 1993. [L] “Social contagion models on hypergraphs” de Arruda et al. 2020.