Q1 & W1

Disentanglement refers to the process of decomposing an entity into multiple independent latent components, or factors. For example, a feature vector can be projected into several distinct latent spaces, each capturing a different aspect of the underlying information, resulting in a set of factor representations.

In the context of hyperedge-level disentanglement, this involves decomposing a hyperedge into multiple types (factors), each corresponding to a distinct interaction mechanism or condition. We will add the explanation in the final version of the paper.

Q2 & W2

Let us begin by analyzing the definition of hyperedge disentanglement in category theoretical viewpoint. Let $V$ be a set of nodes in a hyperedge and let $E$ be the result of group interaction (intuitively, conclusion of group discussion). The transition from $V$ to $E$ is the group interaction mechanism. In category theory, we can represent this as morphism $f$ : $V \rightarrow E$.

By the definition of hyperedge disentanglement, mechanism can be decomposed into several independent mechanisms. (i.e., morphism $f$ can be decomposed as the product of independent morphisms $f_{c}$ and $f_{d}$. $f = f_{c} \times f_{d}$) As can be seen in Figure 13 at the Appendix A.5, objects (nodes $V$ and conclusion $E$) can be decomposed correspondingly : $V = V_{c} \times V_{d}$ and $E = E_{c} \times E_{d}$ such that $f_{c} : V_{c} \rightarrow E_{c}$ and $f_{d} : V_{d} \rightarrow E_{d}$.

Now, let’s examine this from the perspective of factor $c$. We have the commutative diagram for factor $c$. Disentangle first ($V \rightarrow V_{c}$) and then performing interaction $f_{c} : V_{c} \rightarrow E_{c}$ results $E_{c}$. Performing interaction $f : V \rightarrow E$ first, and then disentangling $E \rightarrow E_{c}$ results $E_{c}$. This is category-theoretical representation of hyperedge disentanglement.

Now, let’s bring this abstract concept into the representation space. In category theory, these mappings are carried out through a functor. Functor $F$ will map entities in the definition into representation spaces. Since functors are structure-preserving maps between different domains, the commutative diagram will also hold in the representation space.

Disentangle and then message passing : $F(V) \rightarrow F(V_{c}) \rightarrow F(E_{c})$

Message passing and then Disentangling : $F(V) \rightarrow F(E) \rightarrow F(E_{c})$

Thus, regardless of the sequence of action, we get the same hyperedge factor representation $F(E_{c})$.

Let’s now explain this concept through a more intuitive example. Consider a group discussion involving several people. In this scenario, each factor represents a discussion topic. Suppose there are two independent discussion topics, $c$ and $d$, that do not influence each other. Whether we discuss both topics and then extract the conclusion for $c$, or we focus on topic $c$ alone to derive the conclusion, the conclusion of discussion for $c$ will be the same. This is because $c$ and $d$ are independent, and thus do not affect one another.

Q3 & W3

The results on the hypergraph benchmark datasets in Appendix D include citation networks such as DBLP-coauthorship. For disentanglement to lead to substantial performance improvement, the underlying context or factors must be strongly associated with the labels. If the context exists but is only weakly related to the labels, even a well-disentangled representation may not yield a significant performance gain.

For example, in co-citation and co-authorship networks, hyperedges are formed by grouping all documents cited by a single paper or all papers authored by a single researcher. While citations between a pair of papers may carry meaningful context (e.g., a specific reason for citation), it is difficult to argue that a group of cited documents shares a well-defined or coherent context. Even if we assume that some interaction context exists in such hyperedges, the connection between this context and node labels is often unclear. In such cases, even if contextual information is captured, it may not contribute meaningfully to label prediction performance.

In contrast, in the cancer subtype classification task, pathways are known to carry functional context [2,3] and are closely associated with the labels (i.e., cancer types) [2,4]. To summarize briefly, pathways are associated with specific biological functions (i.e., factors), and whether these functions operate normally—potentially reflected in the factor representations—can determine the type of disease. This indicates that, in this case, the factors are highly relevant to the labels.

$**To assess the applicability of our model beyond genomics**$, we conducted a hyperedge classification experiment on the Chemical Reaction Dataset 1 [1]. In this task, the goal is to classify the reaction type (i.e., the hyperedge label) given a set of molecules (nodes) participating in the reaction.

Due to time constraints, we selected a subset of representative baseline models to evaluate their performance in terms of Macro-F1 score. The results are summarized in the table below. We will evaluate the remaining baseline models and include their performance results in the final version of the paper.

In the chemical reaction dataset, Natural-HNN shows a large performance gap compared to HSDN, which performs hypergraph-structure level disentanglement, and also outperforms AllSetTransformer by approximately 5 percentage points. These results suggest that naturality-based disentanglement can be effectively applied not only in biomedical domain but also in other domains.

Q4

Table 7 presents the performance with and without the factor discrimination loss introduced in Section 4.4. The comparison based on the presence or absence of the naturality constraint (consistency) was shown in Table 11. We will incorporate these results into the main table and add corresponding explanations in the final version of the paper.

[1] DLGNet: Hyperedge Classification through Directed Line Graphs for Chemical Reactions (OpenReview)

[2] Mapping biological process relationships and disease perturbations within a pathway network.

[3] Disentangling the multigenic and pleiotropic nature of molecular function

[4] Identifying cellular cancer mechanisms through pathway-driven data integration

Q1 & W1

Disentanglement is a methodology based on the assumption that data can be decomposed into independent factors, and it aims to identify and isolate these underlying factors. Importantly, the disentanglement process does not introduce new information; rather, it extracts specific factor-related information from the existing data.

Consider the example of people (nodes) participating in group discussions (hyperedges). A node representation corresponds to an individual's opinion. Suppose each factor represents an independent discussion topic (e.g., Factor 1: "Is Hawaiian pizza tasty?" Factor 2: "Do you support a particular political party?"). Then, the factor representation of a node would reflect that individual’s stance on the corresponding topic. In this context, disentanglement refers to the process of extracting the relevant opinions from each person’s overall set of beliefs, focusing only on a specific topic. This extraction does not create new meanings but selectively reveals components of existing information.

Now, consider that a group discussion eventually leads to a conclusion—this can be seen as the hyperedge representation. An entangled hyperedge representation may be thought of as a comprehensive outcome resulting from a discussion that simultaneously covers all possible topics. However, if the topics (factors) are independent of one another, extracting the conclusion related to a specific topic from an entangled hyperedge representation (i.e., message passing and then disentangle) yields the same result as conducting a discussion focused solely on that topic (i.e., disentangle and then message passing).

To make this more concrete, suppose a group discusses two independent topics: Factor 1, "Is Hawaiian pizza tasty?" and Factor 2, "Do you support a particular political party?" These topics are assumed to be independent—opinions on one do not influence opinions on the other. Therefore, the conclusion reached about political party preference after a discussion covering both topics (message passing followed by disentanglement) will be equivalent to the conclusion reached after a discussion focused solely on political preference (disentanglement followed by message passing).

In fact, this is not merely a premise but a fundamental property that must always hold. Let us begin the proof by analyzing the definition of hyperedge disentanglement in category theoretical viewpoint. Let $V$ be a set of nodes in a hyperedge and let $E$ be the result of group interaction (intuitively, conclusion of group discussion). The transition from $V$ to $E$ is the group interaction mechanism. In category theory, we can represent this as morphism $f$ : $V \rightarrow E$.

By the definition of hyperedge disentanglement, mechanism can be decomposed into several independent mechanisms. (i.e., morphism $f$ can be decomposed as the product of independent morphisms $f_{c}$ and $f_{d}$. $f = f_{c} \times f_{d}$) As can be seen in Figure 13 at the Appendix A.5, objects (nodes $V$ and conclusion $E$) can be decomposed correspondingly : $V = V_{c} \times V_{d}$ and $E = E_{c} \times E_{d}$ such that $f_{c} : V_{c} \rightarrow E_{c}$ and $f_{d} : V_{d} \rightarrow E_{d}$.

Now, let’s examine this from the perspective of factor $c$. We have the commutative diagram for factor $c$. Disentangle first ($V \rightarrow V_{c}$) and then performing interaction $f_{c} : V_{c} \rightarrow E_{c}$ results $E_{c}$. Performing interaction $f : V \rightarrow E$ first, and then disentangling $E \rightarrow E_{c}$ results $E_{c}$. This is category-theoretical representation of hyperedge disentanglement.

Now, let’s bring this abstract concept into the representation space. In category theory, these mappings are carried out through a functor. Functor $F$ will map entities in the definition into representation spaces. Since functors are structure-preserving maps between different domains, the commutative diagram will also hold in the representation space.

Disentangle and then message passing : $F(V) \rightarrow F(V_{c}) \rightarrow F(E_{c})$

Message passing and then Disentangling : $F(V) \rightarrow F(E) \rightarrow F(E_{c})$

Thus, regardless of the sequence of action, we get the same hyperedge factor representation $F(E_{c})$.

Q2 & W2

The criterion we derived from category theory establishes that performing message passing followed by disentanglement (disentanglement is implemented as MLP), or disentanglement followed by message passing, should yield consistent results. As demonstrated in our proof-of-concept model, when the hyperedge factor representations obtained through these two paths are similar, we interpret this as the corresponding factor being highly relevant to the hyperedge. Consequently, we assign a higher weight ($\alpha_{i}^{k}$) to that factor, allowing it to contribute more significantly to the final representation.

Q3

The naturality-based criterion is derived directly from the definition of hyperedge disentanglement. By guiding the disentanglement process based on this property, our model was able to capture factors that are closely related to the functional context (i.e., hyperedge context), as clearly demonstrated in the visualizations. Because the proposed criterion effectively captures contextual information, our model achieved the best performance on all datasets except CESC. Notably, on four of the datasets, it outperformed the second-best model by a margin of more than 1.5 point (up to 4.7 point), further highlighting its effectiveness.

Dear reviewer 28h3,

We kindly request you to review our rebuttal, as the author-reviewer discussion period is nearing its end. We hope to know whether our rebuttal adequately addressed your concerns. Let us know if you have any remaining questions.

Q1 & W1

Heuristic-based methods typically rely on assumptions about the data. While such methods may perform well when those assumptions hold, they often struggle when the assumptions are violated. A representative example is the similarity-based criterion, which assumes that instances belonging to the same factor will have similar factor representations. However, this assumption does not always hold in practice. For instance, in genetic pathways, genes within the same pathway or context do not necessarily share similar characteristics or gene expression values.

One of the baseline models, HSDN, adopts a similarity-based criterion. As shown in Figure 5, it failed to disentangle hyperedges effectively, likely due to the limitations of its heuristic assumptions.

In contrast, our goal was to develop a criterion for hyperedge disentanglement that does not rely on heuristics. To achieve this, we sought to identify properties that can be directly derived from the definition of hyperedge-level disentanglement itself. As an analytical tool, we employed category theory, as it provides a mathematically rigorous framework for representing abstract concepts and complex systems.

By formulating hyperedge disentanglement within the language of category theory, we were able to formally express its underlying structure and derive a principled criterion that facilitates effective disentanglement. We will incorporate this motivation into the final version of the paper.

Q2 & W3

Since there is no ground truth for pathway functions, it is not possible to directly compare the learned factors with ground-truth labels. Even if such labels existed, meaningful evaluation would still be difficult without a guarantee that the pathway functions are independent—because disentanglement fundamentally aims to uncover independent factors.

However, evaluating hyperedge-level disentanglement is feasible. Although explicit function labels are unavailable, functional similarity between pathways can be calculated using widely adopted methods in computational biology. By comparing functional similarity from computational biology method and that of ours, we were able to verify that our model captures hyperedge-level factors, rather than hypergraph-structure-level or other types of disentanglement.

Furthermore, as shown in Figure 6, we analyzed the correlation between hyperedge-level factors. This analysis provides supporting evidence that each factor captures distinct information or mechanisms.

Q3

We believe that our approach can also be applied to bipartite GNNs. Our model essentially consists of two phases: node-to-hyperedge message passing and hyperedge-to-node message passing. If nodes and hyperedges are regarded as two distinct node types, the structure can be interpreted as a bipartite graph. Therefore, the proposed naturality-based criterion is also applicable in the context of bipartite GNNs.

W2

The results on the standard hypergraph benchmark datasets (including citation networks) are provided in Appendix D. Although we report performance on the benchmark datasets, it is difficult to conduct in-depth analysis comparable to that of the cancer subtype classification task, as the underlying factors present in these benchmark datasets are not clearly defined or known.

Additionally, during the rebuttal period, we conducted a hyperedge classification experiment on the Chemical Reaction Dataset 1 [1]. In this task, the goal is to classify the reaction type (i.e., the hyperedge label) given a set of molecules (nodes) participating in the reaction.

Due to time constraints, we selected a subset of representative baseline models to evaluate their performance in terms of Macro-F1 score. The results are summarized in the table below. We will evaluate the remaining baseline models and include their performance results in the final version of the paper.

In the chemical reaction dataset, Natural-HNN shows a large performance gap compared to HSDN, which performs hypergraph-structure level disentanglement, and also outperforms AllSetTransformer by approximately 5 percentage points. These results suggest that naturality-based disentanglement can be effectively applied not only in biomedical domain but also in other domains.

[1] DLGNet: Hyperedge Classification through Directed Line Graphs for Chemical Reactions (OpenReview)

Thank you for your constructive review.

Q1 & W3 & W2

While our current evaluation focuses on bioinformatics, our model could be broadly applicable to other domains. We acknowledge the reviewer’s concern that the performance improvements on benchmark datasets in Appendix D are relatively modest compared to those observed in bioinformatics tasks. However, we believe this discrepancy primarily stems from differences in the informativeness of the factors (contexts) encoded in hyperedges across datasets. The more strongly a factor is related to the label, the more critical it becomes to capture that factor, which in turn can have a significant impact on performance. While the degree of informativeness is difficult to quantify, it can be reasonably inferred.

Specifically, in co-citation and co-authorship networks used in Appendix D, hyperedges are formed by simply grouping all documents cited by a single paper or all papers authored by a single researcher. While citations between a pair of papers may carry meaningful context (e.g., a specific reason for citation), it is hard to argue that a group of cited documents shares a well-defined or coherent context. Even if we assume some form of interaction context exists in such hyperedges, the connection between this context and the node labels is often unclear. In such cases, even if the model successfully captures contextual information, it may not significantly contribute to improving label prediction performance.

In contrast, in the cancer subtype classification task, pathways are known to carry functional context [2,3] and are closely associated with the labels (i.e., cancer types) [2,4]. To summarize briefly, pathways are associated with specific biological functions (i.e., factors), and whether these functions operate normally—potentially reflected in the factor representations—can determine the type of disease. This indicates that, in this case, the factors are highly relevant to the labels.

To support our claim that the model is much effective in domains where hyperedges contain rich, label-relevant contextual factors, and to demonstrate its broader applicability, we conducted an additional experiment in the chemical domain—another setting where hyperedges encode meaningful interactions. Specifically, we evaluated our model on the Molecule Reaction Dataset-1 [1], where nodes represent molecules and hyperedges represent chemical reactions annotated with reaction types. The task is to perform hyperedge classification by identifying the reaction type corresponding to each set of participating molecules (i.e., each hyperedge).

While time constraints prevented us from evaluating all baselines during the rebuttal period, we highlight that Natural-HNN substantially outperforms HSDN (a model that also performs structure-level disentanglement) and achieves a 5-point improvement over AllSetTransformer. These results reinforce our claim that our method is particularly effective when hyperedges encode rich, informative contexts, even outside the bioinformatics domain.

We also believe that this response directly addresses the reviewer’s concern regarding the limited data scope (W2) —specifically, the critique that the evaluation was confined to a single domain (bioinformatics) and a single task (hypergraph classification). By presenting results on a new domain (chemistry) and a different task (hyperedge classification), we provide concrete evidence of our model’s broader applicability and generalizability.

Q2

Two pathway representations are $h_{e_{i}}^k$ and $\tilde{h}_{e_{i}}^k$. The difference between the two representations can be used to assess the degree of consistency. However, this approach is not applicable to the cancer subtype classification task, as there are no explicit ground-truth labels for the functional roles of individual pathways.

Q3

Regarding the cancer subtype classification task, we evaluated the performance of our model and the baselines while gradually reducing the number of hypergraphs used during training. The results are presented in Figure 7 of Section 5.4. As shown in the figure, our model maintains strong performance even when the training ratio is reduced.

Additionally, in Figure 8(a), we examined whether our model continues to capture the functional context effectively when the number of training hypergraphs is decreased. The figure shows that our model still captures the functional context well, despite having fewer hypergraphs during training.

For the standard benchmark datasets, please refer to Figure 17 in the Appendix F.3. This figure demonstrates that our model performs well even with a small portion of the training set. Furthermore, in Appendix D.3 (Table 8), we evaluated performance with the training set ratio reduced to as low as 5%, and as shown, our model achieved competitive or superior performance under these conditions.

Therefore, these results demonstrate that the proposed naturality-based criterion enhances the model’s robustness by effectively capturing the functional context or underlying factors, even under limited data conditions.

W1

In the cancer subtype dataset, the goal is to predict each patient’s cancer subtype. All patients share the same set of genes and pathways, but their gene features differ. As a result, although this is a hypergraph classification task, it is not affected by topological differences in the hypergraph.

[1] DLGNet: Hyperedge Classification through Directed Line Graphs for Chemical Reactions (OpenReview)

[2] Mapping biological process relationships and disease perturbations within a pathway network.

[3] Disentangling the multigenic and pleiotropic nature of molecular function

[4] Identifying cellular cancer mechanisms through pathway-driven data integration

Dear reviewer 8x4v,

We kindly request you to review our rebuttal, as the author-reviewer discussion period is nearing its end. We hope to know whether our rebuttal adequately addressed your concerns. Let us know if you have any remaining questions.