Training-Free Message Passing for Learning on Hypergraphs

## Abstract

Abstract Hypergraphs are crucial for modelling higher-order interactions in real-world data. Hypergraph neural networks (HNNs) effectively utilise these structures by message passing to generate informative node features for various downstream tasks like node classification. However, the message passing module in existing HNNs typically requires a computationally intensive training process, which limits their practical use. To tackle this challenge, we propose an alternative approach by decoupling the usage of hypergraph structural information from the model learning stage. This leads to a novel training-free message passing module, named TF-MP-Module, which can be precomputed in the data preprocessing stage, thereby reducing the computational burden. We refer to the hypergraph neural network equipped with our TF-MP-Module as TF-HNN. We theoretically support the efficiency and effectiveness of TF-HNN by showing that: 1) It is more training-efficient compared to existing HNNs; 2) It utilises as much information as existing HNNs for node feature generation; and 3) It is robust against the oversmoothing issue while using long-range interactions. Experiments based on seven real-world hypergraph benchmarks in node classification and hyperlink prediction show that, compared to state-of-the-art HNNs, TF-HNN exhibits both competitive performance and superior training efficiency. Specifically, on the large-scale benchmark, Trivago, TF-HNN outperforms the node classification accuracy of the best baseline by 10 % percent 10 10\% 10 % with just 1 % percent 1 1\% 1 % of the training time of that baseline.

## 1 Introduction

Higher-order interactions involving more than two entities exist in various domains, such as co-authorships in social science (Han et al., ) and spreading phenomena in epidemiology (Jhun, ). Hypergraphs extend graphs by allowing hyperedges to connect more than two nodes, making them suited to capture these complex relationships (Bick et al., ). To utilise such structures for downstream tasks, learning algorithms on hypergraphs have garnered increasing attention.

Inspired by the success of graph neural networks (GNNs) (Wu et al., ), current research focuses on developing hypergraph neural networks (HNNs) with a message passing module (MP-Module) that can be compatible with various task-specific modules. The MP-Module enables information exchange between connected nodes to generate informative node features for specific tasks (Feng et al., ; Wang et al., ; Telyatnikov et al., ). However, similar to other message passing neural networks (Frasca et al., ; Wu et al., ), training the MP-Module makes loss computation interdependent for connected nodes, resulting in a computationally intensive training process for HNNs. This limits their practical applications, especially in the process of large-scale hypergraphs.

To address this challenge, our key solution is to develop a training-free MP-Module that shifts the processing of hypergraph structural information from the learning stage to the data pre-processing phase. This approach is inspired by recent advancements in efficient GNNs (Wu et al., ; Frasca et al., ), where training-free MP-Modules are typically implemented as graph filters (Ortega et al., ). However, directly using existing hypergraph filters (Zhang et al., ; Qu et al., ) presents two major obstacles. Firstly, these filters are specifically designed for $k$-uniform hypergraphs, where all hyperedges must have the same size $k$, limiting their applicability to more general hypergraph structures with varying hyperedge sizes. Secondly, the reliance on an incidence tensor to represent the hypergraph leads to severe memory inefficiencies, as the tensor grows exponentially with $n^{k}$ dimensions, making it unsuitable for practical, large-scale datasets. Thus, instead of adopting these existing hypergraph filters, we introduce a novel training-free MP-Module tailored for hypergraphs.

In this work, we develop a novel, training-free message passing module, TF-MP-Module, for hypergraphs to decouple the usage of hypergraph structure information from the model learning stage. To achieve this, we first construct a theoretical framework that provides a unified view of existing HNNs (Huang & Yang, ; Chen et al., ; Chien et al., ; Wang et al., ). Specifically, this framework identifies the feature aggregation function as the core operator of MP-Modules in existing HNNs. Based on these insights, the proposed framework consists of two stages:
1) We make the feature aggregation functions within the MP-Modules of four state-of-the-art HNNs (Huang & Yang, ; Chen et al., ; Chien et al., ; Wang et al., ) training-free by removing their learnable parameters; and 2) To further enhance efficiency, we remove the non-linear activation functions and consolidate the feature aggregation across $L$ layers into a single propagation step. Remarkably, this two-stage process unifies the chosen MP-Modules into a single formulation, despite their different design philosophies. We refer to this unified formulation as the TF-MP-Module and denote the hypergraph neural network equipped with it as TF-HNN. To demonstrate the efficiency and effectiveness of TF-HNN, we provide a theoretical analysis showing three key advantages: 1) TF-HNN is more training-efficient compared to existing HNNs; 2) TF-HNN can utilise as much information as existing HNNs for generating node features; and 3) TF-HNN is robust against the oversmoothness issue while taking into account the long-range information.

The main contributions of this work are summarised as follows:

$\bullet$ We present an original theoretical framework that identifies the feature aggregation function as the core component of MP-Modules in existing HNNs, which processes the hypergraph structural information. This insight provides a deeper understanding of existing HNNs.

$\bullet$ We develop TF-HNN, an efficient and effective model for hypergraph-structured data. To our knowledge, TF-HNN is the first model that decouples the processing of hypergraph structural information from model learning, significantly enhancing training efficiency.

$\bullet$ We theoretically support the efficiency and effectiveness of TF-HNN by showing that: 1) It leads to remarkably low training complexity when solving hypergraph-related downstream tasks; 2) It utilises the same amount of information as existing HNNs for generating node features; and 3) It is robust against the oversmoothing issue when utilising long-range information.

$\bullet$ We conduct extensive experiments in both node classification and hyperedge prediciton tasks to compare TF-HNN with nine state-of-the-art HNNs. The empirical results show that the proposed TF-HNN exhibits both competitive learning performance and superior training efficiency.

Figure: Figure 1: Training pipeline of HNN vs. TF-HNN for node classification. Top row: HNN uses a hypergraph structure to generate node features by a learnable MP-Module, which are then used by a classifier, with the MP-Module and the classifier being trained together. For brevity, we omit the MLP in HNN for the input node features. Bottom row: TF-HNN comprises only the classifier trained for node classification and the TF-MP-Module that can be recomputed prior to classifier training.
Refer to caption: extracted/5895839/figs/tfnn_v2.png

## 2 Notation

Hypergraph. Let $\mathcal{H}=\{\mathcal{V},\mathcal{E},\mathbf{H}\}$ be a hypergraph, where $\mathcal{V}=\{v_{1},v_{2},\cdots,v_{n}\}$ is the node set, $\mathcal{E}=\{e_{1},e_{2},\cdots,e_{m}\}$ is the hyperedge set, and $\mathbf{H}\in\{0,1\}^{n\times m}$ is an incidence matrix in which $\mathbf{H}_{ik}=1$ means that $e_{k}$ contains node $v_{i}$ and $\mathbf{H}_{ik}=0$ otherwise. Define $\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}\in\mathbb{R}_{\geq 0}^{n\times n}$ as a diagonal matrix of node degrees and $\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}\in\mathbb{R}_{\geq 0}^{m\times m}$ as a diagonal matrix of hyperedge degrees, where $\mathbf{D}_{\mathcal{H}^{\mathcal{V}}_{ii}}$ and $\mathbf{D}_{\mathcal{H}^{\mathcal{E}}_{kk}}$ are the number of hyperedges with $v_{i}$ and the number of nodes in $e_{k}$, respectively. In the following discussions, we assume the hypergraph does not have isolated nodes or empty/duplicate hyperedges. Moreover, we define: 1) The distance between two nodes on a hypergraph is the number of hyperedges in the shortest path between them, e.g., the distance is one if they are directly connected by a hyperedge; and 2) The $k$-hop neighbours of a node $v_{i}$ are all nodes with a distance of $k$ or less.

Graph and clique expansion. Let $\mathcal{G}=\{\mathcal{V},\mathbf{W}\}$ be a graph, where $\mathcal{V}=\{v_{1},v_{2},\cdots,v_{n}\}$ is the node set, and $\mathbf{W}\in\mathbb{R}_{\geq 0}^{n\times n}$ is the adjacency matrix of $\mathcal{G}$ in which $\mathbf{W}_{ij}>0$ means that $v_{i}$ and $v_{j}$ are connected and $\mathbf{W}_{ij}=0$ otherwise. We set $\mathbf{D}\in\mathbb{R}_{\geq 0}^{n\times n}$ as a diagonal node degree matrix for $\mathcal{G}$, where $\mathbf{D}_{ii}$ is the sum of the $i$-th row of $\mathbf{W}$. Moreover, we denote the graph Laplacian of $\mathcal{G}$ as $\mathbf{L}=\mathbf{D}-\mathbf{W}$. Given a hypergraph $\mathcal{H}=\{\mathcal{V},\mathcal{E},\mathbf{H}\}$, its clique expansion is defined as a graph $\mathcal{G}=\{\mathcal{V},\mathbf{W}\}$, where $\mathcal{V}$ remains unchanged, and $\mathbf{W}_{ij}>0$ if and only if $v_{i}$ and $v_{j}$ are connected by a hyperedge on $\mathcal{H}$ and $\mathbf{W}_{ij}=0$ otherwise. Therefore, each hyperedge in $\mathcal{H}$ is a clique in $\mathcal{G}$.

Other notations. We set node features as $\mathbf{X}_{\mathcal{V}}\!=\![\mathbf{x}^{\top}_{v_{1}},\!\mathbf{x}^{\top}_{v
_{2}},\!\cdots,\!\mathbf{x}^{\top}_{v_{n}}]^{\top}\in\mathbb{R}^{n\times d}$, which is a matrix that contains $d$-dimensional features. We denote functions or variables at the $l$-th layer of a model using the superscript $(l)$ and use $\oplus$ for concatenation. We use $\mathbf{\Theta}$ to represent the learnable weight matrix in a model, $\operatorname{MLP}(\cdot)$ for a multilayer perceptron (MLP), and $\sigma(\cdot)$ for non-linear functions (e.g., ReLU). In subsequent sections, we presume that the node features are $d$-dimensional unless otherwise noted.

## 3 Methodology

We present in this section the hypergraph neural network with training-free message passing module (TF-HNN) approach. Subsection 3.1 provides an overview of TF-HNN. In Subsection 3.2, we present a theoretical framework that offers a unified view on existing HNNs. Building on the insights gained from this theoretical framework, Subsection 3.3 details the design of our TF-MP-Module.

### 3.1 Overview

Inspired by the GNNs literature (Wu et al., ; Frasca et al., ), we design the TF-HNN that decouples the usage of hypergraph structural information from the model learning, thus also removing the reliance on it at training. Mathematically, TF-HNN can be described as the following framework:

$$ $\hat{\mathbf{Y}}=\varphi_{\mathbf{\Theta}}(\hat{\mathbf{X}}_{\mathcal{V}}),~{} ~{}\hat{\mathbf{X}}_{\mathcal{V}}=S(\mathbf{X}_{\mathcal{V}},\mathbf{H}),$ (1) $$

where $\varphi_{\mathbf{\Theta}}(\cdot)$ is a task-specific module, $\hat{\mathbf{Y}}$ denotes the task-specific output whose dimension is task-dependent(^1^11For instance, in the node classification task, $\hat{\mathbf{Y}}\in\mathbb{R}^{n\times c}$ contains the logits for $c$ categories; and in the hyperlink prediction task, $\hat{\mathbf{Y}}\in\mathbb{R}^{m_{p}}$ contains the probability of $m_{p}$ potential hyperedges.), $S(\cdot)$ denotes an MP-Module that contains only pre-defined parameters allowing it to be precomputed in the pre-processing phase, and $\hat{\mathbf{X}}_{\mathcal{V}}\in\mathbb{R}^{n\times d}$ denote the features generated by $S(\cdot)$. In the GNN literature, $S(\cdot)$ is implemented as the graph filter Ortega et al. (). However, we cannot directly use existing hypergraph filters (Zhang et al., ; Qu et al., ) for two main reasons. Firstly, existing hypergraph filters are primarily designed for $k$-uniform hypergraphs, where all hyperedges have size $k$. Secondly, these filters require the use of an incidence tensor, which leads to prohibitively high memory usage for practical applications. Specifically, for a hypergraph with $n$ nodes and hyperedges of uniform size $k$, the incidence tensor has $n^{k}$ dimensions. Hence, rather than relying on existing methods, we design a novel function $S(\cdot)$ for hypergraphs, the TF-MP-Module. Since our design is based on existing MP-Modules for hypergraphs, the next section introduces an original framework that offers a unified perspective on current HNNs.

### 3.2 Revisiting Hypergraph Neural Networks

A traditional HNN for a certain downstream task is formulated as the following equation:

$$ $\hat{\mathbf{Y}}=\varphi_{\mathbf{\Theta}}(\hat{\mathbf{X}}_{\mathcal{V}}),~{} ~{}\hat{\mathbf{X}}_{\mathcal{V}}=\Phi_{\mathbf{\Theta}}(\mathbf{X}_{\mathcal{ V}},\mathbf{H}),$ (2) $$

where $\varphi_{\mathbf{\Theta}}(\cdot)$ is a task-specific module, $\hat{\mathbf{Y}}$ denotes the task-specific output whose dimension is task-dependent, $\Phi_{\mathbf{\Theta}}(\cdot)$ is a learnable message passing module, and $\hat{\mathbf{X}}_{\mathcal{V}}\in\mathbb{R}^{n\times d}$ denote the features generated by $\Phi_{\mathbf{\Theta}}(\cdot)$. The modules $\varphi_{\mathbf{\Theta}}(\cdot)$ and $\Phi_{\mathbf{\Theta}}(\cdot)$ are trained together with the supervision given by the downstream tasks. See Figure 1 for the comparison between the training pipelines of traditional HNNs and TF-HNN in the context of node classification. Existing HNNs use the hypergraph structural information by the message passing module $\Phi_{\mathbf{\Theta}}(\cdot)$, we detail this module in the next paragraph.

**Table 1: Overview for four state-of-the-art HNNs and our TF-HNN. In this table, $\gamma_{U},\gamma_{E},\gamma_{D},\gamma^{\prime}_{l}\!\in\!(0,1)$ are hyperparameters and $\mathbf{I}\in\mathbb{R}^{d\times d}$ denotes an identity matrix. Moreover, $M$ is the training computational complexity of the task-specific module, $n$ is the node count, $m$ is the hyperedge count, $m^{\prime}$ is the number of edges in the clique expansion, $\|\mathbf{H}\|_{0}$ is the number of non-zero values in $\mathbf{H}$, $T$ is the number of training epochs, $L$ is the number of layers, and $d$ is the feature dimension.**
| Name | Type | Hypergraph-Wise Feature Aggregation Function | Training Computational Complexity |
| --- | --- | --- | --- |
| UniGCNII (Huang & Yang, ) | Direct | $\mathbf{X}_{\mathcal{V}}^{(l)}\!=\!\sigma\Bigg{(}\Big{(}(\!1\!-\!\gamma_{U}\!)<br>\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1/2}\mathbf{H}\!\tilde{\mathbf{D}}_{<br>\mathcal{H}^{\mathcal{E}}}^{-1/2}\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}^{-1}<br>\mathbf{H}^{\top}\mathbf{X}_{\mathcal{V}}^{(l-1)}\!+\!\gamma_{U}\mathbf{X}_{<br>\mathcal{V}}^{(0)}\Big{)}\mathbf{\Theta}^{(l)}\Bigg{)}$ | $\mathcal{O}(M+TL(n+m+\|\mathbf{H}\|_{0})d+TLnd^{2})$ |
| Deep-HGNN (Chen et al., ) | Direct | $\mathbf{X}_{\mathcal{V}}^{(l)}\!=\!\sigma\Bigg{(}\Big{(}(\!1\!-\!\gamma_{D}\!)<br>\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1/2}\mathbf{H}\!\mathbf{D}_{\mathcal{<br>H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top}\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^<br>{-1/2}\mathbf{X}_{\mathcal{V}}^{(l-1)}\!+\!\gamma_{D}\mathbf{X}_{\mathcal{V}}^<br>{(0)}\Big{)}\Big{(}(1-\gamma^{\prime}_{D})\mathbf{I}+\gamma^{\prime}_{D}<br>\mathbf{\Theta}^{(l-1)}\Big{)}\Bigg{)}$ | $\mathcal{O}(M+TLm^{\prime}d+TLnd^{2})$ |
| AllDeepSets (Chien et al., ) | Indirect | $\mathbf{X}_{\mathcal{V}}^{(l)}=\operatorname{MLP}\Bigg{(}\mathbf{D}_{\mathcal{<br>H}^{\mathcal{V}}}^{-1}\mathbf{H}\operatorname{MLP}\Big{(}\mathbf{D}_{\mathcal{<br>H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top}\operatorname{MLP}(\mathbf{X}_{\mathcal<br>{V}}^{(l-1)})\Big{)}\Bigg{)}$ | $\mathcal{O}(M+TL\|\mathbf{H}\|_{0}d+TL(n+m)d^{2})$ |
| ED-HNN (Wang et al., ) | Indirect | $\mathbf{X}_{\mathcal{V}}^{(l)}=\operatorname{MLP}\Bigg{[}(1-\gamma_{E})<br>\operatorname{MLP}\Bigg{(}\mathbf{D}^{-1}_{\mathcal{H}^{\mathcal{V}}}\mathbf{H<br>}\operatorname{MLP}\Big{(}\mathbf{D}^{-1}_{\mathcal{H}^{\mathcal{E}}}\mathbf{H<br>}^{\top}\operatorname{MLP}(\mathbf{X}_{\mathcal{V}}^{(l-1)})\Big{)}\Bigg{)}+<br>\gamma_{E}\mathbf{X}_{\mathcal{V}}^{(0)}\Bigg{]}$ | $\mathcal{O}(M+TL\|\mathbf{H}\|_{0}d+TL(n+m)d^{2})$ |
| TF-HNN (Ours) | Direct | $\hat{\mathbf{X}}_{\mathcal{V}}=\mathbf{S}\mathbf{X}_{\mathcal{V}}$ | $\mathcal{O}(M)$ |

Message passing module (MP-Module). The MP-Module in existing HNNs is typically based on one of four mechanisms(^2^22Details on these mechanisms can be found in Appendix B): clique-expansion, star-expansion, line-expansion, or incidence-tensor (Huang & Yang, ; Wang et al., ; Chen et al., ; Chien et al., ; Wang et al., ; Yang et al., ; Antelmi et al., ; Kim et al., ). We summarize the MP-Module using these mechanisms in Proposition 3.1, with the proof provided in Appendix C.

###### Proposition 3.1 .

Let $\mathbf{x}_{v_{i}}^{(l)}$ be features of node $i$ in the $l$-th message passing layer of a HNN based on clique-expansion/star-expansion/line-expansion/incidence-tensor, $\phi_{\mathbf{\Theta}}(\cdot)$ denotes a learnable node-wise feature aggregation function, and $\mathcal{N}_{\mathcal{H}_{v_{i}}}$ is the set of neighbours of $v_{i}$ on the hypergraph. For $i=0$, we have $\mathbf{x}_{v_{i}}^{(0)}=\operatorname{MLP}(\mathbf{x}_{v_{i}})$. For $i\in\mathbb{Z}^{+}$, we have $\mathbf{x}_{v_{i}}^{(l)}=\phi_{\mathbf{\Theta}}(\mathbf{x}_{v_{i}}^{(0)},
\mathbf{x}_{v_{i}}^{(l-1)},\oplus_{v_{j}\in\mathcal{N}_{\mathcal{H}_{v_{i}}}}
\mathbf{x}_{v_{j}}^{(l-1)})$.

Specifically, $\phi_{\mathbf{\Theta}}(\cdot)$ defined in Proposition 3.1 can be categorized into two types: direct and indirect feature aggregation functions. The direct approaches aggregate features of neighbouring nodes to that of the target node directly (Huang & Yang, ; Wang et al., ), while the indirect methods aggregate features of neighbouring nodes to that of the target node via some virtual nodes (Chien et al., ; Wang et al., ; Yang et al., ). These approaches can be formulated as:

$$ $\displaystyle\mathbf{x}_{v_{i}}^{(l)}$ $\displaystyle=f_{3}\Big{(}f_{0}(\mathbf{x}_{v_{i}}^{(0)})+f_{1}(\mathbf{x}_{v_ {i}}^{(l-1)})+\sum_{v_{j}\in\mathcal{N}_{\mathcal{H}_{v_{i}}}}f_{2}(\mathbf{x} ^{(l-1)}_{v_{j}})\Big{)},$ (3a) $\displaystyle\mathbf{x}_{v_{i}}^{(l)}$ $\displaystyle=g_{2}\Big{(}g_{0}(\mathbf{x}_{v_{i}}^{(0)})+\mathbf{p}^{\top}g_{ 1}(\mathbf{x}_{v_{i}}^{(l-1)},\oplus_{v_{j}\in\mathcal{N}_{\mathcal{H}_{v_{i}} }}\mathbf{x}_{v_{j}}^{(l-1)})\Big{)},$ (3b) $$

where $f_{0}(\cdot)$, $f_{1}(\cdot)$, $f_{2}(\cdot)$, $f_{3}(\cdot)$, $g_{0}(\cdot)$, $g_{1}(\cdot)$, $g_{2}(\cdot)$ represent seven learnable functions, and $\mathbf{p}\in\mathbb{R}^{n_{v}}$ is a vector used to aggregate the features of virtual nodes generated by $g_{1}(\cdot)$. Here, $n_{v}$ is a hyperparameter corresponding to the number of the virtual nodes, and Eq. (3a) and Eq. (3b) represent the direct and indirect approaches respectively. Based on Propostion 3.1, the hypergraph structure is only used to construct the neighbourhood set $\mathcal{N}_{v_{i}}$ within the MP-Module of HNNs. Consequently, the feature aggregation function $\phi_{\mathbf{\Theta}}(\cdot)$ is the core component of the MP-Module in processing hypergraph structural information. Moreover, based on Eq. (3a) and Eq. (3b), the usage of the hypergraph structural information is not directly related to the learnable parameters and the non-linear function. Therefore, in the next subsection, we design our TF-MP-Module by first removing the learnable parameters for the feature aggregation function from existing HNNs to make it training-free. To further improve its efficiency, we eliminate the non-linear function from the feature aggregation process. Specifically, we design our model based on HNNs shonw in Table 1.

###### Proposition 3.1 .

### 3.3 Training-Free Message Passing

We introduce the TF-MP-Module by removing the learnable parameters and the non-linear function from four state-of-the-art HNNs shown in Table 1, which includes two methods with the direct feature aggregation function and two methods with the indirect feature aggregation function.

Learnable parameters removal. Based on the insights from Subsection 3.2, to develop a training-free MP-Module, we first remove learnable parameters from the feature aggregation functions of the selected HNNs. Specifically, we replace the learnable matrices in these functions with identity matrices. This removal allows us to reformulate the feature aggregation functions as follows:

$$ $\displaystyle\mathbf{X}_{\mathcal{V}}^{(l)}$ $\displaystyle=\sigma\Big{(}(1-\gamma_{U})\mathbf{D}_{\mathcal{H}^{\mathcal{V}} }^{-1/2}\mathbf{H}\!\tilde{\mathbf{D}}_{\mathcal{H}^{\mathcal{E}}}^{-1/2} \mathbf{D}_{\mathcal{H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top}\mathbf{X}_{ \mathcal{V}}^{(l-1)}+\gamma_{U}\mathbf{X}_{\mathcal{V}}^{(0)}\Big{)},$ (4a) $\displaystyle\mathbf{X}_{\mathcal{V}}^{(l)}$ $\displaystyle=\sigma\Big{(}(1-\!\gamma_{D}\!)\mathbf{D}_{\mathcal{H}^{\mathcal {V}}}^{-1/2}\mathbf{H}\!\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}^{-1}\mathbf{H}^ {\top}\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1/2}\mathbf{X}_{\mathcal{V}}^{( l-1)}\!+\!\gamma_{D}\mathbf{X}_{\mathcal{V}}^{(0)}\Big{)},$ (4b) $\displaystyle\mathbf{X}_{\mathcal{V}}^{(l)}$ $\displaystyle=\sigma\Bigg{(}\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1}\mathbf {H}\sigma\Big{(}\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top} \sigma(\mathbf{X}_{\mathcal{V}}^{(l-1)})\Big{)}\Bigg{)},$ (4c) $\displaystyle\mathbf{X}_{\mathcal{V}}^{(l)}$ $\displaystyle=\sigma\Bigg{[}(1-\gamma_{E})\sigma\Bigg{(}\mathbf{D}^{-1}_{ \mathcal{H}^{\mathcal{V}}}\mathbf{H}\sigma\Big{(}\mathbf{D}^{-1}_{\mathcal{H}^ {\mathcal{E}}}\mathbf{H}^{\top}\sigma(\mathbf{X}_{\mathcal{V}}^{(l-1)})\Big{)} \Bigg{)}+\gamma_{E}\mathbf{X}_{\mathcal{V}}^{(0)}\Bigg{]},$ (4d) $$

where Eq. (4a), Eq. (4b), Eq. (4c) and Eq. (4d) represent the simplified formulation of UniGCNII, Deep-HGNN, AllDeepSets and ED-HNN, respectively. Notably, $\sigma(\cdot)$ only denotes a general non-linear function without parameters, not implying that these formulas use the same function.

Linearisation. Inspired by the insights from Subsection 3.2, we further remove the non-linearity from the functions represented by Eqs. (4a$\sim$4d), thereby transforming an MP-Module with $L$ feature aggregation functions into a more efficient single propagation step. For the sake of brevity, we present the simplification results as Proposition 3.2 and provide its mathematical proof in Appendix D.

###### Proposition 3.2 .

Let $\mathbf{X}_{\mathcal{V}}$ be input node features, $\hat{\mathbf{X}}_{\mathcal{V}}$ be the output of the MP-Module of an UniGCNII/AllDeepSets/ED-HNN/Deep-HGNN with $L$ MP layers, and $\alpha\!\in\![\!0,1\!)$. Assume that learnable parameters and the non-linearity are removed from the module. For any hypergraph $\mathcal{H}\!=\!\!\{\!\mathcal{V},\!\mathcal{E},\!\mathbf{H}\!\}$, there exists a clique expansion $\mathcal{G}\!=\!\{\!\mathcal{V},\mathbf{W}\!\}$ to make $\hat{\mathbf{X}}_{\mathcal{V}}\!=\!\Big{(}(1\!-\!\alpha)^{L}\!\mathbf{W}^{L}\!
+\!\alpha\!\sum_{l=0}^{L-1}(1\!-\!\alpha)^{l}\!\mathbf{W}^{l}\Big{)}\mathbf{X}
_{\mathcal{V}}$.

Based on Proposition 3.2, we design the formulation of our TF-MP-Module as the following equation:

$$ $\hat{\mathbf{X}}=\mathbf{S}\mathbf{X}_{\mathcal{V}},$ $$

where $\mathbf{S}=(1-\alpha)^{L}\mathbf{W}^{L}+\alpha\sum_{l=0}^{L-1}(1-\alpha)^{l}
\mathbf{W}^{l}$. Intuitively, in this formula, a larger $\alpha$ value emphasizes the retention of the node’s inherent information, while a smaller $\alpha$ value increases the influence of information from neighbouring nodes. We define that the $L$ used to compute $\mathbf{S}$ is the number of layers of an TF-MP-Module. To understand the behaviour of an $L$-layer TF-MP-Module, we present the Proposition 3.3. For conciseness, we prove it in Appendix E.

###### Proposition 3.3 .

Let $\mathcal{H}=(\mathcal{V},\mathcal{E},\mathbf{H})$ be a hypergraph and $\mathcal{G}=(\mathcal{V},\mathbf{W})$ be its clique expansion with self-loops. Assume that $\alpha\in[0,1)$, and $\mathbf{S}\!=\!(1-\alpha)^{L}\mathbf{W}^{L}+\alpha\sum_{l=0}^{L-1}(1-\alpha)^{
l}\mathbf{W}^{l}$. Then, for $i\neq j$, we have $\mathbf{S}_{ij}>0$ if and only if $v_{i}$ is an $L$-hop neighbour of $v_{j}$ on $\mathcal{H}$, and $\mathbf{S}_{ij}=0$ otherwise.

By Proposition 3.3, an $L$-layer TF-MP-Module enables the information exchange between any node $v_{i}$ and its $L$-hop neighbours on the hypergraph. In the next paragraph, we detail the generation of $\mathbf{S}$.

Operator design. To use the TF-MP-Module in our TF-HNN, the key is designing $\mathbf{W}$ to generate $\mathbf{S}$. We achieve this using a hyperedge-size-based edge weight that is defined as:

$$ $\mathbf{W}_{{H}_{ij}}=\sum_{k=1}^{m}\frac{\delta(v_{i},v_{j},e_{k})}{\mathbf{D }_{\mathcal{H}^{\mathcal{E}}_{kk}}},$ (5) $$

where $\delta(\cdot)$ is a function that returns 1 if $e_{k}$ connects $v_{i}$ and $v_{j}$ and returns 0 otherwise. The main intuition behind the design of this edge weight is that, under certain conditions, the value of $\mathbf{W}_{H_{ij}}$ is positively correlated with the probability of nodes $v_{i}$ and $v_{j}$ having the same label. Consequently, this edge weight design can make the weighted clique expansion fit the homophily assumption, which is that connected nodes tend to be similar to each other (McPherson et al., ). We theoretically elaborate on this intuition in Appendix F. Furthermore, inspired by previous works in the graph domain (Wu et al., ), we generate $\mathbf{S}$ based on a symmetrically normalised $\mathbf{W}_{H}$ with the self-loop. Specifically, we generate the $\mathbf{S}$ by:

$$ $\mathbf{S}=(1-\alpha)^{L}(\tilde{\mathbf{D}}_{H}^{-1/2}\tilde{\mathbf{W}}_{H} \tilde{\mathbf{D}}_{H}^{-1/2})^{L}+\alpha\sum_{l=0}^{L-1}(1-\alpha)^{l}(\tilde {\mathbf{D}}_{H}^{-1/2}\tilde{\mathbf{W}}_{H}\tilde{\mathbf{D}}_{H}^{-1/2})^{l},$ (6) $$

where $\tilde{\mathbf{W}}_{H}=\mathbf{W}_{H}+\mathbf{I}_{n}$ and $\tilde{\mathbf{D}}_{H}\in\mathbb{R}^{n\times n}$ is the diagonal node degree matrix of $\tilde{\mathbf{W}}_{H}$. In the next section, we present the theoretical analysis to support the efficiency and effectiveness of our TF-HNN.

###### Proposition 3.2 .

###### Proposition 3.3 .

## 4 Theoretical Analysis

In this section, we begin by showing the efficiency of the proposed TF-HNN by comparing its training complexity with the traditional HNNs. Additionally, we demonstrate the effectiveness of our TF-HNN by analyzing its information utilisation and its robustness against the oversmoothing issue.

Training complexity. As illustrated by Eq. (1), the TF-HNN in downstream tasks requires training only the task-specific module. Additionally, since the TF-HNN relies on a pre-defined propagation operator, its computations can be handled during data pre-processing. Consequently, the training complexity of the TF-HNN is determined solely by the task-specific module. In contrast, on the basis of Eq. (2), the HNNs involve training both the MP-Module and the task-specific module. The MP-Module in HNNs includes complex learnable operations that facilitate message passing between connected nodes on a hypergraph, inherently making the training of the HNNs more complex than the TF-HNN. See Figure 1 for an example that qualitatively compares the training pipelines of the TF-HNN and HNN in node classification. To provide a quantitative comparison, we summarize the training computational complexity of the TF-HNN and the four state-of-the-art HNNs in Table 1. This table shows that the training complexity of the TF-HNN is consistently lower than that of the HNN, theoretically confirming the training efficiency of the proposed TF-HNN.

Information utilisation. The core strength of existing HNNs lies in their ability to utilise the information embedded within hypergraph structures to generate powerful node features for downstream tasks. Consequently, we analyse the effectiveness of our proposed TF-HNN by comparing its information utilisation capabilities with those of existing HNNs and conclude that our methods utilise the same amount of information as exising HNNs without any loss, which is summarised in Proposition 4.1. For conciseness, we show the detailed proof for Proposition 4.1 in Appendix G.

###### Proposition 4.1 .

Let $H_{0}^{v^{L}_{i}}$ be the entropy of information used by an HNN with $L$ feature aggregation functions in generating the features of a node $v_{i}$, $H_{1}^{v^{L}_{i}}$ be the entropy of information used by an TF-HNN with an $L$-layer TF-MP-Module for the same purpose, and $H_{2}^{v^{L}_{i}}$ denote the entropy of information within node $v_{i}$ and its $L$-hop neighbours on the hypergraph. Then, $H_{0}^{v^{L}_{i}}=H_{1}^{v^{L}_{i}}=H_{2}^{v^{L}_{i}}$.

This proposition theoretically demonstrates that TF-HNN is as effective as existing HNNs in utilising information from pre-defined hypergraphs to generate node features. Specifically, both TF-HNN and existing HNNs achieve this by aggregating neighbourhood information.

Robustness against the oversmoothing issue. According to Proposition 3.3, TF-HNN can use global interactions within the given hypergraph to generate node features by deepening the TF-MP-Module. However, as shown in a recent study (Chen et al., ), the message-passing-based models on hypergraphs may suffer from the oversmoothing issue. This issue refers to the tendency of a model to produce indistinguishable features for nodes in different classes as the model depth increases, which consequently degrades the performance of the framework in downstream tasks. Consequently, to further support the effectiveness of TF-HNN, we theoretically analyse its robustness against the oversmoothing issue. Inspired by previous works in the graph domain (Eswaran et al., ; Yang et al., ; Zhang et al., ), we show Proposition 4.2, with the proof provided in Appendix H.

###### Proposition 4.2 .

Let $\mathcal{H}=\{\mathcal{V},\mathcal{E},\mathbf{H}\}$ denote a hypergraph, $\mathcal{G}=\{\mathcal{V},\mathbf{W}_{H}\}$ be its clique expansion with edge weights computed by Eq. (5), $\mathbf{L}$ be the graph Laplacian matrix of $\mathcal{G}$ computed by a symmetrically normalised and self-loops added $\mathbf{W}_{H}$, $\mathbf{X}_{\mathcal{V}}$ represent the input node features, and $\alpha\in[0,1)$ be a hyperparameter. We define that $F(\mathbf{X})=\operatorname{tr}(\mathbf{X}^{\top}\mathbf{L}\mathbf{X})+\frac{
\alpha}{1-\alpha}\operatorname{tr}[(\mathbf{X}-\mathbf{X}_{\mathcal{V}})^{\top
}(\mathbf{X}-\mathbf{X}_{\mathcal{V}})]$, and $F_{min}$ as the global minimal value of $F(\mathbf{X})$ for $\mathbf{X}\in\mathbb{R}^{n\times d}$. Assume that: 1) $\hat{\mathbf{X}}_{\mathcal{V}}=\mathbf{S}\mathbf{X}_{\mathcal{V}}$; 2) $\mathbf{S}$ is computed by Eq. (6); 3) $\alpha>0$; and 4) $L\to+\infty$. Then, $F(\hat{\mathbf{X}}_{\mathcal{V}})=F_{min}$.

Notably, minimising the first term of $F(\mathbf{X})$ enhances feature similarity among connected nodes on $\mathcal{H}$, and minimising the second term of $F(\mathbf{X})$ encourages the generated features of each node to retain the distinct information from its input features. Therefore, node features that can minimise $F(\mathbf{X})$ capture two key properties: 1) neighbouring nodes on $\mathcal{H}$ have similar features; and 2) each node contains unique information reflecting its individual input characteristics.

Based on Proposition 4.2, a deep TF-MP-Module tends to assign similar features to connected nodes on a hypergraph while maintaining the unique input information of each node. Hence, even with an extremely large number of layers, the TF-MP-Module can preserve the unique input information of each node, making it robust against the oversmoothing issue. This property further ensures the effectiveness of our TF-HNN. Section 6 empirically supports the analysis presented in this section.

###### Proposition 4.1 .

###### Proposition 4.2 .

## 5 Related Work

Recent efforts have leveraged hypergraphs to improve node feature learning for downstream tasks (Bick et al., ). The predominant models in this area are hypergraph neural networks (HNNs) with a message passing module that enables information exchange between connected nodes (Antelmi et al., ). However, as noted for graph counterparts (Wu et al., ; Frasca et al., ) for their graph counterparts, HNNs suffer from low learning efficiency. Two recent works attempt to mitigate this limitation (Feng et al., ; Tang et al., ). In Feng et al. (), an HNN named HGNN (Feng et al., ) is distilled into an MLP during training; while in (Tang et al., ), hypergraph structural information is integrated into an MLP via a loss function computed by the sum of the maximum node feature distance in each hyperedge. These approaches primarily focus on reducing model inference complexity via a dedicated training process. However, these designs still lead to hypergraph machine learning models with computationally intensive training procedures. For instance, the distillation process in (Feng et al., ) necessitates training a HNN with a computational complexity of $\mathcal{O}(M+TLm^{\prime}d+TLnd^{2})$, where with $M$ is the training computational complexity of the task-specific module, $n$ is the node count, $m^{\prime}$ is the number of edges in the clique expansion, $L$ is the number of layers, and $d$ is the feature dimension. Given that many existing models are applied to semi-supervised hypergraph node classification settings, which involve the simultaneous use of training and test data (Huang & Yang, ; Chien et al., ; Wang et al., ; ; Joachims et al., ; Chapelle et al., ), reducing the training complexity of hypergraph-based models remains a critical yet underexplored challenge. We address this challenge with a novel model named TF-HNN, which is the first to decouple the processing of hypergraph structural information from model learning. This decoupling significantly improves training efficiency.

**Table 2: Dataset statistics. More details of these datasets are in Appendix I.**
| Name | # Nodes | # Hyperedges | # Features | # Classes | Node Definition | Hyperedge Definition |
| --- | --- | --- | --- | --- | --- | --- |
| Cora-CA | 2708 | 1072 | 1433 | 7 | Paper | Co-authorship |
| DBLP-CA | 41302 | 22363 | 1425 | 6 | Paper | Co-authorship |
| Citeseer | 3312 | 1079 | 3703 | 6 | Paper | Co-citation |
| Congress | 1718 | 83105 | 100 | 2 | Congressperson | Legislative-Bills-Sponsorship |
| House | 1290 | 340 | 100 | 2 | Representative | Committee |
| Senate | 282 | 315 | 100 | 2 | Congressperson | Legislative-Bills-Sponsorship |
| Trivago | 172738 | 233202 | 300 | 160 | Accommodation | Browsing Session |

## 6 Experiments

### 6.1 Experimental Setup

Task description. We conduct experiments in two tasks: hypergraph node classification and hyperlink prediction. In hypergraph node classification (Wang et al., ; ; Duta et al., ), we are given the hypergraph structure $\mathbf{H}$, node features $\mathbf{X}_{\mathcal{V}}$, and a set of labelled nodes with ground truth labels $\mathbf{Y}_{lab}=\{\mathbf{y}_{v}\}_{v\in\mathcal{V}_{lab}}$, where $\mathbf{y}_{v_{i}}\in\{0,1\}^{c}$ be a one-hot label. Our objective is to classify the unlabeled nodes within the hypergraph. In hyperlink prediction (Chen & Liu, ), we are given node features $\mathbf{X}_{\mathcal{V}}$, a set of observed real and fake hyperedges $\mathcal{E}_{ob}$, and a set of potential hyperedges $\mathcal{E}_{p}$. The task requires the model to distinguish between real and fake hyperedges in $\mathcal{E}_{p}$ using $\mathbf{X}_{\mathcal{V}}$ and $\mathcal{E}_{ob}$.

Dataset and baseline. We conduct experiments on seven real-world hypergraphs: Cora-CA, DBLP-CA, Citeseer, Congress, House, Senate, and Trivago. We use nine HNNs as baselines. Five of these baselines utilize direct feature aggregation, including HGNN (Feng et al., ), HCHA (Bai et al., ), UniGCNII (Huang & Yang, ), PhenomNN (Wang et al., ), and Deep-HGNN (Chen et al., ). The remaining four baselines employ indirect feature aggregation, consisting of HNHN (Dong et al., ), AllDeepSets (Chien et al., ), AllSetTransformer (Chien et al., ), and ED-HNN (Wang et al., ). The dataset statistics are summarized in Table 2.

Metric. Following previous works (Wang et al., ; Chen & Liu, ), we evaluate models for node classification and hyperlink prediction with classification accuracy and the Area Under the ROC Curve (AUC), respectively. Similar to , we employ the relative training time $r_{f}$, defined as $r_{f}=t_{f}/t_{S}$ to evaluate the efficiency of the models, where $t_{f}$ is the training time to achieve optimal performance for the evaluated model, and $t_{s}$ is the training time for optimal performance of TF-HNN.

Implementation. For node classification, we follow previous works (Wang et al., ; Duta et al., ) to use a 50%/25%/25% train/validation/test data split and adapt the baseline classification accuracy from them(^3^33Since there are no reported results for Deep-HGNN under the chosen data split, we use our own results.). Additionally, similar to these works, we implement the classifier based on MLP for our TF-HNN and report the results from ten runs. For hyperlink prediction, existing works (Chen & Liu, ) primarily focus on the design of the prediction head, with no reported results for applying our baseline HNNs to this task; so we report all results from five runs conducted by ourselves. We employ the deep set function implemented by (Chien et al., ) as the prediction head for our method and baseline methods due to its simplicity. We use a 50%/25%/25% train/validation/test data split and ensure each split contains five times as many fake hyperedges as real hyperedges. TF-HNN and HNNs only use real hyperedges in training and validation sets to do message passing. All experiments were on an RTX 3090 with PyTorch, and our code with running commands is at https://anonymous.4open.science/r/TF-HNN. More details are in Appendix L.

### 6.2 Comparison with Baselines

**Table 3: Node classification accuracy (%) for HNNs and TF-HNN. The best result on each dataset is highlighted in bold font. The second and third highest accuracies are marked with an underline.**
|  | Cora-CA | DBLP-CA | Citeseer | Congress | House | Senate | Avg. Mean |
| --- | --- | --- | --- | --- | --- | --- | --- |
| HGNN | 82.64 ± 1.65 | 91.03 ± 0.20 | 72.45 ± 1.16 | 91.26 ± 1.15 | 61.39 ± 2.96 | 48.59 ± 4.52 | 74.56 |
| HCHA | 82.55 ± 0.97 | 90.92 ± 0.22 | 72.42 ± 1.42 | 90.43 ± 1.20 | 61.36 ± 2.53 | 48.62 ± 4.41 | 74.38 |
| HNHN | 77.19 ± 1.49 | 86.78 ± 0.29 | 72.64 ± 1.57 | 53.35 ± 1.45 | 67.80 ± 2.59 | 50.93 ± 6.33 | 68.12 |
| UniGCNII | 83.60 ± 1.14 | 91.69 ± 0.19 | 73.05 ± 2.21 | 94.81 ± 0.81 | 67.25 ± 2.57 | 49.30 ± 4.25 | 76.62 |
| AllDeepSets | 81.97 ± 1.50 | 91.27 ± 0.27 | 70.83 ± 1.63 | 91.80 ± 1.53 | 67.82 ± 2.40 | 48.17 ± 5.67 | 75.31 |
| AllSetTransformer | 83.63 ± 1.47 | 91.53 ± 0.23 | 73.08 ± 1.20 | 92.16 ± 1.05 | 51.83 ± 5.22 | 69.33 ± 2.20 | 76.93 |
| PhenomNN | 85.81 ± 0.90 | 91.91 ± 0.21 | 75.10 ± 1.59 | 88.24 ± 1.47 | 70.71 ± 2.35 | 67.70 ± 5.24 | 79.91 |
| ED-HNN | 83.97 ± 1.55 | 91.90 ± 0.19 | 73.70 ± 1.38 | 95.00 ± 0.99 | 72.45 ± 2.28 | 64.79 ± 5.14 | 80.30 |
| Deep-HGNN | 84.89 ± 0.88 | 91.76 ± 0.28 | 74.07 ± 1.64 | 93.91 ± 1.18 | 75.26 ± 1.76 | 68.39 ± 4.79 | 81.38 |
| TF-HNN (ours) | 86.54 ± 1.32 | 91.80 ± 0.30 | 74.82 ± 1.67 | 95.09 ± 0.89 | 76.29 ± 1.99 | 70.42 ± 2.74 | 82.50 |

Figure: Figure 2: The relative training time required for HNNs and TF-HNN to achieve optimal performance.
Refer to caption: x1.png

Figure: (a) Cora-CA.
Refer to caption: x2.png

Node classification. We summarize the classification accuracy and relative training time of the TF-HNN and HNNs in Table 3 and Figure 2, respectively. Table 3 demonstrates that TF-HNN not only leads to the best results across four datasets (Cora-CA, Congress, House, and Senate) but also results in the highest average mean accuracy overall. These findings confirm the effectiveness of TF-HNN in generating powerful node features for node classification. As discussed in Section 4, we attribute this effectiveness to the ability of TF-HNN to match the power of existing HNNs in utilising neighbourhood information to generate node features, while requiring the optimization of fewer parameters. This makes the TF-HNN more efficient in utilising training data and reduces the risk of overfitting, thereby outperforming more complex HNN counterparts. On the other hand, as shown in Figure 2, the TF-HNN consistently needs less training time to achieve superior performance compared to the HNNs. This highlights the high training efficiency of TF-HNN in downstream tasks.

Hyperlink prediction. Figure 3 shows the results for hyperlink prediction, which exhibit a pattern similar to that observed in the hypergraph node classification task: the TF-HNN outperforms the HNNs while requiring less training time. An interesting observation is that the training efficiency improvement from TF-HNN is smaller in the hyperlink prediction task compared to the node classification task. The intuition is that the task-specific module used for hyperlink prediction is more computationally expensive than the one used for hypergraph node classification. Specifically, the hyperlink predictor needs to generate a prediction for each hyperedge by aggregating the features of the nodes connected by the hyperedge, whereas the node classifier only needs to perform forward propagation for each individual node. Appendix M shows that the efficiency improvement provided by TF-HNN is inversely correlated with the complexity of the task-specific module.

**Table 4: Node classification accuracy (%) and training time for HNNs and TF-HNN on Trivago.**
| Metric / Methods | Deep-HGNN | ED-HNN | TF-HNN (Ours) |
| --- | --- | --- | --- |
| Accuracy (%) | 84.06 ± 1.70 | 48.38 ± 1.35 | 94.03 ± 0.51 |
| Training Time (s) | 3352.35 ± 1039.83 | 143.92 ± 34.64 | 38.09 ± 6.65 |

Figure: (a) Cora-CA.
Refer to caption: x5.png

**Table 6: The impact from the TF-MP-Module of TF-HNN in node classification. The best result on each dataset is highlighted in bold font.**
|  | Cora-CA | Citeseer | Senate | House | Avg. Mean |
| --- | --- | --- | --- | --- | --- |
| SGC with CE | 81.60 ± 1.68 | 71.42 ± 2.15 | 47.18 ± 6.48 | 56.16 ± 3.21 | 64.09 |
| SIGN with CE | 86.04 ± 1.53 | 73.62 ± 1.79 | 64.93 ± 3.15 | 73.75 ± 1.91 | 74.59 |
| TF-MP-Module (ours) | 86.54 ± 1.32 | 74.82 ± 1.67 | 70.42 ± 2.74 | 76.29 ± 1.99 | 77.02 |

**Table 7: The impact from the weighted $\mathbf{S}$ of TF-HNN in node classification. The best result on each dataset is highlighted in bold font.**
|  | Cora-CA | Citeseer | Senate | House | Avg. Mean |
| --- | --- | --- | --- | --- | --- |
| TF-HNN with Weighted $\mathbf{S}$ | 84.90 ± 1.55 | 74.75 ± 1.62 | 68.87 ± 4.57 | 75.57 ± 1.81 | 76.02 |
| TF-HNN without Weighted $\mathbf{S}$ | 86.54 ± 1.32 | 74.82 ± 1.67 | 70.42 ± 2.74 | 76.29 ± 1.99 | 77.02 |

Scalability of state-of-the-art models. We compare the scalability of our TF-HNN against the top two methods from previous experiments, Deep-HGNN and ED-HNN. Deep-HGNN uses a direct feature aggregation function, while ED-HNN uses an indirect feature aggregation function. We evaluate the performance of all methods on node classification and hyperlink prediction using the large-scale hypergraph benchmark, Trivago, with the results detailed in Table 5 and Table 5, respectively. The results show that TF-HNN not only surpasses its competitors in performance but also dramatically reduces training time. Remarkably, in node classification, TF-HNN improves the accuracy by $10\%$ over the best baseline, Deep-HGNN, while using only $1\%$ of its training time. These findings underscore the scalability and value of TF-HNN for large-scale hypergraph applications.(^4^44On Trivago, Deep-HGNN’s training time is much higher for node classification than hyperlink prediction. This is because a 64-layer model largely improves node classification accuracy, while a 2-layer model suffices for hyperlink prediction, as deeper models offer no benefit. The extra layers account for the longer training time.)

### 6.3 Analysis

We conduct a series of analyses on the components within the proposed TF-HNN model in node classification on the Cora-CA, Citeseer, House, and Senate datasets.

Impacts from the TF-MP-Module. In this study, we evaluate the effectiveness of our proposed TF-MP-Module within the TF-HNN framework by comparing it to two training-free message passing methods developed for graphs: SGC (Wu et al., ) and SIGN (Frasca et al., ). To adapt these methods to hypergraphs, we first project the given hypergraph into its clique expansion (CE), as introduced in Section 2, and then apply SGC and SIGN to this projected structure. Notably, we only replace the TF-MP-Module with SGC using CE and SIGN with CE, while keeping all other components of the TF-HNN unchanged. Table 6 show that our TF-MP-Module consistently outperforms the alternatives across all datasets. The superior performance of our method over SGC with CE is due to SGC with CE being a special case of our TF-MP-Module with $\alpha=0$, which, as discussed in Proposition 4.2, can cause node features to become overly similar to their neighbours, reducing distinctiveness. For SIGN with CE, the method concatenates features from different neighbor hops with the initial node features and processes them through a classifier, forcing it to handle both feature merging and classification, which increases complexity and can hinder convergence. In contrast, our TF-MP-Module directly merges neighbouring node information using the operator in Eq. (6), allowing the classifier to focus solely on classification, thereby simplifying the learning process and improving performance. These findings highlight the greater capacity of our TF-MP-Module compared to existing graph-based training-free modules in processing hypergraph structures, further underscoring its unique value for hypergraph machine learning.

Impacts from hyperparameters of TF-MP-Module. We focus on two key hyperparameters of the TF-MP-Module: the number of layers and the $\alpha$. As illustrated in Figure 4, models with $\alpha>0$ exhibit more consistent accuracy as the number of layers increases compared to models with $\alpha=0$. We attribute this consistency to the robustness discussed in Proposition 4.2, where a positive $\alpha$ helps retain the distinct information of each node, thereby mitigating the oversmoothing issue. Additionally, we observe that a positive $\alpha$ significantly enhances performance in the House and Senate datasets, which are more heterophilic, compared to the Cora-CA and Citeseer datasets. This suggests that retaining the distinct information of each node is more beneficial for heterophilic datasets.

Impacts from the weighted $\mathbf{S}$. We define $\mathbf{S}$ based on Eq. (5) as weighted $\mathbf{S}$. In contrast, in this experiment, for the framework without weighted $\mathbf{S}$, $\mathbf{S}$ is generated using an adjacency matrix $\mathbf{W}$ of a clique expansion with all positive values set to one. As shown in Table 7, adding weighted $\mathbf{S}$ further enhances performance, highlighting its informativeness.

## 7 Conclusion

In this paper, we propose a novel model called TF-HNN. The key innovation of TF-HNN is an original training-free message passing module (TF-MP-Module) tailored for data on hypergraphs. We present both theoretical and empirical evidence demonstrating that TF-HNN can efficiently and effectively address hypergraph-related downstream tasks. To our knowledge, TF-HNN is the first model to shift the integration of hypergraph structural information from the model learning stage to the data pre-processing stage, significantly enhancing training efficiency. The proposed TF-HNN can advance the field of hypergraph machine learning research by not only improving the efficiency of using observed hypergraph structures to solve downstream tasks, but also serving as a simple starting point for the development of future hypergraph machine learning models.

Limitation and future work. Learning tasks on hypergraphs can generally be defined at the node, hyperedge, and hypergraph levels. In this paper, we empirically test our TF-HNN on node-level and hyperedge-level tasks, with the main limitation being the lack of empirical results on hypergraph-level tasks. To our knowledge, designing task-specific modules to generate hypergraph-level features based on existing node features itself remains a challenging topic in the literature. Therefore, we consider the application of our TF-HNN to hypergraph-level tasks as future work. Another future direction of our work is to design a training-free module for explicit hyperedge-node importance learning.

## Appendix A Impact Statements

This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.

## Appendix B Background of Hypergraph Message Passing

Clique expansion. Given a hypergraph $\mathcal{H}=\{\mathcal{V},\mathcal{E},\mathbf{H}\}$, its clique expansion is defined as a graph $\mathcal{G}=\{\mathcal{V},\mathbf{W}\}$, where $\mathcal{V}$ remains unchanged, and $\mathbf{W}_{ij}>0$ if and only if $v_{i}$ and $v_{j}$ are connected by a hyperedge on $\mathcal{H}$ and $\mathbf{W}_{ij}=0$ otherwise. Hence, each hyperedge in $\mathcal{H}$ is a clique in $\mathcal{G}$.

Star expansion. Given a hypergraph $\mathcal{H}\!=\!\{\mathcal{V},\mathcal{E},\mathbf{H}\}$, its star expansion is defined as a bipartite graph $\mathcal{G}=\{\mathcal{V}\bigcup\mathcal{V}^{\prime},\mathbf{W}\}$, where $v_{i}\in\mathcal{V}$ is a node in $\mathcal{H}$, hypernode $v_{e_{j}}\in\mathcal{V}^{\prime}$ corresponds to the hyperedge $e_{j}$ in $\mathcal{H}$, and there is an edge between $v_{i}$ and $v_{e_{j}}$ if and only if $e_{j}$ contains $v_{i}$ in $\mathcal{H}$.

Line expansion. Given a hypergraph $\mathcal{H}\!=\!\{\mathcal{V},\mathcal{E},\mathbf{H}\}$, its line expansion is defined as a graph $\mathcal{G}=(\mathcal{V}_{l},\mathbf{W})$. The node set $\mathcal{V}_{l}$ of $\mathcal{G}$ is defined by vertex-hyperedge pair $\{(v,e)\mid v\in e,v\in\mathcal{V},e\in\mathcal{E}\}$ from the original hypergraph. The adjacency matrix $\mathbf{W}\in\{0,1\}^{|\mathcal{V}_{l}|\times|\mathcal{V}_{l}|}$ is defined by pairwise relation with $\mathbf{W}(u_{l},v_{l})=1$ if either $v=v^{\prime}$ or $e=e^{\prime}$ for $u_{l}=(v,e),v_{l}=(v^{\prime},e^{\prime})\in\mathcal{V}_{l}$. In the following sections, we refer the nodes on a line expansion as line-nodes. Notably, a node $v_{i}$ on the original hypergraph corresponds to multiple line-nodes, each representing a vertex-hyperedge pair involving $v_{i}$. These line-nodes are interconnected and also connect to line-nodes corresponding to the 1-hop neighbors of $v_{i}$ on the given hypergraph.

Incidence tensor. A $k$-uniform hypergraph can be represented by a $k$-dimensional supersymmetric tensor such that for all distinct node sets $\{v_{1},\ldots,v_{k}\}\in\mathcal{V}$, $\mathbf{T}_{i_{1},...,i_{d}}=\frac{1}{(k-2)!}$ if hyperedge $e=\{v_{1},...,v_{k}\}\in\mathcal{E}$, and $\mathbf{T}_{i_{1},...,i_{d}}=0$ otherwise.

## Appendix C Proof of Proposition 3.1

###### Proof.

In this section, we provide proofs for the clique-based, star-based, tensor-based and line-based methods respectively.

For the clique-based approaches,there are two specific designs for this type of approaches. The first design is based on the hypergraph convolution operator (Feng et al., ; Yadati et al., ; Bai et al., ; Duta et al., ; Chen et al., ), which is derived from the hypergraph Laplacian (Agarwal et al., ). The second design (Wang et al., ) relies on an optimization algorithm that minimizes a handcrafted hypergraph energy function. Despite the differences in their designs, these methods start by using an MLP to project the given node features to the latent space, namely, for $i=0$, we have $\mathbf{x}_{v_{i}}^{(0)}=\operatorname{MLP}(\mathbf{x}_{v_{i}})$. Moreover, these methods are analogous to using matrix multiplication between the adjacency matrix of a clique expansion, denoted as $\mathbf{W}$, and the node features $\mathbf{X}_{\mathcal{V}}$ to facilitate information exchange among connected nodes on hypergraphs. Therefore, for $i\in\mathbb{Z}^{+}$, the node-wise form of this type of methods can be summarized as:

$$ $\mathbf{x}_{v_{i}}^{(l)}=f_{3}\Big{(}f_{0}(\mathbf{x}_{v_{i}}^{(0)})+f_{1}( \mathbf{x}_{v_{i}}^{(l-1)})+\sum_{v_{j}\in\mathcal{N}_{\mathcal{G}_{v_{i}}}}f_ {2}(\mathbf{x}^{(l-1)}_{v_{j}})\Big{)},$ (7) $$

where $f_{0}(\cdot)$, $f_{1}(\cdot)$, $f_{2}(\cdot)$, $f_{3}(\cdot)$ represent three learnable functions, and $\mathcal{N}_{\mathcal{H}_{v_{i}}}$ is the set of $1$-hop neighbours of $v_{i}$ on the hypergraph. Here the right-hand side of Eq. (7) is a special case of $\phi_{\mathbf{\Theta}}(\mathbf{x}_{v_{i}}^{(0)},\mathbf{x}_{v_{i}}^{(l-1)},
\oplus_{v_{j}\in\mathcal{N}_{\mathcal{H}_{v_{i}}}}\mathbf{x}_{v_{j}}^{(l-1)})$.

For the star-based approaches, three primary designs exist for this type of approach. Despite the differences in their designs, these methods start by using an MLP to project the given node features to the latent space, namely, for $i=0$, we have $\mathbf{x}_{v_{i}}^{(0)}=\operatorname{MLP}(\mathbf{x}_{v_{i}})$. Moreover, the first type of methods, as presented in (Huang & Yang, ), directly applies existing GNNs to a star expansion graph, initializing the features of hypernode $v_{e_{j}}$ as the mean of the features of nodes connected by the hyperedge $e_{j}$. Due to the linear generation of hypernode features, for $i\in\mathbb{Z}^{+}$, the node-wise form of this method can be expressed as Eq.(7). Additionally, (Chien et al., ) integrates set functions with star expansion to create HNNs, while (Wang et al., ) develops a model by studying the information diffusion process on a hypergraph using star expansion. The node-wise forms of the models in (Chien et al., ) and (Wang et al., ) can be summarized as:

$$ $\mathbf{x}_{v_{i}}^{(l)}=g_{2}\Big{(}g_{0}(\mathbf{x}_{v_{i}}^{(0)})+\mathbf{p }^{\top}g_{1}(\mathbf{x}_{v_{i}}^{(l-1)},\oplus_{v_{j}\in\mathcal{N}_{\mathcal {H}_{v_{i}}}}\mathbf{x}_{v_{j}}^{(l-1)})\Big{)},$ (8) $$

$g_{0}(\cdot)$, $g_{1}(\cdot)$, $g_{2}(\cdot)$ represent three learnable functions, $\mathcal{N}_{\mathcal{H}_{v_{i}}}$ is the set of $1$-hop neighbours of $v_{i}$ on the hypergraph, and $\mathbf{p}\in\mathbb{R}^{m_{v_{i}}}$ is a vector used to aggregate the features of hypernodes generated by $g_{1}(\cdot)$. Here, $m_{v_{i}}$ is the number of hyperedges containing $v_{i}$. Here the right-hand side of Eq. (8) is a special case of $\phi_{\mathbf{\Theta}}(\mathbf{x}_{v_{i}}^{(0)},\mathbf{x}_{v_{i}}^{(l-1)},
\oplus_{v_{j}\in\mathcal{N}_{\mathcal{H}_{v_{i}}}}\mathbf{x}_{v_{j}}^{(l-1)})$.

For tensor-based approaches, these approaches facilitate message passing among connected nodes using the incidence tensor. These methods start by using an MLP to project the given node features to the latent space, namely, for $i=0$, we have $\mathbf{x}_{v_{i}}^{(0)}=\operatorname{MLP}(\mathbf{x}_{v_{i}})$. Moreover, as noted by the Theorem 3.3 of (Chien et al., ), practically, they are implemented by combining set functions with a star expansion. Consequently, for $i\in\mathbb{Z}^{+}$, the node-wise forms of these methods are specific instances of Eq. (8), where the right-hand side is a special case of $\phi_{\mathbf{\Theta}}(\mathbf{x}_{v_{i}}^{(0)},\mathbf{x}_{v_{i}}^{(l-1)},
\oplus_{v_{j}\in\mathcal{N}_{\mathcal{H}_{v_{i}}}}\mathbf{x}_{v_{j}}^{(l-1)})$.

For line-based approaches, these approaches (Yang et al., ) generate node features using hypergraph structural information in four steps. Firstly, these methods start by using an MLP to project the given node features to the latent space, namely, for $i=0$, we have $\mathbf{x}_{v_{i}}^{(0)}=\operatorname{MLP}(\mathbf{x}_{v_{i}})$. Then, the hypergraph is projected onto its corresponding line expansion graph, where the features of each line-node are initialised as the features of its corresponding node on the original hypergraph. Next, a GNN is applied to this line expansion graph to facilitate message passing among connected line-nodes. Finally, the features of each node $v_{i}$ on the hypergraph are generated by aggregating the features of the line-nodes corresponding to $v_{i}$. This method can be summarized as Eq. (8), with three key differences from star-based approaches. Firstly, $\mathcal{N}_{\mathcal{H}_{v_{i}}}$ contains the $m$-hop neighbors of $v_{i}$, which includes all nodes with a path to $v_{i}$. Secondly, the line-based method utilises only one layer for generating node features, with the number of feature aggregation functions usually referring to the number of GNNs used to implement $g_{1}(\cdot)$. As a result, for $i\in\mathbb{Z}^{+}$, we still have $\mathbf{x}_{v_{i}}^{(l)}=\phi_{\mathbf{\Theta}}(\mathbf{x}_{v_{i}}^{(0)},
\mathbf{x}_{v_{i}}^{(l-1)},\oplus_{v_{j}\in\mathcal{N}_{\mathcal{H}_{v_{i}}}}
\mathbf{x}_{v_{j}}^{(l-1)})$.

∎

###### Proof.

## Appendix D Proof of Proposition 3.2

**Table 8: Overview for four state-of-the-art HNNs. In this table, $\gamma_{U},\gamma_{E},\gamma_{D},\gamma^{\prime}_{l}\!\in\!(0,1)$ are hyperparameters and $\mathbf{I}\in\mathbb{R}^{d\times d}$ denotes an identity matrix.**
| Name | Type | Message Passing Function |
| --- | --- | --- |
| UniGCNII (Huang & Yang, ) | Direct Message Passing | $\mathbf{X}_{\mathcal{V}}^{(l)}\!=\!\sigma\Bigg{(}\Big{(}(\!1\!-\!\gamma_{U}\!)<br>\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1/2}\mathbf{H}\!\tilde{\mathbf{D}}_{<br>\mathcal{H}^{\mathcal{E}}}^{-1/2}\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}^{-1}<br>\mathbf{H}^{\top}\mathbf{X}_{\mathcal{V}}^{(l-1)}\!+\!\gamma_{U}\mathbf{X}_{<br>\mathcal{V}}^{(0)}\Big{)}\mathbf{\Theta}^{(l)}\Bigg{)}$ |
| Deep-HGNN (Chen et al., ) | Direct Message Passing | $\mathbf{X}_{\mathcal{V}}^{(l)}\!=\!\sigma\Bigg{(}\Big{(}(\!1\!-\!\gamma_{D}\!)<br>\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1/2}\mathbf{H}\!\mathbf{D}_{\mathcal{<br>H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top}\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^<br>{-1/2}\mathbf{X}_{\mathcal{V}}^{(l)}\!+\!\gamma_{D}\mathbf{X}_{\mathcal{V}}^{(<br>0)}\Big{)}\Big{(}(1-\gamma^{\prime}_{D})\mathbf{I}+\gamma^{\prime}_{D}\mathbf{<br>\Theta}^{(l)}\Big{)}\Bigg{)}$ |
| AllDeepSets (Chien et al., ) | Indirect Message Passing | $\mathbf{X}_{\mathcal{V}}^{(l)}=\operatorname{MLP}\Bigg{(}\mathbf{D}_{\mathcal{<br>H}^{\mathcal{V}}}^{-1}\mathbf{H}\operatorname{MLP}\Big{(}\mathbf{D}_{\mathcal{<br>H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top}\operatorname{MLP}(\mathbf{X}_{\mathcal<br>{V}}^{(l-1)})\Big{)}\Bigg{)}$ |
| ED-HNN (Wang et al., ) | Indirect Message Passing | $\mathbf{X}_{\mathcal{V}}^{(l)}=\operatorname{MLP}\Bigg{[}(1-\gamma_{E})<br>\operatorname{MLP}\Bigg{(}\mathbf{D}^{-1}_{\mathcal{H}^{\mathcal{V}}}\mathbf{H<br>}\operatorname{MLP}\Big{(}\mathbf{D}^{-1}_{\mathcal{H}^{\mathcal{E}}}\mathbf{H<br>}^{\top}\operatorname{MLP}(\mathbf{X}_{\mathcal{V}}^{(l-1)})\Big{)}\Bigg{)}+<br>\gamma_{E}\mathbf{X}_{\mathcal{V}}^{(0)}\Bigg{]}$ |

To recap, the overview of UniGCNII, Deep-HGNN, AllDeepSets, and ED-HNN is in Table 8.

Before proving Proposition 3.2, we first prove the following Lemmas.

###### Lemma D.1 .

Let $\mathbf{H}\in\{0,1\}^{n\times m}$ be an incidence matrix of a hypergraph, $\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}\in\mathbb{R}^{n\times n}$ is a diagonal matrix with node degrees, $\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}\in\mathbb{R}^{m\times m}$ is a diagonal matrix with hyperedge degrees, $\tilde{\mathbf{D}}_{\mathcal{H}_{jj}^{\mathcal{E}}}\in\mathbb{R}^{m\times m}$ is a diagonal matrix with $\tilde{\mathbf{D}}_{\mathcal{H}_{jj}^{\mathcal{E}}}\!=\!\frac{\sum_{i=1}^{n}
\mathbf{H}_{ij}\mathbf{D}_{\mathcal{H}^{\mathcal{V}}_{ii}}}{\mathbf{D}_{
\mathcal{H}_{jj}^{\mathcal{E}}}}$, $\gamma_{U}\in(0,1)$ and $\mathbf{W}_{U}=(1-\gamma_{U})\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1/2}
\mathbf{H}\tilde{\mathbf{D}}_{\mathcal{H}^{\mathcal{E}}}^{-1/2}\mathbf{D}_{
\mathcal{H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top}$. We have: $\mathbf{W}_{U}$ is the adjacency matrix of a clique expansion.

###### Proof.

We have:

$$ $\displaystyle\mathbf{W}_{U_{ij}}=(1-\gamma_{U})\sum_{k=1}^{m}\frac{\mathbf{H}_ {ik}\mathbf{H}_{jk}}{\mathbf{D}_{\mathcal{H}_{ii}^{\mathcal{V}}}^{1/2}\mathbf{ D}_{\mathcal{H}^{\mathcal{E}}_{kk}}\tilde{\mathbf{D}}^{1/2}_{\mathcal{H}^{ \mathcal{E}}_{kk}}}.$ $$

Here $\mathbf{W}_{U_{ij}}>0$ if and only if $v_{i}$ and $v_{j}$ are connected by a hyperedge $e_{k}$ on $\mathcal{H}$, otherwise $\mathbf{W}_{U_{ij}}=0$. Therefore, $\mathbf{W}_{U}$ is the adjacency matrix of a clique expansion.
∎

###### Lemma D.2 .

Let $\mathbf{H}\in\{0,1\}^{n\times m}$ be an incidence matrix of a hypergraph, $\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}\in\mathbb{R}^{n\times n}$ is a diagonal matrix with node degrees, $\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}\in\mathbb{R}^{m\times m}$ is a diagonal matrix with hyperedge degrees, $\gamma_{D}\in(0,1)$ and $\mathbf{W}_{D}=(\!1\!-\!\gamma_{D}\!)\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-
1/2}\mathbf{H}\!\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top}
\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1/2}$. We have: $\mathbf{W}_{D}$ is the adjacency matrix of a clique expansion.

###### Proof.

We have:

$$ $\displaystyle\mathbf{W}_{D_{ij}}=(1-\gamma_{D})\sum_{k=1}^{m}\frac{\mathbf{H}_ {ik}\mathbf{H}_{jk}}{\mathbf{D}^{1/2}_{\mathcal{H}^{\mathcal{V}}_{ii}}\mathbf{ D}^{1/2}_{\mathcal{H}^{\mathcal{V}}_{jj}}\mathbf{D}_{\mathcal{H}^{\mathcal{E}} _{kk}}}.$ $$

Here $\mathbf{W}_{D_{ij}}>0$ if and only if $v_{i}$ and $v_{j}$ are connected by a hyperedge $e_{k}$ on $\mathcal{H}$, otherwise $\mathbf{W}_{D_{ij}}=0$. Therefore, $\mathbf{W}_{D}$ is the adjacency matrix of a clique expansion.
∎

###### Lemma D.3 .

Let $\mathbf{H}\in\{0,1\}^{n\times m}$ be an incidence matrix of a hypergraph, $\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}\in\mathbb{R}^{n\times n}$ be a diagonal matrix with node degrees, and $\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}\in\mathbb{R}^{m\times m}$ be a diagonal matrix with hyperedge degrees. We have: $\mathbf{W}_{S}=\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1}\mathbf{H}\mathbf{D}
_{\mathcal{H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top}$ is the adjacency matrix of a clique expansion.

###### Proof.

We have:

$$ $\displaystyle\mathbf{W}_{S_{ij}}=\sum_{k=1}^{m}\frac{\mathbf{H}_{ik}\mathbf{H} _{jk}}{\mathbf{D}_{\mathcal{H}^{\mathcal{V}}_{ii}}\mathbf{D}_{\mathcal{H}^{ \mathcal{E}}_{kk}}}.$ $$

Here $\mathbf{W}_{S_{ij}}>0$ if and only if $v_{i}$ and $v_{j}$ are connected by a hyperedge on $\mathcal{H}$, otherwise $\mathbf{W}_{S_{ij}}=0$. Therefore, $\mathbf{W}_{S}$ is the adjacency matrix of a clique expansion.
∎

With the Lemmas above, we present the proof of Proposition 3.2 as follows.

###### Proof.

For an UniGNN (Huang & Yang, ) with $L$ feature aggregation functions, devoid non-linear activation and by setting the learnable transformation matrix to the identity matrix, its MP-Module can be reformulated as:

$$ $\mathbf{X}_{\mathcal{V}}^{(L)}=\Big{(}(1-\gamma_{U})^{L}(\mathbf{D}_{\mathcal{ H}^{\mathcal{V}}}^{-1/2}\mathbf{H}\tilde{\mathbf{D}}_{\mathcal{H}^{\mathcal{E} }}^{-1/2}\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top})^{L}+ \gamma_{U}\sum_{l=0}^{L-1}(1-\gamma_{U})^{l}(\mathbf{D}_{\mathcal{H}^{\mathcal {V}}}^{-1/2}\mathbf{H}\tilde{\mathbf{D}}_{\mathcal{H}^{\mathcal{E}}}^{-1/2} \mathbf{D}_{\mathcal{H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top})^{l}\Big{)} \mathbf{X}_{\mathcal{V}}.$ $$

Based on Lemma D.1, this equation is a special case of $\hat{\mathbf{X}}_{\mathcal{V}}\!=\!\Big{(}(1\!-\!\alpha)^{L}\!\mathbf{W}^{L}\!
+\!\alpha\!\sum_{l=0}^{L-1}(1\!-\!\alpha)^{l}\!\mathbf{W}^{l}\Big{)}\mathbf{X}
_{\mathcal{V}}$ with $\hat{\mathbf{X}}_{\mathcal{V}}=\mathbf{X}_{\mathcal{V}}^{(L)}$, $\mathbf{W}=\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1/2}\mathbf{H}\tilde{
\mathbf{D}}_{\mathcal{H}^{\mathcal{E}}}^{-1/2}\mathbf{D}_{\mathcal{H}^{
\mathcal{E}}}^{-1}\mathbf{H}^{\top}$ and $\alpha=\gamma_{U}$.

For a Deep-HGNN (Chen et al., ) with $L$ feature aggregation functions, devoid non-linear activation and by setting the learnable transformation matrix to the identity matrix, its MP-Module can be reformulated as:

$$ $\mathbf{X}_{\mathcal{V}}^{(L)}=\Big{(}(1-\gamma_{D})^{L}(\mathbf{D}_{\mathcal{ H}^{\mathcal{V}}}^{-1/2}\mathbf{H}\!\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}^{-1 }\mathbf{H}^{\top}\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1/2})^{L}+\gamma_{D }\sum_{l=0}^{L-1}(1-\gamma_{D})^{l}(\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1 /2}\mathbf{H}\!\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top} \mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1/2})^{l}\Big{)}\mathbf{X}_{\mathcal{ V}}.$ $$

Based on Lemma D.2, this equation is a special case of $\hat{\mathbf{X}}_{\mathcal{V}}\!=\!\Big{(}(1\!-\!\alpha)^{L}\!\mathbf{W}^{L}\!
+\!\alpha\!\sum_{l=0}^{L-1}(1\!-\!\alpha)^{l}\!\mathbf{W}^{l}\Big{)}\mathbf{X}
_{\mathcal{V}}$ with $\hat{\mathbf{X}}_{\mathcal{V}}=\mathbf{X}_{\mathcal{V}}^{(L)}$, $\mathbf{W}=\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1/2}\mathbf{H}\!\mathbf{D}
_{\mathcal{H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top}\mathbf{D}_{\mathcal{H}^{
\mathcal{V}}}^{-1/2}$ and $\alpha=\gamma_{D}$.

For an AllDeepSets (Chien et al., ) with $L$ feature aggregation functions, devoid non-linear activation and by setting the learnable transformation matrix to the identity matrix, its MP-Module can be reformulated as:

$$ $\mathbf{X}_{\mathcal{V}}^{(L)}=(\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1} \mathbf{H}\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top})^{L} \mathbf{X}_{\mathcal{V}}.$ $$

Based on Lemma D.1, this equation is a special case of $\hat{\mathbf{X}}_{\mathcal{V}}\!=\!\Big{(}(1\!-\!\alpha)^{L}\!\mathbf{W}^{L}\!
+\!\alpha\!\sum_{l=0}^{L-1}(1\!-\!\alpha)^{l}\!\mathbf{W}^{l}\Big{)}\mathbf{X}
_{\mathcal{V}}$ with $\hat{\mathbf{X}}_{\mathcal{V}}=\mathbf{X}_{\mathcal{V}}^{(L)}$, $\mathbf{W}=\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1}\mathbf{H}\mathbf{D}_{
\mathcal{H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top}$ and $\alpha=0$.

For an ED-HNN (Wang et al., ) with $L$ feature aggregation functions, devoid non-linear activation and by setting the learnable transformation matrix to the identity matrix, its MP-Module can be reformulated as:

$$ $\mathbf{X}_{\mathcal{V}}^{(L)}=\Big{(}(1-\gamma_{E})^{L}(\mathbf{D}_{\mathcal{ H}^{\mathcal{V}}}^{-1}\mathbf{H}\mathbf{D}_{\mathcal{H}^{\mathcal{E}}}^{-1} \mathbf{H}^{\top})^{L}+\gamma_{E}\sum_{l=0}^{L-1}(1-\gamma_{E})^{l}(\mathbf{D} _{\mathcal{H}^{\mathcal{V}}}^{-1}\mathbf{H}\mathbf{D}_{\mathcal{H}^{\mathcal{E }}}^{-1}\mathbf{H}^{\top})^{l}\Big{)}\mathbf{X}_{\mathcal{V}}.$ $$

Based on Lemma D.1, this equation is a special case of $\hat{\mathbf{X}}_{\mathcal{V}}\!=\!\Big{(}(1\!-\!\alpha)^{L}\!\mathbf{W}^{L}\!
+\!\alpha\!\sum_{l=0}^{L-1}(1\!-\!\alpha)^{l}\!\mathbf{W}^{l}\Big{)}\mathbf{X}
_{\mathcal{V}}$ with $\hat{\mathbf{X}}_{\mathcal{V}}=\mathbf{X}_{\mathcal{V}}^{(L)}$, $\mathbf{W}=\mathbf{D}_{\mathcal{H}^{\mathcal{V}}}^{-1}\mathbf{H}\mathbf{D}_{
\mathcal{H}^{\mathcal{E}}}^{-1}\mathbf{H}^{\top}$ and $\alpha=\gamma_{E}$.

∎

###### Lemma D.1 .

###### Proof.

###### Lemma D.2 .

###### Proof.

###### Lemma D.3 .

###### Proof.

###### Proof.

## Appendix E Proof of Proposition 3.3

We first introduce some concepts about neighbours on hypergraphs and graphs: 1) The $k$-th hop neighbours of a node $v_{i}$ on a hypergraph are exactly $k$ hyperedges away from $v_{i}$; and 2) The $k$-hop neighbours of a node $v_{i}$ on a graph are all nodes with a distance of $k$ or less, while the $k$-th hop neighbours are exactly $k$ edges away from $v_{i}$. Before proving Proposition 3.3, we introduce some Lemmas.

Based on the definition of clique expansion, we present the following Lemma without a proof:

###### Lemma E.1 .

Let $\mathcal{H}=(\mathcal{V},\mathcal{E},\mathbf{H})$ be a hypergraph and $\mathcal{G}=(\mathcal{V},\mathbf{W})$ be its clique expansion with self-loops. For any node $v_{i}$, let $\mathcal{N}_{\mathcal{H}_{v_{i}}}^{1}$ be the node set containing the $1$-hop neighbours of $v_{i}$ on $\mathcal{H}$, and $\mathcal{N}_{\mathcal{G}_{v_{i}}}^{1}$ be the node set containing the $1$-hop neighbours of $v_{i}$ on $\mathcal{G}$. Then, $\mathcal{N}_{\mathcal{G}_{v_{i}}}^{1}=\mathcal{N}_{\mathcal{H}_{v_{i}}}^{1}
\bigcup\{v_{i}\}$.

Moreover, we present the following Lemma about $\mathbf{S}$:

###### Lemma E.2 .

Let $\mathcal{H}=(\mathcal{V},\mathcal{E},\mathbf{H})$ be a hypergraph and $\mathcal{G}=(\mathcal{V},\mathbf{W})$ be its clique expansion with self-loops. Assume that $\alpha\in[0,1)$, and $\mathbf{S}\!=\!(1-\alpha)^{L}\mathbf{W}^{L}+\alpha\sum_{l=0}^{L-1}(1-\alpha)^{
l}\mathbf{W}^{l}$. Then, for $i\neq j$, we have $\mathbf{S}_{ij}>0$ if and only if $v_{i}$ is a $L$-hop neighbour of $v_{j}$ on $\mathcal{G}$, and $\mathbf{S}_{ij}=0$ otherwise.

###### Proof.

According to previous works in graph theory (West et al., ), the element $\mathbf{W}^{l}_{ij}$ represents the number of walks of length $l$ from node $v_{i}$ to node $v_{j}$ on the graph $\mathcal{G}$. Since $\mathbf{W}$ includes self-loops, $\mathbf{W}^{l}_{ij}$ also accounts for paths where nodes can revisit themselves. Hence, for $i\neq j$, we have $\mathbf{W}^{l}_{ij}>0$ if and only if $v_{i}$ is a $l$-hop neighbour of $v_{j}$ on $\mathcal{G}$, and $\mathbf{S}_{ij}=0$ otherwise. As a result, for $\mathbf{S}\!=\!(1-\alpha)^{L}\mathbf{W}^{L}+\alpha\sum_{l=0}^{L-1}(1-\alpha)^{
l}\mathbf{W}^{l}$ and $i\neq j$, we have $\mathbf{S}_{ij}>0$ if and only if $v_{i}$ is a $L$-hop neighbour of $v_{j}$ on $\mathcal{G}$, and $\mathbf{S}_{ij}=0$ otherwise.
∎

On the basis of Lemmas E.3 and E.2, the proof of Proposition 3.3 can be transformed into the proof of the following Lemma:

###### Lemma E.3 .

Let $\mathcal{H}=(\mathcal{V},\mathcal{E},\mathbf{H})$ be a hypergraph and $\mathcal{G}=(\mathcal{V},\mathbf{W})$ be its clique expansion with self-loops. For any node $v_{i}$, let $\mathcal{N}_{\mathcal{H}_{v_{i}}}^{l}$ be the node set containing the $l$-hop neighbours of $v_{i}$ on $\mathcal{H}$, and $\mathcal{N}_{\mathcal{G}_{v_{i}}}^{l}$ be the node set containing the $l$-hop neighbours of $v_{i}$ on $\mathcal{G}$. Then, for $l\in\mathbb{Z}_{+}$, $\mathcal{N}_{\mathcal{G}_{v_{i}}}^{l}=\mathcal{N}_{\mathcal{H}_{v_{i}}}^{l}
\bigcup\{v_{i}\}$.

###### Proof.

We prove this Lemma based on the mathematical induction.

Firstly, for the base case with $l=1$, according to Lemma E.3, this case can be proved by the definition of the clique expansion.

Secondly, for the inductive step, we assume that $\mathcal{N}_{\mathcal{G}_{v_{i}}}^{k}=\mathcal{N}_{\mathcal{H}_{v_{i}}}^{k}
\bigcup\{v_{i}\}$. Let $\widehat{\mathcal{N}}_{\mathcal{G}_{v_{i}}}^{k}$ be the $1$-hop neighbours of the $k$-hop neighbours of $v_{i}$ on the graph $\mathcal{G}$, and $\widehat{\mathcal{N}}_{\mathcal{H}_{v_{i}}}^{k}$ be the $1$-hop neighbours of the $k$-hop neighbours of $v_{i}$ on the hypergraph $\mathcal{H}$. Then, we have $\mathcal{N}_{\mathcal{G}_{v_{i}}}^{k+1}=\widehat{\mathcal{N}}_{\mathcal{G}_{v_
{i}}}^{k}\bigcup\mathcal{N}_{\mathcal{G}_{v_{i}}}^{k}$ and $\mathcal{N}_{\mathcal{H}_{v_{i}}}^{k+1}=\widehat{\mathcal{N}}_{\mathcal{H}_{v_
{i}}}^{k}\bigcup\mathcal{N}_{\mathcal{H}_{v_{i}}}^{k}$. According to the result in the base case, we have $\mathcal{N}_{\mathcal{H}_{v_{i}}}^{k+1}=\mathcal{N}_{\mathcal{H}_{v_{i}}}^{k+1
}\bigcup\{v_{i}\}$.

By induction, for $l\in\mathbb{Z}_{+}$, $\mathcal{N}_{\mathcal{G}_{v_{i}}}^{l}=\mathcal{N}_{\mathcal{H}_{v_{i}}}^{l}
\bigcup\{v_{i}\}$.
∎

###### Lemma E.1 .

###### Lemma E.2 .

###### Proof.

###### Lemma E.3 .

###### Proof.

## Appendix F Discussion about Eq. ( 5 )

To recap, for a given hypergraph $\mathcal{H}\!=\!\{\mathcal{V},\mathcal{E},\mathbf{H}\}$, the edge weight between $v_{i}$ and $v_{j}$ in our clique expansion is defined as:

$$ $\mathbf{W}_{{H}_{ij}}=\sum_{k=1}^{m}\frac{\delta(v_{i},v_{j},e_{k})}{\mathbf{D }_{\mathcal{H}^{\mathcal{E}}_{kk}}},$ $$

where $\delta(\cdot)$ is a function that returns 1 if $e_{k}$ connect $v_{i}$ and $v_{j}$ and returns 0 otherwise. We summarise the relationship between this edge weight and the probability of nodes $v_{i}$ and $v_{j}$ being in the same category as the following Lemma:

###### Lemma F.1 .

Let $p_{i,j}$ denote the probability of $v_{i}$ and $v_{j}$ having the same label, $p^{\prime}_{i,j}$ represent the probability of $v_{i}$ and $v_{j}$ being connected by a hyperedge that contains only nodes with the same label, and $\hat{p}_{e_{k}}$ be the probability of $e_{k}$ containing nodes with different labels. Then, $p_{i,j}$ is positively correlated with $\mathbf{W}_{{H}_{ij}}$ if the following conditions hold:

- a)
Hyperedges with higher degrees are more likely to connect nodes with different labels, and $\hat{p}_{e_{k}}=g_{1}(\frac{1}{\mathbf{D}_{\mathcal{H}^{\mathcal{E}}_{kk}}})$, where $g_{1}(\cdot)$ is a function, for any $x,y>0$, with $g_{1}(x)\cdot g_{1}(y)=g_{1}(x+y)$, $\frac{\operatorname{d}g_{1}}{\operatorname{d}x}<0$, and $g_{1}(x)\in(0,1)$.
- b)
$p_{i,j}$ is positively correlated with $p^{\prime}_{i,j}$, namely, $p_{i,j}\!=\!g_{2}(p^{\prime}_{i,j})$, where $g_{2}(\cdot)$ is a function for any $x>0$ with $\frac{\operatorname{d}g_{2}}{\operatorname{d}x}\!>\!0$ and $g_{2}(x)\in(0,1)$.

###### Proof.

According to the pre-defined condition a), we can have:

$$ $p^{\prime}_{i,j}\!=\!1\!-\!\underset{e_{k}\in\hat{\mathcal{E}}_{v_{i},v_{j}}}{ \Pi}\hat{p}_{e_{k}}\!=\!1\!-\!\underset{e_{k}\in\hat{\mathcal{E}}_{v_{i},v_{j} }}{\Pi}g_{1}(\frac{1}{\mathbf{D}_{\mathcal{H}^{\mathcal{E}}_{kk}}})=1-g_{1}( \underset{e_{k}\in\hat{\mathcal{E}}_{v_{i},v_{j}}}{\sum}\frac{1}{\mathbf{D}_{ \mathcal{H}^{\mathcal{E}}_{kk}}})=1-g_{1}(\sum_{k=1}^{m}\frac{\delta(v_{i},v_{ j},e_{k})}{\mathbf{D}_{\mathcal{H}^{\mathcal{E}}_{kk}}})=1-g_{1}(\mathbf{W}_{{ H}_{ij}}),$ (9) $$

where $\hat{\mathcal{E}}_{v_{i},v_{j}}$ is a set containing hyperedges connecting $v_{i}$ and $v_{j}$. Based on Eq. (9) and the pre-defined condition b) we can have:

$$ $p_{i,j}=g_{2}(p^{\prime}_{i,j})=g_{2}(1-g_{1}(\mathbf{W}_{{H}_{ij}})).$ (10) $$

The derivative of Eq. (10) with respect to $\mathbf{W}_{{H}_{ij}}$ is:

$$ $\frac{\operatorname{d}p_{i,j}}{\operatorname{d}\mathbf{W}_{{H}_{ij}}}=\frac{ \operatorname{d}g_{2}}{\operatorname{d}\mathbf{W}_{{H}_{ij}}}=\frac{ \operatorname{d}g_{2}}{\operatorname{d}p^{\prime}_{i,j}}\cdot\frac{ \operatorname{d}p^{\prime}_{i,j}}{\operatorname{d}g_{1}}\cdot\frac{ \operatorname{d}g_{1}}{\operatorname{d}\mathbf{W}_{{H}_{ij}}}=-1\cdot\frac{ \operatorname{d}g_{2}}{\operatorname{d}p^{\prime}_{i,j}}\cdot\frac{ \operatorname{d}g_{1}}{\operatorname{d}\mathbf{W}_{{H}_{ij}}}.$ $$

On the basis of our pre-defined conditions a) and b), we have $\frac{\operatorname{d}g_{2}}{\operatorname{d}p^{\prime}_{i,j}}>0$ and $\frac{\operatorname{d}g_{1}}{\operatorname{d}\mathbf{W}_{{H}_{ij}}}<0$, thereby we have:

$$ $\frac{\operatorname{d}p_{i,j}}{\operatorname{d}\mathbf{W}_{{H}_{ij}}}=-1\cdot \frac{\operatorname{d}g_{2}}{\operatorname{d}p^{\prime}_{i,j}}\cdot\frac{ \operatorname{d}g_{1}}{\operatorname{d}\mathbf{W}_{{H}_{ij}}}>0.$ $$

As a result, $p_{i,j}$ is positively correlated with $\mathbf{W}_{{H}_{ij}}$.
∎

###### Lemma F.1 .

###### Proof.

## Appendix G Proof of Proposition 4.1

Before proving Proposition 4.1, we first present two Lemmas.

###### Lemma G.1 .

Let $H_{0}$ denote the entropy of the information that an HNN utilises by $L$ feature aggregation function $\phi_{\mathbf{\Theta}}(\cdot)$ introduced in Propostion 3.1 for generating features of $v_{i}$, and $H_{2}^{v^{L}_{i}}$ denote the entropy of information within node $v_{i}$ and its $L$-hop neighbours on the hypergraph. Then, $H^{v^{L}_{i}}_{0}=H^{v^{L}_{i}}_{2}$.

###### Proof.

Based on the mathematical induction, we start by proving this Using mathematical induction, we first prove this lemma for the case where $\mathcal{N}_{\mathcal{H}_{v_{i}}}$ includes only the 1-hop neighbours of $v_{i}$, which applies to all models except the line-based models discussed in Appendix B.

Firstly, for the base case with $L=1$, the message passing layer for generating the features of $v_{i}$ can be reformulated as:

$$ $\mathbf{x}_{v_{i}}^{(1)}=\phi_{\mathbf{\Theta}}(\mathbf{x}_{v_{i}}^{(0)}, \mathbf{x}_{v_{i}}^{(0)},\oplus_{v_{j}\in\mathcal{N}_{\mathcal{H}_{v_{i}}}} \mathbf{x}_{v_{j}}^{(0)}),$ (11) $$

where $\mathcal{N}_{\mathcal{H}_{v_{i}}}$ is a set containing the $1$-hop neighbours of $v_{i}$ on the hypergraph. Hence, the entropy of the input information utilised to generate $\mathbf{x}_{v_{i}}^{(1)}$ is $H^{v^{1}_{i}}_{2}$, namely, we have $H^{v^{1}_{i}}_{0}=H^{v^{1}_{i}}_{2}$.

Secondly, for the inductive step, we assume that $H^{v^{k}_{i}}_{0}=H^{v^{k}_{i}}_{2}$. Moreover, we have the feature generation function as:

$$ $\mathbf{x}_{v_{i}}^{(k+1)}=\phi_{\mathbf{\Theta}}(\mathbf{x}_{v_{i}}^{(k)}, \mathbf{x}_{v_{i}}^{(0)},\oplus_{v_{j}\in\mathcal{N}_{\mathcal{H}_{v_{i}}}} \mathbf{x}_{v_{j}}^{(k)}).$ (12) $$

Based on the assumption about $L=k$, for $L=k+1$, $\oplus_{v_{j}\in\mathcal{N}_{\mathcal{H}_{v_{i}}}}\mathbf{x}_{v_{j}}^{(k)}$ contain the information from the $(k+1)$-hop neighbours of $v_{i}$ and $\mathbf{x}_{v_{i}}^{(k)}$ contain the information from the $k$-hop neighbours of $v_{i}$. Let $\mathcal{N}^{l}_{\mathcal{H}_{v_{i}}}$ denote a set containing $l$-hop neighbours of $v_{i}$. Based on information theory (Kullback, ), we have the information entropy of $\mathcal{N}^{k}_{\mathcal{H}_{v_{i}}}$ is zero with the information of $\mathcal{N}^{k+1}_{\mathcal{H}_{v_{i}}}$ is given, as $\mathcal{N}^{k}_{\mathcal{H}_{v_{i}}}\subseteq\mathcal{N}^{k+1}_{\mathcal{H}_{
v_{i}}}$. As a result, based on the feature aggregation function introduced in Eq. (12), we have $H^{v^{k+1}_{i}}_{0}=H^{v^{k+1}_{i}}_{2}$.

By induction, for $L\in\mathbb{Z}_{+}$, $H^{v^{L}_{i}}_{0}=H^{v^{L}_{i}}_{2}$.

Now, we conduct the proof for the line-based models.

Let $H^{v^{L}_{i}}_{L}$ denote the entropy of the information contained by the $L$-hop neighbours of the line-nodes corresponding to $v_{i}$. According to previous papers in the GNN literature (Wu et al., ), the entropy of information used by an $L$-layer GNN to generate the features of line-nodes corresponding to $v_{i}$ is $H^{v^{L}_{i}}_{L}$. Based on the discussion in Appendix B and Appendix 3.1, we have $H^{v^{L}_{i}}_{0}=H^{v^{L}_{i}}_{L}$. By definition, the line-nodes corresponding to $v_{i}$ are only connected to each other and the line-nodes corresponding to the $1$-hop neighbours of $v_{i}$. Therefore, we have $H^{v^{L}_{i}}_{L}=H^{v^{L}_{i}}_{2}$. Accordingly, we have $H^{v^{L}_{i}}_{0}=H^{v^{L}_{i}}_{2}$.
∎

###### Lemma G.2 .

Let $H_{1}$ denote the entropy of the information that an TF-HNN with an $L$-layer TF-MP-Module utilises to generate features of $v_{i}$, and $H_{2}^{v^{L}_{i}}$ denote the entropy of information within node $v_{i}$ and its $L$-hop neighbours on the hypergraph. Then, $H^{v^{L}_{i}}_{1}=H^{v^{L}_{i}}_{2}$.

###### Proof.

For the feature generation of $v_{i}$, an $L$-layer TF-MP-Module can be directly reformulated as:

$$ $\hat{\mathbf{x}}_{v_{i}}=f_{l_{0}}(\mathbf{x}_{v_{i}})+f_{l_{1}}(\oplus_{v_{j} \in\mathcal{N}^{L}_{\mathcal{H}_{v_{i}}}}\mathbf{x}_{v_{j}}),$ (13) $$

where $f_{l_{0}}$ and $f_{l_{1}}$ are two linear functions, and $\mathcal{N}^{L}_{\mathcal{H}_{v_{i}}}$ is a set containing $L$-hop neighbours of $v_{i}$ on the hypergraph. Hence, we have the entropy of the input information utilised by an $L$-layer TF-MP-Module for generating features of $v_{i}$ is $H^{v^{L}_{i}}_{2}$, namely, $H^{v^{L}_{i}}_{1}=H^{v^{L}_{i}}_{2}$.
∎

After having the Lemmas above, we prove Proposition 4.1 as follows:

###### Proof.

Based on Lemma G.1, we have $H^{v^{L}_{i}}_{0}=H^{v^{L}_{i}}_{2}$. Moreover, according to Lemma G.2, we have $H^{v^{L}_{i}}_{1}=H^{v^{L}_{i}}_{2}$. As a result, $H^{v^{L}_{i}}_{0}=H^{v^{L}_{i}}_{1}=H^{v^{L}_{i}}_{2}$.
∎

###### Lemma G.1 .

###### Proof.

###### Lemma G.2 .

###### Proof.

###### Proof.

## Appendix H Proof of Proposition 4.2

Before proving Proposition 4.2, we first prove the following Lemma.

###### Lemma H.1 .

Let $\mathcal{H}=\{\mathcal{V},\mathcal{E},\mathbf{H}\}$ denote a hypergraph, $\mathcal{G}=\{\mathcal{V},\mathbf{W}_{H}\}$ be its clique expansion with edge weights computed by Eq. (5), $\mathbf{L}$ be the graph Laplacian matrix of $\mathcal{G}$ computed by a symmetrically normalised and self-loops added $\mathbf{W}_{H}$, $\mathbf{X}_{\mathcal{V}}$ represent the input node features, and $\alpha\in(0,1)$. We set $F(\mathbf{X})=\operatorname{tr}(\mathbf{X}^{\top}\mathbf{L}\mathbf{X})+\frac{
\alpha}{1-\alpha}\operatorname{tr}[(\mathbf{X}-\mathbf{X}_{\mathcal{V}})^{\top
}(\mathbf{X}-\mathbf{X}_{\mathcal{V}})]$, and $\mathbf{X}^{\star}$ as the global minimal point for $F(\mathbf{X})$. Then, we have:

$$ $\displaystyle\mathbf{X}^{\star}=\Big{(}\mathbf{I}_{n}-(1-\alpha)\tilde{\mathbf {D}}_{H}^{-1/2}\tilde{\mathbf{W}}_{H}\tilde{\mathbf{D}}_{H}^{-1/2}\Big{)}^{-1} \alpha\mathbf{X}_{\mathcal{V}},$ (14) $$

where $\tilde{\mathbf{W}}_{H}=\mathbf{W}_{H}+\mathbf{I}_{n}$ and $\tilde{\mathbf{D}}_{H}\in\mathbb{R}^{n\times n}$ is the diagonal node degree matrix of $\tilde{\mathbf{W}}_{H}$.

###### Proof.

Taking the partial derivative of $F$ with the respective of $\mathbf{X}$, we have:

$$ $\displaystyle\frac{\partial F}{\partial\mathbf{X}}=2\mathbf{L}\mathbf{X}+\frac {2\alpha}{1-\alpha}(\mathbf{X}-\mathbf{X}_{\mathcal{V}}).$ $$

By setting $\frac{\partial f}{\partial\mathbf{X}}=0$, we have:

$$ $\displaystyle\mathbf{X}^{\star}=(\mathbf{L}+\frac{\alpha}{1-\alpha})^{-1}\frac {\alpha}{1-\alpha}\mathbf{X}_{\mathcal{V}}=\Big{(}\mathbf{I}_{n}-(1-\alpha) \tilde{\mathbf{D}}_{H}^{-1/2}\tilde{\mathbf{W}}_{H}\tilde{\mathbf{D}}_{H}^{-1/ 2}\Big{)}^{-1}\alpha\mathbf{X}_{\mathcal{V}},$ $$

where $\tilde{\mathbf{W}}_{H}=\mathbf{W}_{H}+\mathbf{I}_{n}$ and $\tilde{\mathbf{D}}_{H}\in\mathbb{R}^{n\times n}$ is the diagonal node degree matrix of $\tilde{\mathbf{W}}_{H}$.

Since the second-order derivative of $F$ with the respect of $\mathbf{X}$ is:

$$ $\displaystyle\frac{\partial^{2}f}{\partial^{2}\mathbf{X}}=2\mathbf{L}+\frac{2 \alpha}{1-\alpha}\mathbf{I}_{n}.$ (15) $$

Given $\mathbf{L}\in\mathbb{R}^{n\times n}$ is a graph Laplacian matrix, which is positive semi-definite, and, for $\alpha\in(0,1)$, $\frac{2\alpha}{1-\alpha}\mathbf{I}_{n}\in\mathbb{R}_{+}^{n\times n}$ is a diagonal matrix, which is positive definite. Accordingly, we have the righthand side of Eq.(15) is a positive definite matrix for $\alpha\in(0,1)$. As a result, for $\alpha\in(0,1)$, $\mathbf{X}^{\star}$ in Eq. (14) is the global minimal point for $F$.
∎

With the Lemma above, we present the proof of Proposition 4.2 as follows.

###### Proof.

Let $\mathcal{H}=\{\mathcal{V},\mathcal{E},\mathbf{H}\}$ denote a hypergraph and $\mathcal{G}=\{\mathcal{V},\mathbf{W}_{H}\}$ be its clique expansion with edge weights computed by Eq. (5). According to Eq. (6), the output of an $L$-layer TF-MP-Module can be reformulated as:

$$ $\displaystyle\hat{\mathbf{X}}_{\mathcal{V}}=\Big{(}(1-\alpha)^{L}(\tilde{ \mathbf{D}}_{H}^{-1/2}\tilde{\mathbf{W}}_{H}\tilde{\mathbf{D}}_{H}^{-1/2})^{L} +\alpha\sum_{l=0}^{L-1}(1-\alpha)^{l}(\tilde{\mathbf{D}}_{H}^{-1/2}\tilde{ \mathbf{W}}_{H}\tilde{\mathbf{D}}_{H}^{-1/2})^{l}\Big{)}\mathbf{X}_{\mathcal{V }},$ $$

where $\tilde{\mathbf{W}}_{H}=\mathbf{W}_{H}+\mathbf{I}_{n}$ and $\tilde{\mathbf{D}}_{H}\in\mathbb{R}^{n\times n}$ is the diagonal node degree matrix of $\tilde{\mathbf{W}}_{H}$. When $L\to+\infty$, the left term tends to 0 and the right term becomes a convergent geometric series (Meyer & Stewart, ). Hence, for $L\to+\infty$, this equation can be further represented as:

$$ $\displaystyle\hat{\mathbf{X}}_{\mathcal{V}}=\Big{(}\mathbf{I}_{n}-(1-\alpha) \tilde{\mathbf{D}}_{H}^{-1/2}\tilde{\mathbf{W}}_{H}\tilde{\mathbf{D}}_{H}^{-1/ 2}\Big{)}^{-1}\alpha\mathbf{X}_{\mathcal{V}}.$ $$

Based on Lemma H.1, we have $F(\hat{\mathbf{X}}_{\mathcal{V}})=F_{min}$.
∎

###### Lemma H.1 .

###### Proof.

###### Proof.

## Appendix I Details of Benchmarking Datasets

Our benchmark datasets consist of existing six datasets (Cora-CA, DBLP-CA, Citeseer and House from (Chien et al., ), Congress and Senate from (Wang et al., )). In co-citation networks (Citeseer), all documents cited by a particular document are connected by a hyperedge. For co-authorship networks (Cora-CA, DBLP-CA), all documents co-authored by a single author are grouped into one hyperedge. The node features in these co-citation and co-authorship networks are represented using bag-of-words models of the corresponding documents, with node labels corresponding to the paper classes. In the House dataset, each node represents a member of the US House of state-of-the-arts, with hyperedges grouping members of the same committee. Node labels denote the political party of the state-of-the-arts. In the Congress dataset, nodes represent US Congresspersons, with hyperedges linking the sponsor and co-sponsors of bills introduced in either the House or the Senate. In the Senate dataset, nodes also represent US Congresspersons, but hyperedges link the sponsor and co-sponsors of Senate bills only. Nodes in both datasets are labeled by political party affiliation.

## Appendix J Additional Experimental Results

**Table 9: The hyperlink prediction AUC (%) for HNNs and TF-HNN. The best result on each dataset is highlighted in bold font.**
|  | Cora-CA | DBLP-CA | Citeseer | Congress | House | Senate | Avg. Mean |
| --- | --- | --- | --- | --- | --- | --- | --- |
| AllSetTransformer | 95.51 ± 0.46 | 95.77 ± 1.01 | 95.98 ± 0.73 | 71.56 ± 0.07 | 80.28 ± 3.33 | 63.56 ± 1.15 | 83.78 |
| PhenomNN | 97.60 ± 0.74 | 96.11 ± 0.86 | 96.13 ± 0.92 | 70.68 ± 0.34 | 74.57 ± 3.02 | 58.46 ± 1.98 | 82.26 |
| ED-HNN | 97.61 ± 0.83 | 95.71 ± 2.10 | 96.04 ± 0.85 | 71.48 ± 0.12 | 81.09 ± 3.47 | 65.96 ± 1.79 | 84.64 |
| Deep-HGNN | 97.00 ± 0.50 | 97.60 ± 0.18 | 96.45 ± 0.55 | 71.34 ± 0.28 | 78.84 ± 6.20 | 62.06 ± 3.13 | 83.88 |
| TF-HNN (ours) | 97.87 ± 0.65 | 97.96 ± 2.43 | 96.95 ± 0.97 | 72.33 ± 0.14 | 82.18 ± 3.70 | 65.04 ± 0.97 | 85.39 |

**Table 10: The training time for HNNs and TF-HNN in hyperlink prediction. The best result on each dataset is highlighted in bold font.**
|  | Cora-CA | DBLP-CA | Citeseer | Congress | House | Senate | Avg. Mean |
| --- | --- | --- | --- | --- | --- | --- | --- |
| AllSetTransformer | 20.82 ± 5.27 | 64.07 ± 5.75 | 9.18 ± 2.75 | 2086.49 ± 244.37 | 51.76 ± 13.03 | 33.15 ± 8.26 | 377.58 |
| PhenomNN | 14.34 ± 2.72 | 365.47 ± 144.07 | 25.87 ± 9.60 | 1029.77 ± 200.26 | 130.82 ± 18.67 | 30.89 ± 3.50 | 266.19 |
| ED-HNN | 8.03 ± 1.71 | 65.84 ± 8.52 | 11.28 ± 2.71 | 899.00 ± 162.75 | 75.01 ± 10.62 | 53.81 ± 8.34 | 185.50 |
| Deep-HGNN | 12.31 ± 2.72 | 107.97 ± 37.50 | 8.49 ± 1.68 | 795.25 ± 173.54 | 36.33 ± 7.74 | 7.82 ± 1.32 | 161.36 |
| TF-HNN (ours) | 3.22 ± 1.99 | 29.36 ± 6.31 | 2.75 ± 1.26 | 501.66 ± 111.99 | 17.83 ± 4.98 | 3.19 ± 1.31 | 93.00 |

**Table 11: The node classification accuracy (%) for TF-HNN, HyperGCL, and HDS^ode. The best result on each dataset is highlighted in bold font.**
|  | Cora-CA | DBLP-CA | Citeseer | Congress | House | Senate | Avg. Mean |
| --- | --- | --- | --- | --- | --- | --- | --- |
| HyperGCL | 83.02 ± 1.36 | 91.12 ± 0.28 | 70.70 ± 1.77 | 94.30 ± 0.75 | 65.25 ± 7.93 | 48.76 ± 4.73 | 75.53 |
| HDS^ode | 85.60 ± 1.07 | 91.55 ± 0.33 | 73.68 ± 1.99 | 90.58 ± 1.72 | 71.58 ± 1.60 | 59.72 ± 5.60 | 78.79 |
| TF-HNN (ours) | 86.54 ± 1.32 | 91.80 ± 0.30 | 74.82 ± 1.67 | 95.09 ± 0.89 | 76.29 ± 1.99 | 70.42 ± 2.74 | 82.50 |

**Table 12: The training time (s) for TF-HNN, HyperGCL, and HDS^ode. The best result on each dataset is highlighted in bold font.**
|  | Cora-CA | DBLP-CA | Citeseer | Congress | House | Senate | Avg. Mean |
| --- | --- | --- | --- | --- | --- | --- | --- |
| HyperGCL | 33.94 ± 3.86 | 722.25 ± 137.08 | 7.94 ± 1.72 | 1384.02 ± 160.35 | 108.08 ± 6.38 | 14.67 ± 1.23 | 378.48 |
| HDS^ode | 2.05 ± 0.66 | 531.43 ± 144.96 | 49.70 ± 4.39 | 213.87 ± 0.15 | 5.05 ± 0.99 | 72.23 ± 32.53 | 145.72 |
| TF-HNN (ours) | 0.22 ± 0.12 | 4.39 ± 0.45 | 1.12 ± 0.30 | 0.98 ± 0.35 | 1.01 ± 0.51 | 0.19 ± 0.10 | 1.32 |

## Appendix K Negative Sampling for Hyperlink Prediction

We adapt the algorithm from Chen & Liu () for this purpose. For each (positive) hyperedge $e\in E$, we generate a corresponding negative hyperedge $f$, where $\alpha\times 100\%$ of the nodes in $f$ are from $e$ and the remaining are from $V\textbackslash\{e\}$. Denote the negative hyperlink set as $F$. The number $\alpha$ controls the genuineness of the negative hyperlinks, i.e., higher values of $\alpha$ indicate that the negative hyperlinks are closer to the true ones. Additionally, we define $\beta$ to be the number of times of negative sampling which controls the ratio between positive and negative hyperlinks. In practice, we set $\alpha=0.5$ and $\beta=5$.

## Appendix L Reproducibility Statement

Our code and data are available at: https://anonymous.4open.science/r/TF-HNN.
To ensure the reproducibility of our results, we provide detailed hyperparameter settings used in our experiments. The configurations for the TF-HNN model on the hypergraph node classification task and the hyperlink prediction task are listed below.

**Table 13: Hyperparameter settings for TF-HNN on the hypergraph node classification task.**
| Dataset | TF-HNN Layers | MLP Layers | MLP Hidden Dimension | Learning Rate | Alpha | Dropout | Weight Decay |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Cora-CA | 2 | 3 | 1024 | 0.001 | 0.3 | 0.7 | 0.0 |
| DBLP-CA | 2 | 3 | 1024 | 0.0006 | 0.15 | 0.7 | 0.0 |
| Citeseer | 8 | 3 | 1024 | 0.001 | 0.65 | 0.9 | 0.0 |
| House | 16 | 3 | 128 | 0.005 | 0.7 | 0.9 | 0.0 |
| Congress | 1 | 3 | 1024 | 0.0001 | 0.05 | 0.8 | 0.0 |
| Senate | 2 | 7 | 512 | 0.005 | 0.6 | 0.5 | 0.0 |

**Table 14: Hyperparameter settings for TF-HNN on the hyperlink prediction task.**
| Dataset | TF-HNN Layers | Predictor Layers | Predictor Hidden Dimension | Learning Rate | Alpha | Dropout |
| --- | --- | --- | --- | --- | --- | --- |
| Cora-CA | 64 | 3 | 256 | 0.0005 | 0.5 | 0.0 |
| Citeseer | 2 | 3 | 256 | 0.0005 | 0.5 | 0.0 |
| House | 4 | 3 | 128 | 0.0002 | 0.0 | 0.0 |

## Appendix M Efficiency Improvement from TF-HNN

In this section, we theoretically discuss that the efficiency improvement provided by TF-HNN is inversely correlated with the complexity of the task-specific module. Specifically, we prove a Lemma.

###### Lemma M.1 .

Let $M$ denote the training complexity of the task-specific module and, $J$ denote the training complexity of an HNN. Since an TF-HNN only requires the training of the task-specific module, we approximate its training complexity with the function $t_{s}(M)=M$. Similarly, we approximate the training complexity of an HNN with the function $t_{h}(M,J)=M+J$. Furthermore, We quantify the efficiency improvement brought by TF-HNN using the ratio $r(M,J)=\frac{t_{h}(M,J)}{t_{s}(M)}$, where a larger $r(M,J)$ indicates a greater efficiency improvement provided by the TF-HNN. Then we have $\frac{\partial r}{\partial M}<0$.

###### Proof.

We can reformulate the $r(M,J)$ as:

$$ $r(M,J)=\frac{M+J}{M}.$ $$

Taking the partial derivative of $r$ with the respective of $M$, we have:

$$ $\frac{\partial r}{\partial M}=-\frac{J}{M^{2}}<0.$ $$

∎

###### Lemma M.1 .

###### Proof.

## Peer Reviews

confidence: 4
contribution: 1
initial_rating: 3
presentation: 3
questions: See weaknesses.
soundness: 2
strengths: 1. The paper is well-structured and easy to follow.
2. The summary of hypergraph neural networks is comprehensive, particularly with the insights provided in Table 1 and the related analysis.
summary: The paper proposes TF-HNN, a training-free hypergraph neural network that removes the need for computationally intensive message passing during training. By shifting hypergraph structural processing to the preprocessing stage, TF-HNN reduces computational complexity. The model achieves efficient, robust node feature generation without oversmoothing, utilizing as much structural information as traditional HNNs. Experiments show that TF-HNN outperforms state-of-the-art HNNs in accuracy and training speed, especially on large-scale benchmarks.
weaknesses: Major weaknesses:
1  Proposition 4.2 shows the similarity between the proposed method and APPNP [1], yet the paper does not cite APPNP. Specifically, Eq (6) and Eq (14) appear to apply APPNP after performing clique expansion on the hypergraph. This connection should be discussed more thoroughly, referencing APPNP's Eq (7) and Eq (8) to clarify the relationship and implications of this similarity in the context of hypergraph learning. While the experimental results demonstrate superiority, this oversight is a significant limitation on the paper's originality.

2  Equations (4a) to (4d) result from removing learnable parameters from several baselines, corresponding to the different scenarios in the authors' proposed framework. While these modifications reasonably reduce training time, there is insufficient ablation study to explain the performance improvements. The authors' comparison of the impact from the weighted $S$ of TF-HNN in node classification is commendable. However, they should also present the performance of these modified baselines to clearly demonstrate the impact of removing the learnable parameters.

Minor weakness:

Chien et al.'s work [2] provides the challenging YELP dataset, where many baseline methods yield unsatisfactory performance. Including results on this dataset would enhance the paper's quality and offer a more comprehensive evaluation of the proposed method.
[1] You are allset: A multiset function framework for hypergraph neural networks. Chien  et al. ICLR 2022 

[2] Predict then Propagate: Graph Neural Networks meet Personalized PageRank
Gasteiger et al. ICLR 2019

---

confidence: 4
contribution: 4
initial_rating: 10
presentation: 4
questions: 1. An assumption is made about the structure of hypergraph i.e., absence of isolated nodes or empty hyperedges, for the theoretical results. What happens if isolated nodes or empty hyperedges are present? I am not able to see why this assumption is required, and what breaks if it is violated?
2. It is commendable that the proposed TF-HNN performs significantly better than the baselines, but it is also a bit strange to see the baselines performing so poor, particularly on trivago. I understand the boost in training time, but not able to fully understand why there is a 10% improvement, it seems to me  that the learning ability of any SOTA HNN should be similar to TF-HNN. I may have missed something, but curios to hear what the authors have to say on this.
soundness: 4
strengths: 1. The paper makes a significant contribution by addressing the issue of high computational complexity of Hypergraph learning algorithms. 
2. The proposed solution, TF-HNN is novel and elegant, which decouples the processing of structural information from the model training stage. 
3. Authors provide a strong theoretical foundation for TF-HNN, the unified framework presented in the paper links all the popular HNN approaches, which shows that TF-HNN is designed by keeping many existing methodologies in mind, and hence provides a comprehensive mechanism for efficient training.
4. Extensive experiments on diverse real-world datasets for node classification and hyperedge prediction tasks demonstrate the competitive performance of TF-HNN against state-of-the-art HNN baselines while requiring significantly less training time.
5. The paper is well-written, with a clear motivation, rigorous theoretical analysis, and thorough empirical evaluation supporting the proposed method's effectiveness and efficiency.
summary: The paper proposes a novel approach called TF-HNN (Training-Free Hypergraph Neural Network) to address the high computational complexity during training in existing hypergraph neural networks (HNNs). The key innovation is a training-free message passing module (TF-MP-Module) that decouples the processing of hypergraph structural information from the model learning stage. The authors first derive a theoretical framework that provides a unified view of existing HNN approaches, identifying the feature aggregation function as the core component processing hypergraph structure. Based on this insight, they remove the learnable parameters and non-linear activations from the feature aggregation functions of four state-of-the-art HNNs to make them training-free. Further, they consolidate the feature aggregation across layers into a single propagation step, resulting in the proposed TF-MP-Module. Extensive experiments on seven real-world datasets for the tasks of node classification and hyper-link prediction demonstrate the competitive performance of TF-HNN, with very less training time. TF-HNN is the first approach to shift the processing of structure to pre-processing stage, which significantly enhances training efficiency.
weaknesses: I do not see any weak points in this paper. This is a very well written paper, with significant contributions. Please refer to the questions sections for the questions I have.

---

confidence: 3
contribution: 3
initial_rating: 5
presentation: 3
questions: Isn’t too much time being spent on hyperparameter search due to the extensive hyperparameter search range?

If the hyperparameters were indeed selected based on the validation set, could you demonstrate this by providing heatmaps of the hyperparameters across the various validation and test sets used in the experiments?

**minor comments**
- line 209: shonw -> shown
- line 144: Ortega et al. (2018) -> (Ortega et al. (2018))
soundness: 2
strengths: 1. The design of the proposed method is very interesting.
2. The proposed method is both highly efficient and effective.
3. The paper is clearly written and well-organized, making the research easy to follow.
summary: This paper introduces an efficient hypergraph learning scheme that performs message passing prior to learning neural network parameters. The proposed framework, called TF-HNN, includes a TF-MP-Module in which training-free message passing is performed. Using the updated features from the TF-MP-Module, TF-HNN learns an MLP model without a heavy computational burden. Despite its high learning efficiency, TF-HNN demonstrates either superior or competitive performance in hypergraph learning tasks. Theoretical analysis supports the design of the proposed method.
weaknesses: It appears that there is an issue regarding the hyperparameters. The combinations of hyperparameters used in the experiments shown in Table 13 and Table 14 are quite diverse. For example, the value of alpha ranges from 0.05, 0.15, 0.3, 0.6, 0.65, to 0.7. The learning rate also varies, with values like 0.0006, 0.0001, 0.005, 0.001, and 0.0002. What method was used for hyperparameter search? Additionally, upon reviewing the attached anonymous GitHub link, it appears that the optimal hyperparameters were selected based on performance on the test set rather than the validation set. Were the hyperparameters selected in a fair manner? An analysis of hyperparameter sensitivity should be added.

---

confidence: 4
contribution: 3
initial_rating: 8
presentation: 2
questions: Refer to the Weakness section.
soundness: 3
strengths: - S1. Given that real-world hypergraphs are often large, scalable HNNs are necessary in practice. 
- S2. Although the model design is quite trivial, the theoretical motivation (i.e., building a general unified framework and simplifying it) of the model design is systematic and interesting.
- S3. Experiments are comprehensive.
summary: The authors propose a scalable hypergraph neural network (HNN) that reduces the computational cost associated with message passing at each training epoch by decoupling the message passing step.

To achieve this, they (1) establish a general framework that explains popular HNN models, and (2) simplify this framework by removing learnable components and non-linearity, resulting in a single linear operator.

They demonstrate the effectiveness of their approach on both small- and large-scale hypergraphs, showing improvements over several existing HNN models.
weaknesses: ***Major comments.***

- ***W1. Regarding Proposition 3.2.*** My understanding is that the key idea is the existence of a clique expansion that satisfies the property outlined in Proposition 3.2. 
In practice, however, the authors use a fixed clique expansion as described in Equation 5. 
To what extent does this chosen clique expansion align with the one referenced in Proposition 3.2? 
While the exact formulations may differ, it would be helpful to know if the high-level characteristics of these clique-expanded graphs are similar. 
This is essential, in my view, as it clarifies whether the theoretical analysis effectively supports the proposed method.

- ***W2. Regarding Proposition 4.1.*** Could you clarify what is meant by the "entropy of information"? Does this refer to mutual information between features and node labels? Further elaboration on this point would help in understanding the key takeaway from this proposition.

- ***W3. Regarding the Initial Message Passing Operator Computation Complexity*** Although the message passing operator incurs a one-time computation cost, the time required for this process should be reported. If this initial computation is substantial and exceeds the typical training time of existing HNNs, it could limit the practical efficiency of the proposed method.

***Minor comments.***
- In Lines 52-53, the text mentions that $n^{k}$ memory is required. While this is accurate for dense tensor formats, typical tensor representations are stored in a sparse format, and sparse operations are well-supported in modern deep-learning libraries. Thus, storing a dense incidence tensor is generally not necessary in practice. It may be helpful to revise this part to reflect real-world scenarios.
- In Lines 79-80, the period "." is missing.
- Please provide clarification on the source of the datasets used.

Please let me know if I have misunderstood any parts. Thank you.

Meta-review: The authors propose a hypergraph neural network that decouples message passing from training by precomputing message passing during preprocessing. To achieve this, the authors unify popular hypergraph neural network designs into a single theoretical framework and systematically remove learnable parameters. The experimental results demonstrate that this method significantly reduces training overhead with little to no loss in performance.

The reviewers recognized the paper's practical significance, theoretical contributions, and the comprehensiveness of its experimental evaluation. However, opinions were divided on the novelty of the work. Some reviewers pointed out notable similarities between the proposed model and decoupled GNN architectures, but the authors clearly clarified differences in their responses. 

The overall reviews leaned towards acceptance, and the strengths of the work, particularly as it introduces the first decoupled approach in hypergraph neural networks, justify its acceptance.

Decision: Accept (Poster)