Parameter-Free Hypergraph Neural Network for Few-Shot Node Classification
Authors: Chaewoon Bae
Doyun Choi
Jaehyun Lee
Jaemin Yoo
Venue: NeurIPS 2025 poster ()
Keywords: Hypergraph neural network
Few-shot node classification
Linear model
Primary area: general_machine_learning
TL;DR: In this work, we propose ZEN (Zero-Parameter Hypergraph Neural Network), a fully linear and parameter-free model that achieves both expressiveness and efficiency.

Abstract: Few-shot node classification on hypergraphs requires models that generalize from scarce labels while capturing high-order structures. Existing hypergraph neural networks (HNNs) effectively encode such structures but often suffer from overfitting and scalability issues due to complex, black-box architectures. In this work, we propose ZEN (Zero-Parameter Hypergraph Neural Network), a fully linear and parameter-free model that achieves both expressiveness and efficiency. Built upon a unified formulation of linearized HNNs, ZEN introduces a tractable closed-form solution for the weight matrix and a redundancy-aware propagation scheme to avoid iterative training and to eliminate redundant self-information. On 11 real-world hypergraph benchmarks, ZEN consistently outperforms eight baseline models in classification accuracy while achieving up to 696x speedups over the fastest competitor. Moreover, the decision process of ZEN is fully interpretable, providing insights into the characteristic of a dataset. Our code and datasets are fully available at https://github.com/chaewoonbae/ZEN.

Parameter-Free Hypergraph Neural Network for Few-Shot Node Classification
Chaewoon Bac
Doyun Choi Jachyun Lee Jaemin Yoo
School of Electrical Engineering Korea Advanced Institute of Science and Technology (KAIST)
{chaewoon. bae, doyun choi, jaehyun.lee, lee, jaemin) kr
Abstract
Few-shot node classification on hypergraphs requires models that generalize from scarce labels while capturing high-order structures. Existing hypergraph neural networks (HNNs) effectively encode such structures but often suffer from overfit- ting and scalability issues due to complex, black-box architectures. In this work, we propose ZEN Kero-Parrmeter Hypergraph Neural Network), a fully linear and parameter-free free model that achieves both expressiveness and efficiency. Built upon a unified formulation of linearized HNNs, ZEN introduces a tractable closed- propagation scheme
form solution for the weight matrix and redundancy- to avoid iterative training and to eliminate redundant self-information. On 11 real-world hypergraph benchmarks, ZEN consistently outperforms eight baseline models in classification accuracy while achieving up to 696x speedups over the fastest competitor. Moreover, the decision process of ZEN is fully interpretable, providing insights into the characteristic of a dataset. Our code and datasets are fully available https://gthuu.ccom//chewwooonnn
Introduction
Many real-world datasets naturally modeled as hypergraphs, which generalize ordinary graphs by capturing higher-order interactions involving more than two nodes [3 16]. Unlike traditional graphs that represent pairwise relationships, hypergraphs allow for the modeling of multilateral con- nections, making them particularly suitable for complex data structures. Hypergraphs are especially useful in representing multifaceted relationships across range of domains, such as document-wood associations, gene-disease correlations, and user-item interactions 9 33
Few-shot node classification predicting node labels with only handful of annotated examples- is a fundamental yet challenging task, especially the context hypergraphs [4 The complexity of higher-order structures, combined with the scarcity of labeled data, makes difficult to design models that are both generalizable and efficient. Although a number of hypergraph neural networks (HNNs) have been proposed to effectively capture high-order relationships between nodes, many of them rely on complex, nonlinear architectures with numerous parameters. Such models often suffer from overfitting and poor scalability in few-shot scenarios 21 32).
In contrast, linear models offer strong generalization, low complexity, and are particularly effective in few-shot scenarios [5] 18 20 25 Despite these advantages, existing work on HNNs has largely overlooked fully linear formulations, likely due the perception that linear models cannot sufficiently capture the structural richness of hypergraphs. This motivates us to explore if a carefully designed linear HNN can overcome these limitations while preserving both expressiveness and efficiency.
In this paper, we propose ZEN (Zero-Parameter Hypergraph Neural Network), a model designed capture high-order relationships while maintaining strong generalization ability and scalability. We begin by linearizing existing HNN models, yielding a general form characterized by a single
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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propagation matrix and a single weight matrix. This formulation enables us to design two optimization strategies tailored this structure: tractable closed-form solution (TCS) for the weight matrix and redundoncy-awart propagation (RAP). TCS allows us to avoid burdensome iterative training by deriving an efficient closed-form solution, while RAP eliminates redundant self-information across multi-hop propagation. ZEN achieves the best average rank in classification performance across 1 real-world hypergraphs, while also exhibiting exceptional computational efficiency.
To summarize, our main contributions are as follows:
We present general form of linearizing representative hypergraph neural networks, and exhibit similarities and differences between different HNNs excluding their nonlinearity. To the best of our knowledge, this is the first comprehensive study on linear HNNs.
We derive a tractable closed-form approximation for the weight matrix, the only parameter in linear HNN. Our solution eliminates the need for computing matrix pseudoinverses, significantly improving computational efficiency while maintaining high classification accuracy.
We introduce redundancy-awann aware propagation that effectively reduces the structural overlap across multi-hop neighborhoods when building the propagation matrix. This leads to more efficient and expressive information aggregation, especially uesful for complex real hypergraphs.
We conduct comprehensive experiments on 11 real-world hypergraphs and demonstrate that our method outperforms existing HNNs in few-shot classification while showing exceptional scalability; our ZEN is up to 696 faster than the fastest competitor. In addition, we conduct a case study that shows the interpretability of ZEN arising from its linear decision process.
Problem definition and related work
2.1 Problem definition
We consider a hypergraph G = (V,E), where V is the set of nodes and E is the set of hyperedges. The incidence matrix H E IR Mix encodes hypergraph connectivity, with H. = 1 if node belongs to hyperedge j. and otherwise. Each node is associated with feature vector, forming feature matrix X E IR 1ixd where d is the feature dimension. The class label for each node is encoded in a one-hot matrix Y E {0, 1 where e is the number of classes.
For node classification, we are given set Vin C V of training nodes and a set Viest C. V of test nodes such that Vir n Vtest = 0. We define two diagonal indicator matrices Dum E RMV and Diest E RM where (Dum)) hs = iff i E Vim and (Drest)s = iff i E Viest- In the k-shot setting, assume that exactly & labeled nodes are available per class, i.c., tr tr(Dua) ur) = kc. The goal of k-shot node classification is to find model f that produces prediction Y IR [ such that:
Drest D Drest Y,
where Y= (H, DEm Y). That is, the model is trained using only the labeled nodes in Dum Y, and evaluated on its ability to generalize to test nodes specified by D Dtest- Note that a general representation of both training-baseer based and training- ffree node classifiers.
2.2 Related work
Hypergraph neural networks (HNNs) HNNs have emerged as powerful tools for modeling higher- order relationships, where each hyperedge can connect an arbitrary number of nodes 66 (2). This
expressive capacity enables applications in diverse domains such as recommendation systems, biolog- ical networks, and knowledge graphs. Early models such as HGNN (6) extend graph convolution to the hypergraph domain via spectral approximation. UniGNN 10 provides a unified message-passing permutation-invariin design that separates set encoding from message propagation. More recently, ED-HNN 23 explores the connection between the general class of hypergraph diffusion algorithms and the design of HNNs to improve expressiveness in hypergraph settings. Despite their representational power, these models often suffer from overfitting or scalability issues when training labels are scarce.
framework for both graphs and hypergraphs, while AllSet 3
introduces
Linear graph neural networks (GNNs) GNNs are highly effective for learning node representations from relational data [8] [4 28). However, their reliance repeated non non-lincar transformations across

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Table 1: Linearized forms of five representative HNNs. All parameters in each layer are integrated into a single weight matrix W due the linearization, and the 1-hop adjacency matrix A generated from the incidence matrix H without involving any additional parameters.
Method
Linearized form
Adjacency matrix
HGNN
Y A
A = (D, HD. H D. b1
HNHN
Y = ALXW
A = (D v. HD:D D
1
UniGCNII
Y ELLO a(1 - o. (1-0)AAL) xw
A, = (D, 'HD Hy
AllDeepSet
Y A
A (D,'HD:
ED-HNN
Y = Elzo a(1 - a) + +(1a))AA XW
A. (D, 'HD. '
layers leads high computational costs and limited scalability. To mitigate this, several linear GNN variants have been proposed, which simplify architecture by removing intermediate non-linearity and decoupling transformation from propagation [7]. For instance, SGC [26) eliminates nonlinearities and applies fixed propagation matrix multiple times after single feature transformation. APPNP 77) adopts a personalized PageRank-base propagation scheme that retains residual connection to the input features, thereby mitigating over-smoothing without increasing parameter count. SCGC (35] improved SGC by manually adjusting the strength self-loops, increasing the number of propagation steps. More recent models such as SlimG [3]] explore the linearized form of GNNs and further improve generalizability and interpretability. These approaches demonstrate that linear architectures can match or surpass nonlinear GNNs in performance, especially with sparse labels.
3 Proposed method: ZEN
We introduce ZEN, a linear hypergraph neural network (HNN) for fast, scalable. and generalizable node classification in few-shot settings. Our method builds on a unified linear formulation of existing HNNs (Section 3.11 Leveraging this formulation, we develop a closed-form approximation for the weight matrix (Section 3.2) to eliminate iterative training and propose redundancy- design for to mitigate structural redundancy.
the propagation matrix (Section 3.3
3.1 General form linearized HNN
Linearization
20 25 26
can effectively simplify the formulation of machine learning models
and reduce their computational complexity, resulting in improved robustness. scalability, and in- terpretability. We conduct the linearization of five representative HNNs: HGNN (6), HNHN [4] UnIGCNII [10), AllDeepSets [3] ED-HNN [23], and introduce their lincarized forms in Table We provide the full process of linearization in Appendix [ We remove all non-linear functions, including the activation functions, and integrate multiple weight matrices multiplied together into the single equivalent matrix W of size c.
Then, we generalize the linearized forms of HNNs as follows:
= ( xW,
(2)
where A; denotes the I-hop adjacency matrix created from H. and ar is a hyperparameter. The matrix A₁ determines how the incidence matrix is converted into an adjacency matrix between nodes, while ar determines how much information is propagated from the (-hop neighbors. It is noteworthy that A does not contain any learnable parameters, since all parameters are already integrated into W. For simplicity, we denote P Σ, A, and call it propagation matrix in the rest of this paper.
The formulation in Eq. (2) reveals that the performance of a linear HNN is primarily determined by two factors: (a) the design of the propagation matrix P, which aggregates multi-hop structures A with appropriate coefficients on, and (b) the optimization of the weight matrix W. We elaborate on
the principled design of the propagation matrix Section (3: In Section 3.2 we propose a tractable closed-fome solution for W, which climinates the need for iterative training via backpropagation.

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3.2 Tractable closed-form solution for the weight matrix W
There are two approaches to obtain the optimal weight matrix for linear HNNs. The first approach is to iteratively update W through backpropagation until it reaches steady state. Although it is popular and reasonable approach, we argue that the linear characteristic of the model is not fully exploited in this way. The second approach is derive closed-form solution. Given training labels Dun Y, we directly compute the optimal W as a function of Dru Y without having any iterative process. The limitation is its large computational complexity; it is rarely used in practice due the cubic time complexity (d)) required to compute the pseudoinverse of matrix.
Therefore, we propose tractable approximation of the cloesd-form solution to eliminate the need for a pseudoinverse, while maximizing its scalability by removing dependence on iterative learning. To fully linearize the objective function, consider the squared error (SSE) loss instead of the typical cross-entropy loss which involves the non-linear softmax function. The closed-form solution of W, without any approximation optimization, is given Lemma] I
Lemma 1. Given a linear HNN Y = PXW and the SSE loss LSsE = ||Dm Y -Dun Y P the closed-form solution W that minimizes LSSE is given by
W (PX) Dim (PX)) DumY,
(3)
where denotes the Moore-Penrose inverse (or the pseudoinverse) of matrix.
Proof. The full proof is provided in Appendix A.1
The problem of Eq. (3) is the pseudoinverse of matrix (PX) Dtrn (PX) E RRdxd, whose compu- tational complexity O(d)). To avoid the pseudoinverse by safely approximating the closed-form solution, we introduce two assumptions on the matrix PX which we call the embedding matrix of nodes before being mapped to the class space by the matrix W.
for all Assumption 2. For any two row vectors of PX, their intru-class cosine similarity is approximately 1 - while their inter-class cosine similarity is approximately That is, (PX)s, (PX(P) R3 for i,j in the same class, while (PX... Rs e efor i,j in different classes, with small c.
Assumption 1. Each row vector of PX has a unit norm, i.e. (,X),,
Assumption] I can be easily achieved by normalizing the embedding matrix PX. Assumption is harder to satisfy, but is reasonable to assume in many cases where the node feature matrix X provides sufficient information for classification when combined with the structural information encoded in P. especially when we loosen the error bound Then, we propose Theorem | for approximating the closed-form solution of W with the introduced assumptions.
Theorem 1. Under Assumption 7 and the following holds:
w = ((PX) ( (PX) Dum Y (PX) D
D Y.
Proof. The full proof is provided in Appendix A.2
By Theorem 1 the burdensome pseudoinverse in the optimal weight matrix be approximated as II which is a constant matrix. Since the constant e is multiplied to all dimensions, it does not change the result prediction; we can safely remove it from our model.
To satisfy Assumption 1 we apply the row-wise L2 normalization for the embedding matrix PX as Prow (PX) where the function Srow denotes the row-wise normalization operator. Consequently, we normalize the weight matrix W as well, since it represents the embedding matrix of labels; each column of it can be understood the d-dimensional embedding of each label. Without normalizing W*, the class with a large-norm embedding gets an advantage in the prediction stage. Therefore, we normalize both node and class embeddings, and our final method is expressed follows:
Y grow (PX) (grow (PX) DtrnY)
(4)

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3.3 Redundancy-awarr propagation for eliminating self-information
A key design objective in linear HNNs is to flexibly control the influence of each I-hop neighborhood on node representations. This is achieved through a propagation matrix of the form
P
A
(5)
2=0
which denotes weighted combination of (-hop adjacency matrix A, multiplied with the coefficient ag. Since PX is row-normalized as shown in Eg.
44 without loss of generality, we can constrain
the coefficients to lie on the probability simplex: اا= 04 1, where ay 2 for all By carefully controlling the coefficients for each dataset, determining how much information to take from each 1-hop neighborhood, linear HNN can exhibit superior performance. For example, on can be large for homophilic graphs, but small for heterophilic graphs following typical assumption.
However, the formulation in Eq. (5) inevitably contains redundant relf-information, which hinders the precise adjustment of neighborhood influence. We formalize this phenomenon as follows:
Definition 1. Given A with 1 > 0, we define its residual relf-infommaton as the diagonal matrix
RST(Ai) == diag(()) R
which quantifies the amount of information returning to each self-node.
The self inffiinformatinn of Ay can arise not only from the self-loops, but also from cycles or return paths in hypergraph walks if 2 2. Such self-information is not a notable problem, but even promoted in nonlinear HNNs which aim to avoid losing the information of initial node features when deep layers are adopted. However, in linear HNNs, self-informin adjacency matrices redundant and prevents one from fully optimizing the propagation function for each dataset with unique characteristic. Another problem is that the self-information in Ar, if it is used for creating Ay with /' > I, results in boosting the effect of local neighborhoods in Ar, making the self-information even stronger in later adjacency matrices.
repeatedly appearing in different-hop
To overcome these limitations, we propose redundancy-awan propagation that explicitly eliminates redundant self-information from the I-hop adjacency matrix A,. Specifically, we define the (-hop adjacency matrix Ai without redundant ooff-informatio as follows:
A A. - RSI(A
where À : A is the normalized adjacency matrix, Di is the degree matrix designed specifically for the normalization method used in creating A,, and A := I. Then, we replace all A, with A in Eq. (5) and use it as our propagation matrix P = Σ, A, This ensures that the self-information in P is exactly determined by co. and is not confounded by higher-hop structures. As result, it allows a precise control over self- versus neighbor- information, which is critical in few-shot scenarios where overfitting to redundant self-signals can hinder generalization.
Normalization of adjacency matrices All lincarized HNNs in Table I normaalze the adjacency matrix A either by symmetric or row normalization to make it numerically stable in deep layers. However, we apply the same original normalization and remove self- elf-infoormttin from normalized A, it makes ovrr-noomaiizztion and gradually diminishes feature magnitudes across layers. Thus, we re- normalize A, and derive the following matrices for 1,2:
Å D. H(D. -I ' D
(8)
Å A D DD, - Ai
(9)
Avoiding dense adjacency matrices The main challenge for Eq. 7 deriving the dense matrix À, of size | x |V] for the computation of its self-information. We provide explicit expressions of the RSI terms for l = 1 and / = 2, since we set I. to 2 in our experiments. Our derivation is based on symmetric normalization, but the framework readily extensible to row normalization. Detailed derivations for the row-normalized variant are included in Appendix D
5

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Lemma 2. Given A in Eq. 88. RSI (1) is given by
(RST(A))) dā (EnjENtonsdas 10l0a,11111 -1)-1,
(10)
where dı denotes the degree of node I or the number of nodes in hyperedge x, based on the type of and (vv)) denotes the set of hyperedges incident to node V/.
Proof. The full proof is provided in Appendix
A.3
Lemma 3. Given A = Ai - RSI(Ai) and A2 in Eq. () RSI(AA) is given by
(RSI(A2)) = dr, (Eeserco, 1 -1
(En (Freee,
1 -1)
(1I)
where dig denotes the degree of node the number of nodes in hyperedge x. based on the type of and (mz) denotes the set of hyperedges incident to node V;-
Proof. The proof follows by applying the same reasoning as in Lemma
From the 3-hop neighborhood onward, self-information can return through cycles rather than simple backtracking paths, complicating both its identification and principled removal during propagation. Additionally, deeper propagation exacerbates the computational burden and enlarges the hyperparam- eter space. For these reasons, we restrict our model to 2-hop propagation, which strikes a balance between expressive power and efficiency. This design choice is supported by two key observations: (a) Increasing the number of hops introduces a linearly growing number of propagation coefficients ag resulting in the quadratic expansion of the hyperparameter space. (b) Empirically, most HNNs attain competitive performance with only 1-2 propagation layers.
3.4 Summary of ZEN and its computational complexity
Putting all together, the ZEN classifier f" can be summarized follow:
f (H, Dim Y) = grow Jooo (P"X) Durn Y),
(12)
where P = Ag + a AjAi= + A2 with three hyperparameters clg. d11 and 022, and A denotes the refined I-hop adjacency matrix which climinates the redundant self-information.
The function f is a closed-form prediction formula that directly produces label predictions for all nodes given small set of labeled nodes. Crucially, it eliminates the need for iterative training: the labels of the training nodes are injected as explicit inputs rather than being implicitly encoded via gradient-based optimization. This design enables extremely fast and stable inference, particularly under low-resource scenarios such as few-shot settings, where traditional saining-based models often suffer from instability due to limited supervision. More importantly, by removing self-information explicitly, ZEN avoids redundancy and enables fully optimized combinations of multi-hop structure within the probability simplex. This capability is fundamentally absent in conventional propagation schemes, where self-information is entangled with higher-hop structures.
The computational complexity of f" depends on the two main stages. The first is the dense matrix multiplication between frow (P*X) and (frow (P*X) Deru Y), which O(D)lde), linear in the number of nodes, feature dimension, and number of classes. The second is the construction of P', where we remove self-information from each A, for all 2 This has the same time complexity as standard message passing schemes, ((nz((H) dL). where nnz(H) denotes the number of non-zero entries in the hypergraph structure, and the number L of layers is set to in our experiments.
Experiments
We conduct comprehensive experiments on real-world hypergraphs to verify the effectiveness of ZEN. We show that ZEN consistently outperforms existing HNNs in few-shot node classification tasks while exhibiting remarkable scalability. We then present case study highlighting the interpretability of ZEN, which comes from its linear decision mechanism, a real hypergraph.

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Table 2: Statistics of datasets. The first ten datasets are used as the main benchmark for evaluating aceuracy, while the Zoo dataset is used for interpretability analysis.
Cora
Citeseer
Pubmed
Cora-CA
20News
MN40
Congress
Walmart
Senate
House
Zoo
# nodes [V]
2,708
3,312
19.717
2,708
16,242
12.311
1,718
88,860
282
1,290
101
# odges [E]
1,579
1.079
7,963
1,072
100
12.311
83,105
69,906
315
340
43
# classes
7
6
3
4
40
2
11
2
2
# features d
1,433
3,703
500
1,433
100
100
100
100
100
100
16
density (177)
[8
0.5835
0.3258
0.4041
0.3959
0.0062
1.0000
48.3946
0.7868
1.1660
0.2636
0.4257
Table 3: Classífication accuracy (%) for 5-shot node classification on real-world hypergraphs. We report the mean and standard deviation over 10 runs. Boldfaced letters indicate the best accuracy, and underlined letters indicate the second. ZEN achieves the highest average rank.
Methods
Cora
Citesser
Pahmedd
Cora CA
20News
MN40
Congress
Wallmart
Senate
House
Rank
Avg.
HGNN
44.45
40.1
32.5gn
54.3g14
73. 1ea
94.7308
86.7.11
30
56.8
63.4g
IINIIN
36.7
36.0ga
51.8
39.2gsa
41.2
90.81.
S1.1
15.9,a
69.7
07.4gs.
7,9
HCHA
44.4877
41.2ga.5
12.0
54.5g
72.9
94.7g0s
86.6
30.3
63.0g5.
63.5g14
5.9
UniGCN
48.5gs
41.6ga7
64.2gms
55.3
70.4
96 96.9gos
91. 1.
40.1
61.4244
67.0
3.9
UniOCNII
13.3,00
38.9
54.5,a4
42.0g
66.5
96.4.n
83.5
23.5ga4
70.4
70.7
5.5
AllDeepSets
48.0g1
42.6gg44
53,2g58
55.3g5
514414
947700
09.5g47
24.5ga7
63.3gi0a
63.4g5
5.7
ArrSeTTramoommer
50.5
44.8ea
60/4ea
50.6g:14
70.3
96.5gu2
88 2
33.3gn
63.1gm07
06.3
3.6
18.4264
44.5gas
56.5g44
58.8
67.7
96.0gea
89.1440
42.
63.1ga:
62.8
ZEN (propused)
51.9_10.1
49.1
02.0asa
60.0gwa
68 63.
97. 07.8
87.0gax
43.9.a1
70.4g30.
73.2
4.1 Experimental setup
We evaluate ZEN on a total of 11 real-world hypergraph graphs. To assess predictive performance and computational efficiency, use 10 standard benchmarks: Cora, Citeseer, Pubmed, Cora CA. 20News 100, ModelNet40, Congress. Walmart, Senate, and House, following prior work 15 23 For interpretability analysis, we use Zoo (15), small dataset whose feature attributes are semantically interpretable. Detailed dataset statistics are provided in Table [
Datasets.
Baselines and hy hyperparameters. We compare ZEN with 8 representative hypergraph neural net- works (HNN) models: HGNN (6). HNHN [4), HCHA (2), UniGCN, UniGCNII [10), AleDeepSets, AllSeTTransformer B1 and ED-HNN 23 All basclines are implemented based the official codebase of ED-HNN, which provides a unified framework for fair comparison. All baseline are trained using the Adam optimizer with no weight decay, and we conduct grid search over 72 hyper- parameter configurations:
E (10-2, 10-3 3,101),, epochs E (50, 100, 150, 200), num num_layers E (1, hidden_diig dim E (64, 128, 256). In contrast, ZEN requires no training hyperparameters. In- stead, we search over 55 combinations of propagation coefficients (oo, 01 (2) uniformly sampled from the 2-simplex, yielding a comparable hyperparameter space size that of baselines. For each dataset split, we report the test accuracy corresponding to the best validation performance. All our experiments are conducted with NVIDIA RTX A6000 and AMD EPYC 9354.
Evaluation. We evaluate the accuracy of all models on 10 random data splits per dataset. For cach split, we allocate 5 labeled nodes per class for training. and another 5 nodes per class for validation 12 27), making 5-shot node classification. The remaining nodes are used for testing. We report the average classification accuracy and standard deviation across the ten splits [1]
4.2 Classification accuracy
Table compares the accuracy of ZEN and the baseline HNNs 10 hypergraphs. ZEN demonstrates competitive or superior performance across all datasets, showing the highest average rank. Despite its simple linear architecture, ZEN achieves high accuracy even complex hypergraph structures, being competitive with complicated nonlinear methods. This validates the effectiveness of ZEN's architectural design in capturing high-order relationships, and highlights its strong generalization ability in fho-shot node classification scenarios, where model robustness is crucial.
The results also highlight intriguing trends among baseline models. In particular, early models such as HGNN and UniGCN remain competitive, particularly on datasets such as 20News and Congress. Their relatively simple architectures may contribute to stronger generalization in sho-shot settings,
7

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Table 4: The running time of ZEN and the baseline models, including both training and inference. Each time represented as ratio over the running time of ZEN. Therefore, the lower is the better. ZEN consistently shows the fastest runtime, up to 696 faster than the best competitor.
Methods
Cora
Citeseer
Pubmed
Cora_ CA
20News
MN40
Congress
Wallmart
Semate
House
HGNN
8.65
3.00
8.95
10.54
20.05
30.16
717.14
388.05
HNHN
7.61
2.71
3.56
7.23
15.51
13.78
42.11
21.47
696.85
345.93
HCHA
10.19
12:21
20.63
71.62
12.73
1008.85
609.41
UnIGCN
17.12
2.63
8.29
8.97
28.79
21.86
91.16
32.90
716.43
292.95
UniGCNII
15.77
5.18
2.56
17.25
16.98
19.41
50.17
36.37
63
369.10
AllDeepSets
58.11
24.92
11.01
57.56
35.32
47.51
273.58
90.16
40488.55
1748.67
AllSeTTransfomer
9.76
4.30
10.98
12.37
60.21
21.94
65.59
76.89
999.43
524.15
ED-HNN
16.01
6.06
5.16
23.55
46.71
28.70
126.91
16.31
714.73
379.11
ZEN (proposed)
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
where overfitting is a common challenge. In contrast, more complex methods such as HNHN and AeeDeepSets tend underperform, likely due to higher complexity and reduced robustness under limited supervision. These observations further underscore the strength of the simple yet effective design of ZEN, producing consistently high performance in diverse hypergraph structures.
4.3 Running time
Table shows the running time ZEN and the baseline models, including both training and inference time. All existing HNNs exhibit significantly higher computational costs over ZEN. The speedup stands out for large complicated HNNs, such as AllDeepSets and ED-HNN, as ZEN is over 1700 faster than AltDeepSets in the House dataset. Even for relatively simple models such as HGNN, HCHA, UniGCN, and UnIGCNII, ZEN shows a consistent improvement ranging from 2.5x even to x In summary, ZEN demonstrates overwhelming speed superiority across all datasets while maintaining competitive accuracy, establishing itself as a highly efficient solution.
The efficiency of ZEN comes from its lightweight architecture, where the computational cost is linear in the number of nodes, feature dimension and number classes. This design leads to substantial runtime advantages House have feature dimensions that are at least 5x and up to 37x smaller, node counts up to 9x fewer, and class counts between 2 and 20 fewer than other datasets. These characteristies make them ideal for highlighting the scalability of ZEN, which achieves remarkable speedups of up to 292x on House and 696x Senate while maintaining competitive accuracy.
datasets with compact input dimensions. In particular, Congress, Senate, and
4.4 Interpretability
ZEN is inherently interpretable, thanks to its linear decision process that directly maps the feature space to the prediction. There are two ways to interpret the Iearned knowledge of ZEN, First, the A)th element of the weight matrix W can be understood as the importance of the i-th feature for predicting the k-th class. Second, each column of W can be understood as the embedding of class k lying on the feature space. In this way, the relationships between class embeddings and node features provide deeper insight the nature of the given dataset, along with the graphical structure.
To verify the interpretability of ZEN, we conduct case study on the Zoo dataset, whose node features have clear semantic meanings: each feature attribute represents a characteristic of an animal, e.g., hair or milk, while the target class is its species. The nodes represent animals, and the hyperedges group together animals that share common feature, c.g., all animals having the same hair.
Table visualizes the weight matrix W where each value color-coded: darker red indicates higher relative value compared to other classes for that feature. Since the initial input features are all nonnegative, the resulting weight elements also remain nonnegative. For instance, the Mammal class shows significantly higher values in Hair, Milk, and Capsize, suggesting that these features play a key role distinguishing mammals from other animal groups. Similarly, the Bird class exhibits prominent values in Feathers, Eggs, Airborne, etc., reflecting biologically distinctive traits of birds. These results demonstrate that ZEN not only achieves high predictive performance but also yields representations that align with domain knowledge in transparent and interpretable manner.
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Table 5: Relative feature importance values learned by ZEN on the Zoo dataset under 3-shot setting. Darker red cells indicate higher values relative other classes, with cells having darkness of 80% or higher highlighted by black boxes. Refer to Section 4.4 4 for detailed information.
Manmal
Bird
Reptile
Fish
Amphibian
Bug
Invertebrate
Hair
0.1735
0.0719
0.0778
0.0819
0.0661
0.0800
0.0465
Feathers
0.0251
0.1726
0.0383
0.0414
0.0305
0.0276
Eggs
0.1072
0.2440
0.2255
0,2931
0.1979
0.1792
0,1850
Milk
0.1720
0.0664
0,0740
0.0826
0.0626
0.0432
0.0409
Airbome
0.0310
0.1417
0.0450
0,0456
0.0376
0.0979
0.0361
Aquatic
0.0807
0.0688
0.0727
0,2522
0. 0.1006
0.0192
0.1437
Predetor
0.1796
0.1054
0.1878
0,2841
0.1601
0.1006
0.1688
Toothed
0.1707
0.1001
0.1946
0.2921
0.1959
0.0682
0.0745
Backhone
0.2280
0.2768
0,2687
0.3380
0.2304
0.1012
0.1049
Breathes
0.2239
0,2769
0,2597
0.1652
0.2235
0.1939
0,1061
Venomous
0.0109
0.0141
0,0628
0.0217
0.0465
0.0415
0.171
0.0246
0.0272
0.0337
0.2027
0.0273
Legs 0,8380
0.7794
0.7956
0,5875
0.8520
0. 0.9333
0.9310
Tail 0.1836
0.2642
0,2542
0.3204
0.1522
0.0913
0.0963
Domestic
0.0225
0.0266
0.0234
0.0253
0.0197
0.0179
0.0146
Capsize
0.1416
0.1515
0,1134
0.1887
0.0687
0.0487
0.0744
Table 6: Ablation study of ZEN with four baselines: three variants with selective removal of the ZEN components, and the linearized HGNN that lacks multi-hop message combination. Our full model consistently outperforms all ablations, demonstrating the effectiveness of both components.
Methods
Citeszer
Pubmed
Cora, CA
MN40
Congress
Wallmart
House
Linearized BGNN No TCS. No RAP
12.7
34.ass
51.7aus
49.6.a9
68 68.0eaa
94.5ga.s
63.72
29.0.s..
55.8
57 57.8
4.5
44.3g7
25.0g74
82.8
48.836
61.8g75
97.8ga.
88.5ga
22.3_..
71.8
09.0g14
3.7
TCS
16 16.9
37.8
52.5
69.8 8
97. 8.0
87.0gg
26.0.s.
67 67.2
2.7
RAP
50.6
08.6g44
62.6
60.0g54
61.7g44
97.7ga.
68.4gas
40.6
73.8
71.4ga
ZKN (proposed)
51.0_101
19.
6.30
60.0gaa
97.6ga.3
87.0gam
70. 70.4.30.
73 2,0.
4.5
Ablation study
In Table we conduct an ablation study to assess the individual and combined impact of tractable closed-form solution (TCS) and redundancy-awar propagation (RAP), two core modules of ZEN, in the same setting as in T The results show that the removal of both components leads to notable performance drops, while isolating TCS or RAP yields moderate gains, with TCS generally offering more stable improvements. Our full model ZEN consistently achieves the best results, outperforming all ablated variants with the highest average rank. These findings highlight the complementary roles of TCS and RAP-TCS provides stability and tractability through a closed-form solution, while RAP enhances representation by mitigating redundancy.
We observe that removing RAP or TCS can improve performance on certain datasets (Congress and Senate for RAP, 20News and MN40 for TCS). RAP tends to degrade performance high-density datasets such as Congress and Senate, likely because the degree normalization reduces each node's relative contribution, limiting the effect of self-information. For TCS, which assumes informative node embeddings, the impact of removal is more significant on 20News than MN40. The extremely low density of 20News may hinder ZEN, which relies solely on propagation to refine embeddings, from producing sufficiently informative representations. In contrast, MN40's s large number of classes increases the relative Frobenius norm error of TCS, suggesting that the approximation becomes less accurate as the number of classes grows, partially explaining the observed results.
5 Conclusion
In this work, we propose ZEN -Zero-Paramett Hypergraph Neural Network), a parameter-free model for few-shot node classification on hypergraphs. By reformulating existing HNNs into unified

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linear framework with a tractable closed-form weight solution and redundancy-awee aware propagation, ZEN achieves strong generalization. fast inference, and interpretable representations without iterative training. Extensive experiments demonstrate its superior accuracy and scalability. One limitation of our work is that ZEN is specifically designed for node classification and not tailored to other hypergraph tasks such as hyperedge prediction, local clustering, or hyperedge disambiguation, In future work, we plan to extend our framework to support broader range of hypergraph learning tasks by designing more general-purpose and efficient linear HNN architecture.
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) RRS-2024-00341423 and RS-2024-00406985) This work was supported by BK21 FOUR (Connected AI Education & Research Program for Industry and Society Innovation, KAIST EE, No. $120200113769) Jaemin Yoo is the corresponding author.
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Answer: [NA]
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The core method development in this research does not involve LLMs.
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17

---

A Proofs
A.1 Proof of lemma
ÖLsS
PX) (Dum Y -im
(13)
OW
where Y = PXW. The optimal W, minimizes LSSE:
-2(D PX) (Drm Y PXW.)
(14)
(Dim D DmmP. = (m PX) T Dim
(15)
(PXX)
W. (PX) Dem
(16)
W, = ((PX) (PX)) (PX) D Y
(17)
A.2
Proof of theorem
Without loss of generality, ordered with their labels. Let D [kke 0] Rkex Vl. We can easily accept D D tr Dun
set the first ks diagonal elements in Dtru to be and they
(Dun)² by definition.
We first prove the following four lemmas for proving the main theorem:
Lemma 4. By definition, the following holds:
D DTT Dí Dum (Dum)²
(18)
Proof. The proof straightforward.
Lemma 5. Under the assumptions, (D (D ( can be expressed as:
(P (PX)) = (1 (E) & Jb) + < + Jkc)
(19)
where J is an all-one matrix, iee. Jkc 1kel ko and & denotes Kronecker product.
Proof. The proof straightforward.
Lemma 6. When > (1 2<) Jx) + E(Iee + Jkc) has inverse matrix as follows:
1
1
1
- ((1-2c) X J&)= e(Ikc + Jkc))
Mi
+
M2
+ Ms
(20)
A
A
where d A2 (1 - 2c)k + d, k(1 2c + cc) M1
I
| @ J2)
[I,
J&)
J key M3 kc.
kc
Proof. We consider the matrix M -1-2c) @ Jx) + c[ke + Jkc). We define three mutually
orthogonal projection matrices:
1 M I 1 (L. K.
1 (T 2 Jv)
1
1
M2
Jkcs
M2
Jkc
(21)
b.
Rc
&c
It is easily verified that these satisfy
M = M,,
M,M, = 0 (i f1),
M, + M2 + M3 = Ipc-
(22)
We now compute the action of M on each subspace:
MM, EMI,
MM, ((1 - (11-2s)k k + E) M2,
MM, (k(1 28 + ec) + 8) M3.
Thus, M admits the spectral decomposition
M = M + AgM2 + AgMs.
(23)
18

---

where
A2= =
26)1+ +c, X = = 2E + EC) +c.
(24)
Since e > 0, all eigenvalues are strictly positive, so M is invertible. The inverse is given by
M = M1 X2 M2 M2 M3.
(25)
Lemma 7. Under the small c < 1, the following holds:
(26)
Proof.
1
1
1
(iD tm (D (PX))
M1 +
M2 + M3
(27)
A
A2
A3
((D tra (PX) PX -)-² Mi
+
M2
+ M3
(28)
X
X
X
When = ,, = 2 Therefore,
(D (XX D (PX)) 3=-M -
(29)
1
(DD (PX) (PX)) () Mi 6
(30)
Thus, ((D)., im (P (PX)) (D (PX)) (PX)) with small e < 1.
The rest of this section proves the theorem based on above lemmas. First, we can reformulate Kt = ((PX) Deru ( as:
Kt ((XX) "D D' (PX)) = (PX)) (PX)))
(31)
tra
by Lemma 4.
By definition of Moore-Penrose pseudoinverse matrix,
(PX)) (D' tra (PX)))) = (Pim(P))) (PX))(( (D'rr.(X))
(PX)
(32)
where (D/ (PX)) tm ( has inverse matrix by Lemma
By Lemma 7,
K (D' mm (PX)) (PX) (D' (D'm (PX))
(33)
Therefore,
W kiz arizT. Y tm DD (XX) (III (Pmm
(D'm,
T Dum
(34)
By Lemma 4,
(Dim ( (PX))) (PX))))) Y
(35)
(D' (PX))
(D' (D', (PX)
D trm
Dur (36)
trm
(D'r (PX)) (D (D (PX) (PX) P
D
uru Y
(37)
1
D DD tm (PX)) D D Y - PPX
D T D' Dum Y = - (PX) D Y
(38)
trn tra
Therefore.
W (XX D
(39)
19

---

Table 7: Layer-wise formulations of five HNNs, assuming single-layer MLPs.
Method
Each layer
HGNN
( D
HDD "0
H D w)
HNHN
( D.HHD D H D D ( w
X = ((1-a) - D HD H X (1-1) + ax ()
UniGCNII
AllDeepSet
X = Ho (D.'H H
X W W W
ED-HNN
X = a ((1 - ajo (D, Ho
D
H (X
(x()))
w) W) + +xx W)
A.3
Proof of lemma 2
We analyze the diagonal entries of the A D, H,'HD-'H D. Specifically,
(A1)
HD
H )xt
HDD 'H )
(40)
dde.
Vde.
du
Since H.; = if and only node v, belongs to hyperedge ej, i),t) entry of HD corre- sponds to the sum of edge-normalized weights over all hyperedges incident node u; Therefore, obtain:
HD H
Σ
(41)
dey
e.E)((ss)
B Linearization of five representative HNNs
Table presents the layer-wise formulations of five hypergraph neural networks (HNNs), where each MLP block is assumed to be a single-layer perceptron. By removing nonlinear activation functions, all weight matrices
network can be merged into single equivalent weight matrix, since multi-layer perceptrons reduce to linear transformation without nonlinearity.
Let D. and De be the diagonal degree matrices of nodes and hyperedges, respectively. We define the variants of degree matrices as follows:
(D.).
Σ (())((
duys (Dv,a)i
Σ
(De,a)u
Σ
d
9, (E(()
CEV())
We denote by a a nonlinear function (e.g., ReLU), and by W = (1- B) a convex combination of the identity and learnable weight matrix. Since W is free parameter being updated during the training. we can safely replace W with W under lincarization.
C Error bounds in Theorem 1
We present the relative Frobenius norm error, confirming it remains sufficiently low as decreases. In TCS, our approximation is given follows:
1
1 M, A1
1 M2
1
Mi +
M2
+
M3
+
M3
(42)
X
X
X
xa
A
where \ e,
26)k + ,, k(1 2e Ec) e, Mi I (L, JR), M2
(I. J&) te Jkc, kc, M3 :.- Jkc. denotes an all-one matrix and the Kronecker product.
To evaluate the approximation quality, we compute the relative Frobenius norm error as follows:
Mi + M2 + 1 M3 - ( M + + X
As M2
F
(43)
Mi + X3 M X Ms
x
20

---

Table 8: Relative error for different values of e.
Relative Error
0.1
1.14%
0.01
0.10%
0.001
0.01 %
For representative case with k = and c = 10, the relative errors are summarized in Table
These results indicate that the approximation becomes increasingly accurate decreases. This supports the validity of our approach, particularly in the regime where c sufficiently small.
D Row-normalized adjacency matrices
Eq. and Eq. (45 present the row-normalized adjacency matrices Å and Åz The corresponding
residual self-information terms, RST(A1) and RST (2)) are shown to coincide with those obtained under symmetric normalization. This equivalence is formally established Lemma g and Lemma
A = H - - -H
(44)
A2 Ai ((D, - -I) -D)",,, Ai.
(45)
Lemma 8. Given A Eq. (44), (A)) is given by
(RST(A))) = dr. (de - -1)-))
(46)
where dug denotes the degree of node the number of nodes in hyperedge x. based on the type of and N (va)) denotes the set of hyperedges incident to node Us-
Proof. We analyze the diagonal entries of the A = D. HD. HH Specifically,
HDD''' (A )i HHD H )
(47)
Since Hij if and only if node v; belongs hyperedge ej. (i,i)-th entry of HD sponds to the sum of edge-normalized weights over all hyperedges incident to node u; Therefore, we obtain:
H HD corre-
(HD H Ja =
(48)
des
e,EN(
Lemma 9. Given A; A -RST and A₂ in Eq.
45
RST(Az) is given by
(RSI(A2)) = d (Eee - 1 (EoLEN(e)) (du, -1)
(49)
where d2 denotes the degree of node x or the number of nodes in hyperedge x., based on the type of x, ((v)) denotes the set of hyperedges incident node Us and ((e)) denotes the set of nodes incident to hyperedge ej.
Proof. The proof follows by applying the same reasoning as in Lemma
E
Possible approximations for deeper propagation
Some datasets may require models with higher-hop propagations for better expressivity. Without approximation, the exact computation of RSI requires high computational cost. One possible approximation is to estimate the probability that random walker returns to the starting node after steps, where the number hops that we aim to model. A simple pseudocode for this strategy provided in Algorithm I
21

---

Algorithm Approximation via random walks
Require: Node i, walk length I
Ensure: Return probability of node
1: count
2: for each trial do
3: current
4: for 1= 1 to do
5: Sample hyperedge incident to current uniformly at random
6: Sample a node j connected to uniformly at random
7: current j
8: end for
9: if current then
10: count t count +
end
12: end for return counl/((nubber of trials)
13:
Another possible approximation to use Hutchinson's ' Estimator, which provides an unbiased stochastic estimate of the diagonal entries. The equation is given by:
diag(A) sd
z
(A)
(50)
where m the number of random probe vectors, each (k) R" a random vector with en- tries independently sampled from the Rademacher distribution (i.e., each entry is +1 or -1 with equal probability). and @ denotes the element-wise (Hadamard) product. The expectation satisfies diag(A) E[z (Az)).
Hutchinson's 's Estimator does not require explicit storage of the entire matrix A; instead, it relies solely on matrix- multiplications with randomized vectors z. This characteristic renders Hutchinson's 's Estimator particularly suitable and computationally efficient for hypergraph structures. We consider these approaches to be promising directions for extending ZEN scenarios where deeper propagation may provide additional benefits.
Additional experiments
In this section, we present additional experimental results omitted from the main paper due to space limitations.
F11 Evaluation with increasing shots
ZEN a parameter-free free model, which gives it particular advantage in settings where labeled data scarce and training is challenging. While ZEN orginally designed for few-shot setting, results in TablePand Table show it performs strongly with more training samples as well. ZEN achieves top average ranks in both 10-shot and 20-shot settings, suggesting it scales well beyond few-shot scenarios.
F.2 Evaluation against additional baselines
We have additionally included eight baselines; three representative linear GNNs (SGC (6). APPNP 77 and SSGC (35)). three linearized hypergraph models based on UniGCNII I0] AllDeepSets and ED-HNN (23), and two semi-supervised hypergraph models (LEGCN 30 and HyperND
)2).. LEGCN utilizes a line expansion approach to adapt hypergraphs to conventional GNN architectures, while HyperND introduces diffusion-bacee mechanism for improved label propagation in hypergraphs. For GNNs, we applied clique expansion to convert the hypergraph into pairwise graph. As shown in the Table IT ZEN demonstrates consistently strong performance, outperforming all linear GNN and lincarized hypergraph baselines on the majority of datasets.
22

---

Table 9: Classification accuracy (%) for 10-shot node classification on real-world hypergraphs. We report the mean and standard deviation over 10 runs. Boldfaced letters indicate the best accuracy, and underlined letters indicate the second. ZEN achieves the highest average rank.
Methods
Cora
Citeseer
Puhmed
CA
20News
MIN40
Congress
Wallmurt
Senate
House
Raak
Ang.
HGNN
69.33asa
49.28
61158ana
66.9444
70.54.1.1
94.90usa
N7.1810
46.80a1.2
68.01.25
64.001.44
5.4
HNHN
42.47a30
42 2
68.92
51.28
98.5kgge
47.77g1
17.18
69.74g91
32
8.0
HCHA
68.96
49.00gne
60,02244
64 64.28
75.09.1.
94,89
27
47.12
87.8
63.86g
UniGCN
6088711
51. 18ga0
61.20aca
68.57a30
73.60
96.24
91. 7
45.88
62
62.57a25
70:27aa0
8.7
UniGCNI
asa
1821
61.12
05.17aa
70:91,2:
07.03..4
78-7012.
14.14
75.05122
AllDeepSets
27.3
50.6
61.89pa.1
02.78ga1
02.10am
90.25
7268
85.044277
69.53
00 08.96gg:1
6.3
AllSeffrassoomerr
87,00g50
54.07ans
60.41464
67.63
72.07.14
96.87
83000g0
45.0944-7
78.12g55
40.99
ED-INN
58.84
St. 12aas
60,50
64.70as
11.13
96.30
00/7Bar
47 47.17,33
06.4
40gro
4.7
7IN (proposed)
61 44ga2
59.173a8
08. 4
05 05:00ga
73.00gm
97 92
66.54gm
76.72
73.90.a
2.2
Table 10: Classification accuracy (%) for 20-shot node classification on real-world hypergraphs. We report the mean and standard deviation over 10 runs. Boldfaced letters indicate the best accuracy, and underlined letters indicate the second. ZEN achieves the highest average rank.
Methods
Coru
Citeseer
Pohmed
CA
20Nems
MN40
Congress
Wallmurt
Senate
House
Avg.
HGNN
02.46.11
96.57a1
69. 85gg.
70.53gr
76.00
95.07.20
87 27.80
49. 46.6
60 70
64 88g
6.1
HNHN
48.28.19
5000842s
62.60
62.88
52.60
56.20
20 20.5
21.82
78 78.51a3
72:00.4.
8.0
HCHA
64.19.19
$6.81123
68.4413.3
70.0
76.08.10
96.09
87.91
40.0412.2
61:07:13
64.1122
UnIGCN
65.44-13
58. 47a1.
70.84424
71.20.18
72.50.22
96.63
91. 33.6230.
49.00a2.1
02.0aza
72 2200
3.9
UniGCNII
04.29
56.97
68.0243-4
09:56
74.05213
97.5 07.56e81
52.04
77.59.00
76.054ne
AllDeepSets
02.37437
57. 28.s
67.10.s.a
60.90a1a
07. 17.611
96,09
77.60
48.97.2.4
78.0612
6.4
AllSeffrassoomer
63.63.1
28.49,23
80.70
69 20.82:42
74.22.11
96.02:24
89.20205
40.29
76.14.18
75.58127
ED-HNN
64.24
27.27
69.56gg.
69 69.20
7
96009
50.02
72.30
71.27
4.8
ZEN (pcoposed)
67.87
64.00114
71.32.1.6
71.99.2a
70.74.13
98. 0..
58.04129
51.30.1.1
74.10a12
76.70.0.
2.0
Table 11: Comparison of classification accuracy (%) with eight additional baselines for 5-shot node classification on real-world hypergraphs. We report the mean and standard deviation over 10 runs. Boldfaced letters indicate the best accuracy, and underlined letters indicate the second. ZEN achieves the highest average rank.
Methods
Cora
Citeseer
Puhmed
Coru, CA
20News
MN40
Congress
Wallmart
Senate
Hlusse
Rank
Asg.
SGC
44.00gaa
40 12gas
17.1
30.79asa
72. 72.08
26.1gga
51 51:00gar
50:37
APPNP
46. 46.77a9a
39 50
56.8 55.88gma
60.34
50.00g3
52. 70202
68 Rh.0P 2..
24.70
71 71.24
70.001.k4
5.8
12.60_.0
14.00.44
57.07443
52.58
50:91,52
50.75202
SL.14
26.56.14
73.00.ur
71.74
lin URIGCNTI
30. 30.44am4
60.00.
56.85ga
10. 30.75
50-45
58
73.35
17.91a17
71,22,10
68 20
5.8
Tin AlDoosSSess
40.4944a
40 20.70
56.1
51.01
08.62250
02
80.39...s
20.45ga0
00.05
65. 13
lo ED-HNN
41,97a5
29.44as
50.82
47,87asa
61.47a4
97.28.044
78.35a2.7
18.8433
71.10laa
70.201m.
LEGEN
37. 37,59ana
37 28
58. 1la62
37.39.5.2
49 29.41.43
90, 90.27zut
72.14
O.0.M
71.28
72,85
6,5
HypenND
39. 00.00
35 90gss
56.54444
41.74gr
54 64.70as
01.40
73. 73.57444
13.55
73 78.8
72.84g9
ZRN (proposod)
51.885
40.08
62 02.02,..
60.04
68 08.57.4x
97. 97.68.a0
86 16.96
48.88.a
70.40,i04
73.22..a
1.7
F33
Evaluation under standard n-way k-shot setting
To examine the generality of our approach, we further evaluate it under the standard n-way k-shot setting. To the best of our knowledge, this setup has not been explicitly explored for general hypergraphs. Accordingly, we designed new evaluation protocol by drawing inspiration from the experimental setup of Meta-GNI [54] and adopting hy hyperparameter configurations aligned with those used in ED-HNN 2 The results, summarized in Table [2 demonstrate the strong performance of ZEN in this setting, underscoring its effectiveness for few-shot learning on hypergraphs. Notably, ZEN achieves competitive results even when applied solely to test episodess revealing promising and underexplored direction for future research.
G Running times
The actual running times are reported in Table 13 with all values measured in seconds. ZEN runs in less than a second across all datasets, achieving as fast as 0.003s on the Senate dataset.
23

---

Table 12: Classification accuracy (%) under the standard way k-shot setting. Results are averaged over 10 runs. ZEN achieves superior performance across all configurations.
Methods
Cora (2-way 3-shot)
Cora (2-way 1-shol)
Citeseer (2-way 3-shot)
Citeseer (2-way 1-shol)
HGNN
64.4g0.1
55.9
60.1 60.1
54.9
UniGCNII
68.0go.1
58.9 58.920.2
25.2
56.821.
AllDeepSets
52.310.2
48.710.2
51.1
50.210.2
ZEN (proposed)
73.3_0.
62.610.
71.5101
62.31o.1
Table 13: The actual running time of ZEN and the baseline models, including both training and inference. Each time is measured seconds. ZEN exhibits a significant speed advantage over all baselines.
Methods
Cora
Citeseer
Pubmed
Cora CA
20News
MN40
Congress
Walmart
Senute
House
HGNN
2.301
2,500
2.123
2.506
2.199
3.958
1.950
20.277
2.5-10
2.513
HNHN
2.023
1.996
2,877
2.024
3.676
2.720
3.925
14.435
2.278
2.267
HCHA
3.290
3.485
4.448
2,850
2,894
4.072
6.675
8.358
3.297
4.584
UniGCN
4.553
1.932
6.693
2.511
6.825
4.315
8.497
22
1.920
UniGCNII
1.193
3.812
2.069
1.826
4.025
3.831
7.472
21 132
2.277
2.419
AllDeepSets
15.456
18.331
8.889
16.108
8:420
9.378
25.499
60.619
13:232
11.461
AllSefTransfomme
2.597
3.167
8.865
3.462
14.273
6.114
51.702
3.260
3.435
ED-HNN
4.267
4.458
4.407
6.588
11.072
39.791
31.137
2.487
ZEN (proposed)
0.266
0.736
0.807
0.280
0.237
0.197
0.093
0.672
0.003
0.007
24


Reviews:
[{"review_id": "Ab9VCrnuhF", "reviewer": "Reviewer_xaQe", "summary": "This paper introduces ZEN (Zero-Parameter Hypergraph Neural Network), a linear and training-free model for few-shot node classification on hypergraphs. ZEN is built upon a unified linearized formulation of existing HNNs and introduces two key innovations: (1) a tractable closed-form solution (TCS) for computing the weight matrix without iterative optimization, and (2) a redundancy-aware propagation (RAP) mechanism to eliminate self-information in multi-hop neighborhoods. The method shows consistent performance improvements over eight strong baselines across 11 real-world datasets. The model is interpretable and the paper includes a detailed ablation and case study.", "questions": "Did the authors consider involving more baselines for evaluation?", "limitations": "See weaknesses.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n1. The formulation of a fully parameter-free HNN using closed-form solutions and a redundancy-aware propagation is both original and impactful, especially for low-resource, few-shot learning scenarios.\n2. The theoretical contributions, including the derivation of the closed-form approximation (Theorem 1) and its assumptions, are carefully motivated and reasonably justified.\n3. The theoretical contributions, including the derivation of the closed-form approximation (Theorem 1) and its assumptions, are carefully motivated and reasonably justified.\n\nWeaknesses:\n1. As acknowledged by the authors, ZEN is specifically designed for node classification. Extensions to other hypergraph tasks (e.g., hyperedge prediction, clustering) are left as future work. This means the scope of applicability of this work is limited. The authors may consider analyzing the effectiveness of the work in other tasks.\n2. The paper discusses the success of linear GNNs such as SGC and APPNP, but does not directly compare ZEN to their hypergraph counterparts or generalizations. A direct comparison with a naïve linearized hypergraph model (beyond HGNN) would provide a stronger baseline.\n3. ZEN is limited to 2-hop propagation. While this is justified by computational cost and redundancy, it may restrict representational power on deeper hypergraph structures. A discussion on possible approximations or sampling strategies for deeper propagation would be a valuable addition.\n4. Lack of baselines. The authors focus on the task of few-shot node classification. However, they only conduct experiments with hypergraph neural networks (HNN) models. There are multiple few-shot node classification works that can act as baselines.", "was_revised": true, "final_justification": "I appreciate the additional experiments and clarifications, especially the extended baseline comparisons. I will keep my original score.", "rebuttal": "Thank you for your valuable comments. Your feedback is greatly appreciated and helps us improve the clarity and quality of our work. We will do our best to address them in the planned revision.\n\n**Response to Weakness 1:**\n> **Weakness 1:** As acknowledged by the authors, ZEN is specifically designed for node classification. Extensions to other hypergraph tasks (e.g., hyperedge prediction, clustering) are left as future work. This means the scope of applicability of this work is limited. The authors may consider analyzing the effectiveness of the work in other tasks.\n\nWe fully agree on the importance of assessing our method on other tasks. Some tasks, such as hyperedge prediction, can be reformulated as node classification, allowing ZEN to be applied without modification—for example, by treating hyperedges as nodes using $X_e=D_e^{-1}H^\\top X, H_e=H^\\top$. For tasks not easily reducible to this setting, ZEN may require further adaptation. We plan to explore such extensions in future work.\n\n**Response to Weakness 2:**\n> **Weakness 2:** The paper discusses the success of linear GNNs such as SGC and APPNP, but does not directly compare ZEN to their hypergraph counterparts or generalizations. A direct comparison with a naïve linearized hypergraph model (beyond HGNN) would provide a stronger baseline.\n\nThank you for your detailed feedback. We have additionally included three representative linear GNNs (SGC [1], APPNP [2], and SSGC [3]) and three linearized hypergraph models based on UniGCNII [4], AllDeepSets [5], and ED-HNN [6]. For GNNs, we applied clique expansion to convert the hypergraph into a pairwise graph. As shown in the table below, ZEN demonstrates consistently strong performance, outperforming all linear GNN and linearized hypergraph baselines on the majority of datasets.\n\n|Method|Cora|Citeseer|Pubmed|Cora\\_ca|20News|MN40|Congress|Walmart|Senate|House|\n|:-|-|-|-|-|-|-|-|-|-|-|\n|SGC|44.1±9.8|40.4±5.3|57.2±5.5|50.8±5.9|58.1±5.1|91.6±0.7|72.5±3.1|24.1±3.2|51.3±2.7|52.3±1.4|\n|APPNP|45.8±9.5|39.6±4.7|55.4±7.1|50.0±6.7|59.9±5.8|92.8±0.3|69.1±6.8|24.7±3.3|71.2±5.4|70.0±8.4|\n|SSGC|42.6±10.9|40.7±4.7|57.1±4.9|52.6±6.7|59.9±5.2|93.8±0.2|81.1±2.4|26.6±3.4|**73.0±0.7**|71.7±3.2|\n|Lin HGNN|42.7±8.4|34.3±9.4|51.7±6.6|49.6±6.9|68.0±6.0|94.5±0.5|83.7±5.0|29.0±6.4|55.8±5.1|57.8±6.5|\n|Lin UniGCNII|39.4±7.4|40.0±4.0|56.9±4.9|49.8±6.1|59.5±5.2|96.6±0.4|73.4±3.1|17.9±1.7|71.2±10.8|68.2±10.6|\n|Lin AllDeepSets|40.5±8.2|40.8±4.5|56.1±6.0|51.0±5.8|66.6±5.0|96.6±0.3|**90.4±1.5**|26.4±3.0|61.0±4.7|65.1±13.1|\n|Lin ED-HNN|42.0±5.6|39.6±4.2|55.8±5.4|47.4±5.8|61.5±6.0|97.3±0.4|78.4±2.7|18.8±1.6|71.1±9.9|70.2±9.3|\n|ZEN (**ours**)|**51.9±10.1**|**49.1±4.8**|**62.6±3.9**|**60.0±6.2**|**68.6±4.8**|**97.6±0.3**|87.0±4.8|**43.9±3.1**|70.4±10.0|**73.2±6.3**|\n\n**Response to Weakness 3:**\n> **Weakness 3:** ZEN is limited to 2-hop propagation. While this is justified by computational cost and redundancy, it may restrict representational power on deeper hypergraph structures. A discussion on possible approximations or sampling strategies for deeper propagation would be a valuable addition.\n\nThank you for your suggestion. We agree that some datasets may require models with higher-hop propagations for better expressivity. Without approximation, the exact computation of RSI requires high computational cost.\n\nOne possible approximation is to estimate the probability that a random walker returns to the starting node after $l$ steps, where $l$ is the number of hops that we aim to model. A simple pseudocode for this strategy is as follows:\n- Start at node $i$.  \n- At each step:  \n  1. Uniformly sample a hyperedge incident to the current node.  \n  2. Uniformly sample a node connected to that hyperedge.  \n- Repeat this process for $l$ steps.  \n- Record whether the walk ends at the original node $i$.\nThe return probability may serve as an approximation of node-wise self-information at a given hop.\n\nAnother possible approximation is to use Hutchinson's Estimator, which provides an unbiased stochastic estimate of the diagonal entries. The equation is given by:\n$$\n\\operatorname{diag}(A) \\approx \\frac{1}{m} \\sum_{k=1}^m z^{(k)} \\odot \\left( A z^{(k)} \\right)\n$$\nwhere:\n- $m$ is the number of random probe vectors,\n- each $z^{(k)} \\in \\mathbb{R}^n$ is a random vector with entries independently sampled from the Rademacher distribution (i.e., each entry is $+1$ or $-1$ with equal probability),\n- $\\odot$ denotes the element-wise (Hadamard) product.\n\nThe expectation satisfies\n$$\n\\operatorname{diag}(A) = \\mathbb{E} \\left[ z \\odot (A z) \\right].\n$$\n\nHutchinson's Estimator does not require explicit storage of the entire matrix $A$; instead, it relies solely on matrix-vector multiplications with randomized vectors $z$. This characteristic renders Hutchinson's Estimator particularly suitable and computationally efficient for hypergraph structures.\n\nWe consider these approaches to be promising directions for extending ZEN to scenarios where deeper propagation may provide additional benefits.\n\n**Response to Weakness 4:**\n> **Weakness 4:** Lack of baselines. The authors focus on the task of few-shot node classification. However, they only conduct experiments with hypergraph neural networks (HNN) models. There are multiple few-shot node classification works that can act as baselines.\n\nWe appreciate the suggestion to strengthen our evaluation by including additional baselines. To the best of our knowledge, there has been no hypergraph model explicitly addressing few-shot node classification as a direct keyword. In response, we have included two semi-supervised hypergraph models—LEGCN [7] and HyperND [8]—as additional baselines. LEGCN utilizes a line expansion approach to adapt hypergraphs to conventional GNN architectures, while HyperND introduces a diffusion-based mechanism for improved label propagation in hypergraphs. As shown in the table below, ZEN consistently demonstrates strong performance, outperforming all additional baselines across the majority of datasets.\n\n|Method|Cora|Citeseer|Pubmed|Cora\\_ca|20News|MN40|Congress|Walmart|Senate|House|\n|:-|-|-|-|-|-|-|-|-|-|-|\n|LEGCN|37.6±5.2|37.3±3.8|58.1±4.0|37.6±5.2|49.4±4.3|93.3±0.7|72.1±3.4|O.O.M|71.3±8.5|72.9±7.0|\n|HyperND|39.1±6.2|35.3±4.8|56.5±4.5|41.7±4.6|54.7±3.8|91.4±0.5|73.6±4.9|13.6±1.7|**73.9±7.5**|72.8±7.1|\n|ZEN (**ours**)|**51.9±10.1**|**49.1±4.8**|**62.6±3.9**|**60.0±6.2**|**68.6±4.8**|**97.6±0.3**|**87.0±4.8**|**43.9±3.1**|70.4±10.0|**73.2±6.3**|\n\n**References**\n\n[1] Felix Wu, Amauri Souza, Tianyi Zhang, Christopher Fifty, Tao Yu, and Kilian Weinberger. Simplifying graph convolutional networks. In PMLR, 2019. URL https://proceedings.mlr.press/v97/wu19e.html.\n\n[2] Johannes Gasteiger, Aleksandar Bojchevski, and Stephan Günnemann. Predict then propagate: Graph neural networks meet personalized pagerank, 2022. URL https://arxiv.org/abs/1810.05997.\n\n[3] Hao Zhu and Piotr Koniusz. Simple spectral graph convolution. In International conference on learning representations, 2021.\n\n[4] Jing Huang and Jie Yang. Unignn: a unified framework for graph and hypergraph neural networks, 2021. URL https://arxiv.org/abs/2105.00956.\n\n[5] Eli Chien, Chao Pan, Jianhao Peng, and Olgica Milenkovic. You are allset: A multiset function framework for hypergraph neural networks, 2022. URL https://arxiv.org/abs/2106.13264.\n\n[6] Peihao Wang, Shenghao Yang, Yunyu Liu, Zhangyang Wang, and Pan Li. Equivariant hypergraph diffusion neural operators, 2023. URL https://arxiv.org/abs/2207.06680.\n\n[7] Yang, Chaoqi and Wang, Ruijie and Yao, Shuochao and Abdelzaher, Tarek. Semi-supervised hypergraph node classification on hypergraph line expansion. CIKM, 2022. URL https://arxiv.org/abs/2005.04843.\n\n[8] Konstantin Prokopchik, Austin R Benson, and Francesco Tudisco. Nonlinear feature diffusion on hypergraphs. PMLR, 2022. URL https://arxiv.org/pdf/2103.14867."}, {"review_id": "33UyMh1X0w", "reviewer": "Reviewer_W1xk", "summary": "This paper focuses on the few-shot node classification task on hypergraphs, and proposes a model, ZEN, to capture high-order relationships while maintaining strong generalization ability and scalability. Experiments on 11 real-world datasets demonstrate the effectiveness and efficiency of the proposed method.", "questions": "Q1. More experiments are needed to evaluate the effectiveness of the proposed method under the $n$-way $k$-shot setting.\n\nQ2. The ablation study (Table 6) shows that ZEN underperforms compared to the No TCS variant on the 20News and MN40 datasets, and compared to the No RAP variant on the Senate dataset. Thus, the limitations of ZEN need to be clarified in the revised manuscript.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "S1. This paper presents a comprehensive study on liner hypergraph neural networks.\n\nS2. Two techniques, tractable closed-form approximation and redundancy-aware propagation, are developed to improve the efficiency and the corresponding time complexity analysis is provided.\n\nS3. Experiments on real-world datasets have been conducted and the results show the superiority of the proposed method.\n\nW1. While the paper addresses the problem of few-shot node classification, it does not follow the conventional $n$-way $k$-shot learning paradigm in the evaluation setup. This deviation from standard practices should be discussed or justified.\n\nW2. The ablation study shows that removing certain components of ZEN may lead to improved performance, which requires further discussion.", "was_revised": true, "final_justification": "The authors have conducted additional experiments in the n-way k-shot paradigm and provided a more thorough ablation study. Thus, I will maintain my original score.", "rebuttal": "Thanks for your thoughtful feedback. We appreciate your constructive review, which has helped us improve the clarity and quality of the paper.\n\n**Response to Weakness 1:**\n> **Weakness 1:** While the paper addresses the problem of few-shot node classification, it does not follow the conventional n-way k-shot learning paradigm in the evaluation setup. This deviation from standard practices should be discussed or justified.\n\nTo address the concerns, we would like to respectfully provide clarification on our task formulation. We define our few-shot node classification task as a transductive, semi-supervised node classification under severe label constraints. In standard n-way k-shot learning, the entire set of classes is typically partitioned into disjoint training and test sets. From each set, n classes are sampled, and k labeled examples per class are selected to construct support sets for episodic training. The model is trained on a variety of such tasks in order to evaluate its capacity for adaptation and generalization to novel, unseen tasks. In contrast, our setting can be regarded as a more challenging variant, where the class set remains fixed and identical to the test classes, and n corresponds to the total number of classes.\n\nTo avoid potential confusion with n-way k-shot learning from the meta-learning literature, we are open to revising our terminology in the revised version of our paper to reflect this distinction more clearly; e.g., hard few-shot node classification or semi-supervised node classification with hard label scarcity.\n\nOn the other hand, it would be interesting to see how our approach fits into the standard n-way k-shot setting. To the best of our knowledge, this setup has not been explicitly explored for general hypergraphs. Accordingly, we designed a new evaluation protocol by drawing inspiration from the experimental setup of Meta-GNN [1] and adopting hyperparameter configurations aligned with those used in ED-HNN [2]. The results are reported in the table below.\n\n|Method|Cora 2-way 3-shot|Cora 2-way 1-shot|Citeseer 2-way 3-shot|Citeseer 2-way 1-shot|\n|:-|-|-|-|-|\n|HGNN|64.4±0.1|55.9±0.1|60.1±0.2|54.9±0.2|\n|UniGCNII|68.0±0.1|58.9±0.2|65.2±0.1|56.8±1.6|\n|AllDeepSets|52.3±0.2|48.7±0.2|51.1±0.2|50.2±0.2|\n|ZEN (**ours**)|**73.3±0.1**|**62.6±0.1**|**71.5±0.1**|**62.3±0.1**|\n\nThe strong performance of ZEN in this setting highlights its potential as an effective approach for standard few-shot tasks on hypergraphs. Interestingly, ZEN yields promising results even when applied solely to test episodes, suggesting a potentially valuable and underexplored direction. We appreciate the opportunity to highlight this aspect and will consider it an important avenue for future research.\n\n**Response to Weakness 2:**\n> **Weakness 2:** The ablation study shows that removing certain components of ZEN may lead to improved performance, which requires further discussion.\n\nIt is true that removing RAP or TCS leads to performance improvements on some datasets (Congress and Senate for RAP, 20News and MN40 for TCS). To analyze this behavior, we examine the ratio of hyperedges to nodes (i.e., density), which we believe an important factor to understand the characteristic of a dataset. The results are reported in the table below.\n\n|Dataset|Cora|Citeseer|Pubmed|Cora-CA|20News|MN40|Congress|Walmart|Senate|House|\n|:-|-|-|-|-|-|-|-|-|-|-|\n|Density|0.5835|0.3258|0.4041|0.3959|0.0062|1.0000|48.3946|0.7868|1.1160|0.2636|\n\nWe observe that RAP improves performance on datasets with low density, while it slightly degrades performance on Congress and Senate, which have densities greater than 1. This aligns with the intuition: as the number of neighbors increases, each node's contribution diminishes due to degree normalization, reducing the relative influence of self-information. In such cases, removing self-information via RAP may have limited or even adverse effects.\n\nAs for TCS, it assumes *good* node embeddings. In the ablation study, the impact of removing TCS is more significant on 20News than on MN40. This may due to the extremely low density of 20News, where ZEN, which relies solely on propagation to refine embeddings, may have struggled to produce sufficiently informative representations.\n\nOn the other hand, the result on MN40 may have been influenced by its large number of classes — 40, the highest among all datasets. The relative Frobenius norm error of TCS is given by:\n\n$$\n\\frac{\\left\\|\\frac{1}{\\lambda_1^2}M_1+\\frac{1}{\\lambda_2^2}M_2+\\frac{1}{\\lambda_3^2}M_3 - \\frac{1}{\\epsilon}\\left(\\frac{1}{\\lambda_1}M_1+\\frac{1}{\\lambda_2}M_2+\\frac{1}{\\lambda_3}M_3\\right)\\right\\|_F}{\\left\\|\\frac{1}{\\lambda_1^2}M_1+\\frac{1}{\\lambda_2^2}M_2+\\frac{1}{\\lambda_3^2}M_3\\right\\|_F}\n$$\n\nAs shown in the table below, this approximation error slightly increases with the number of classes:\n\n|$ε$|$c=4$|$c=40$|\n|-|-|-|\n|0.1|1.1064%|1.1753%|\n|0.01|0.1006%|0.1010%|\n|0.001|0.0100%|0.0100%|\n\nAlthough the error remains small overall, the increase suggests that the approximation may become less accurate as the number of classes grows, which could partially explain the results observed on MN40.\n\n**References**\n\n[1] Fan Zhou, Chengtai Cao, Kunpeng Zhang, Goce Trajcevski, Ting Zhong, and Ji Geng. Meta-gnn: On few-shot node classification in graph meta-learning. CIKM, 2019. URL https://doi.org/10.1145/3357384.3358106.\n\n[2] Peihao Wang, Shenghao Yang, Yunyu Liu, Zhangyang Wang, and Pan Li. Equivariant hypergraph diffusion neural operators, 2023. URL https://arxiv.org/abs/2207.06680."}, {"review_id": "J0dTQ50mEK", "reviewer": "Reviewer_SfcC", "summary": "This paper proposes ZEN, which is a parameter-free hypergraph neural network. They theoretically derive their parameter-free solution. The experimental study shows the effectiveness and efficiency of ZEN compared with 8 baselines in ten datasets.", "questions": "Q1. Why do the authors focus on few-shot classifications? Is it possible to use ZEN for non-few-shot settings?\n\nQ2. Could you add the actual running time in ZEN?", "limitations": "Yes, but very short.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strong points.\n\nS1. It is surprising that the parameter-free method ZEN outperforms hypergraph neural networks with learning parameters.\n\nS2. Their theoretical analysis supports their structural design of ZEN.\n\nS3. The experimental study is sufficient, validating efficiency and effectiveness in ten datasets, analyzing the effectiveness of components in ZEN as an ablation study, and testing interpretability.\n\nWeak points.\n\nW1. It is unclear why the authors focus only on few-shot node classification. ZEN seems distinct from the few-shot node classification. And, it is unclear what the performance is if the size of the training data varies.\n\nW2. They do not show the actual running time in ZEN.", "was_revised": true, "final_justification": "This paper provides a theoretical analysis of parameter-free models and empirical experiments that showed their performance. Their rebuttals clarified the issues of presentations. I believe that this paper is ready to be accepted in NeurIPS.", "rebuttal": "Thank you for your recognition of our work and providing the valuable suggestions and constructive comments. Your thoughtful feedback is deeply appreciated and will greatly assist us in refining and strengthening the manuscript.\n\n**Response to Question 1:**\n> **Question 1:** Why do the authors focus on few-shot classifications? Is it possible to use ZEN for non-few-shot settings?\n\nZEN is a parameter-free model, which gives it a particular advantage in settings where labeled data is scarce and training is challenging. While ZEN was orginally designed for few-shot setting, our results in the tables below show it performs strongly with more training samples as well. ZEN achieves top average ranks in both 10-shot and 20-shot settings, suggesting it scales well beyond few-shot scenarios.\n\n### 10-shot classification (10 training nodes per class)\n\n|Method|Cora|Citeseer|Pubmed|Cora_ca|20News|MN40|Congress|Walmart|Senate|House|Avg. Rank|\n|:-|-|-|-|-|-|-|-|-|-|-|:-:|\n|HGNN|59.1±5.8|49.3±3.1|60.5±7.0|64.9±4.4|75.5±1.2|94.9±0.2|87.2±0.9|46.9±1.2|58.0±2.5|64.3±4.4|6.1|\n|HNHN|42.5±5.0|42.8±2.8|58.9±3.3|46.7±3.5|51.2±4.9|94.0±0.6|47.8±1.8|17.1±2.2|69.7±6.1|70.3±7.6|9.9|\n|HCHA|59.0±5.4|49.7±2.9|60.0±5.6|64.3±5.6|**75.6±1.3**|94.9±0.2|87.2±0.9|47.1±1.7|57.9±2.8|63.9±4.4|5.8|\n|UniGCN|60.9±5.8|51.1±3.0|61.2±6.9|66.9±3.0|73.7±1.7|96.2±0.2|**91.4±0.7**|45.4±1.4|62.6±2.5|70.3±3.0|4.1|\n|UniGCNII|56.8±5.1|49.2±2.9|61.1±6.4|63.2±3.8|70.9±2.1|97.0±0.2|87.7±1.9|28.7±2.5|**75.1±4.3**|73.7±2.9|4.8|\n|AllDeepSets|57.3±4.4|50.6±2.6|61.4±5.1|62.8±4.1|62.2±2.3|95.3±0.3|72.6±5.8|35.5±2.7|69.5±6.7|70.0±3.1|6.9|\n|AllSetTransformer|57.5±5.0|54.1±2.4|63.4±6.6|**67.6±4.2**|72.7±1.4|95.9±0.1|83.1±2.2|45.4±3.7|73.1±5.5|71.0±4.5|3.9|\n|ED-HNN|58.8±4.3|51.1±2.8|60.6±7.0|64.7±3.7|71.1±2.2|96.3±0.2|90.7±1.8|**47.2±2.3**|66.4±9.3|62.5±7.9|4.9|\n|ZEN (**ours**)|**61.4±4.2**|**59.2±2.3**|**68.8±3.4**|65.6±3.9|73.0±2.4|**97.9±0.1**|86.5±2.6|47.1±3.2|74.7±1.9|**73.9±6.4**|**1.7**|\n\n### 20-shot classification (20 training nodes per class)\n\n|Method|Cora|Citeseer|Pubmed|Cora_ca|20News|MN40|Congress|Walmart|Senate|House|Avg. Rank|\n|:-|-|-|-|-|-|-|-|-|-|-|:-:|\n|HGNN|62.5±3.1|56.6±1.9|69.7±3.9|70.3±1.9|**76.7±1.1**|95.1±0.2|87.8±0.5|49.5±2.4|60.7±3.2|64.8±2.2|5.4|\n|HNHN|48.3±3.9|50.1±2.3|62.6±1.6|52.9±2.9|52.7±6.3|95.2±0.5|49.9±3.0|21.8±1.8|73.5±3.9|72.0±4.1|9.7|\n|HCHA|64.2±3.9|56.8±2.1|69.4±3.3|70.0±1.9|**76.7±1.0**|95.1±0.1|87.9±0.4|49.4±2.2|61.1±3.3|64.9±2.3|4.9|\n|UniGCN|65.4±3.3|58.5±1.2|70.8±2.6|71.3±1.8|72.6±2.1|96.6±0.1|**91.6±0.3**|49.0±2.1|62.0±2.3|72.7±0.9|3.2|\n|UniGCNII|64.3±2.9|57.0±1.2|68.9±3.4|68.6±2.5|74.1±1.5|97.6±0.1|88.3±1.9|32.0±1.5|**77.6±2.0**|76.1±0.9|3.6|\n|AllDeepSets|62.4±3.7|57.8±1.8|67.1±2.2|65.9±1.3|67.6±2.5|96.7±0.3|77.7±5.8|44.0±2.4|73.1±2.2|73.3±2.5|6.6|\n|AllSetTransformer|63.7±1.6|59.5±2.1|69.7±2.3|69.8±2.2|74.2±1.2|96.0±0.2|89.2±0.5|49.3±1.8|76.1±3.8|75.6±2.7|3.4|\n|ED-HNN|64.2±3.0|57.3±1.6|69.6±3.4|69.3±1.6|70.9±1.9|96.7±0.2|90.1±2.1|50.6±3.3|72.3±6.7|71.3±3.7|4.3|\n|ZEN (**ours**)|**67.9±1.4**|**64.0±1.4**|**71.3±1.6**|**72.0±2.0**|73.7±1.2|**98.2±0.1**|88.0±2.9|**51.3±1.1**|74.1±5.2|**76.7±0.4**|**1.9**|\n\n**Response to Question 2:**\n\n> **Question 2:** Could you add the actual running time in ZEN?\n\nThank you for the suggestion. The actual running times are reported in the table below, with all values measured in seconds (s). ZEN runs in less than a second across all datasets, achieving as fast as 0.003s on the Senate dataset.\n\n|Method|Cora|Citeseer|Pubmed|Cora_ca|20News|MN40|Congress|Walmart|Senate|House|\n|:-|-|-|-|-|-|-|-|-|-|-|\n|HGNN|2.301|2.800|2.423|2.506|2.499|3.958|4.950|20.277|2.540|2.543|\n|HNHN|2.023|1.996|2.877|2.024|3.676|2.720|3.925|14.435|2.278|2.267|\n|HCHA|3.290|3.485|4.448|2.850|2.894|4.072|6.675|8.558|3.297|4.584|\n|UniGCN|4.553|1.932|6.693|2.511|6.825|4.315|8.497|22.121|2.342|1.920|\n|UniGCNII|4.193|3.812|2.069|4.826|4.025|3.831|7.472|24.452|2.277|2.419|\n|AllDeepSets|15.456|18.331|8.889|16.108|8.420|9.378|25.499|60.619|13.232|11.461|\n|AllSetTransformer|2.597|3.167|8.865|3.462|14.273|4.924|6.114|51.702|3.260|3.435|\n|ED-HNN|4.267|4.458|4.407|6.588|11.072|5.664|39.791|31.137|2.336|2.487|\n|ZEN (**ours**)|**0.266**|**0.736**|**0.807**|**0.280**|**0.237**|**0.197**|**0.093**|**0.672**|**0.003**|**0.007**|"}, {"review_id": "FWxgdaQF9Z", "reviewer": "Reviewer_UPmr", "summary": "The paper proposes a novel parameter-free hypergraph neural network for few-shot node classification. Tractable Closed-form Solution (TCS) eliminates iterative training, and Redundancy-Aware Propagation (RAP) explicitly removes self-information redundancy. Comprehensive tests show improved accuracy and efficiency compared to previous methods.", "questions": "* Can the authors provide error bounds for closed-form approximation in Theorem 1?\n* Can the authors clarify the unit used in Table 4?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n* The method shows improved accuracy and efficiency compared to previous methods.\n* The paper is technically solid with rigorous theoretical analysis.\n* The paper is well-written and easy to read.\n\n**Weakness**\n* There is no error bounds for closed-form approximation in Theorem 1.\n* Table 4 presents values without specifying units. If the time for the proposed method is used as the unit of time, please state this clearly and maybe provide the original data in the appendix.\n* Typos exist. For example, $W_*$ should be $W^*$ in Proof of lemma 1.", "was_revised": true, "final_justification": "The rebuttal answers all my questions, so I keep my score.", "rebuttal": "We greatly appreciate your high recognition of our work, and are eager to share our thoughts with you.\n\n**Response to Question 1:**\n> **Question 1:** Can the authors provide error bounds for closed-form approximation in Theorem 1?\n\nYes, we believe more discussion on the error bounds can make our theoretical claim stronger. We present a summary of our analysis as follows, which will be included in the final version of our paper as well.\n\nIn TCS, our approximation is given as follows:\n$$\n\\frac{1}{\\lambda_1^2}M_1+\\frac{1}{\\lambda_2^2}M_2+\\frac{1}{\\lambda_3^2}M_3 \\approx \\frac{1}{\\epsilon}\\left(\\frac{1}{\\lambda_1}M_1+\\frac{1}{\\lambda_2}M_2+\\frac{1}{\\lambda_3}M_3\\right),\n$$  \nwhere  \n- $\\lambda_1 = \\epsilon$  \n- $\\lambda_2 = (1 - 2\\epsilon)k + \\epsilon$  \n- $\\lambda_3 = k(1 - 2\\epsilon + \\epsilon c) + \\epsilon$\n\nand the matrices form an orthogonal decomposition:  \n- $M_1 = I_{kc} - \\frac{1}{k}(I_c \\otimes J_k)$  \n- $M_2 = \\frac{1}{k}(I_c \\otimes J_k) - \\frac{1}{kc}J_{kc}$  \n- $M_3 = \\frac{1}{kc} J_{kc}$\n\nHere, $J$ denotes an all-one matrix and $\\otimes$ is the Kronecker product.\n\nTo evaluate the approximation quality, we compute the relative Frobenius norm error:  \n$$\n\\frac{\\left\\|\\frac{1}{\\lambda_1^2}M_1+\\frac{1}{\\lambda_2^2}M_2+\\frac{1}{\\lambda_3^2}M_3 - \\frac{1}{\\epsilon}\\left(\\frac{1}{\\lambda_1}M_1+\\frac{1}{\\lambda_2}M_2+\\frac{1}{\\lambda_3}M_3\\right)\\right\\|_F}{\\left\\|\\frac{1}{\\lambda_1^2}M_1+\\frac{1}{\\lambda_2^2}M_2+\\frac{1}{\\lambda_3^2}M_3\\right\\|_F}\n$$\n\nFor a representative case with $k = 5$, $c = 10$, the relative errors are as follows:\n\n| $\\epsilon$ | Relative Error |\n|------------|----------------|\n| 0.1        | 1.14%          |\n| 0.01       | 0.10%          |\n| 0.001      | 0.01%          |\n\nThese results indicate that the approximation becomes increasingly accurate as $\\epsilon$ decreases. This supports the validity of our approach, particularly in the regime where $\\epsilon$ is sufficiently small.\n\n**Response to Question 2:**\n> **Question 2:** Can the authors clarify the unit used in Table 4?\n\nThank you for your suggestion. As shown in Table 4, the values represent the running times relative to ZEN, and thus no explicit unit is indicated. The actual running times are reported in the table below, with all values measured in seconds (s). ZEN runs in less than a second across all datasets, achieving as fast as 0.003s on the Senate dataset.\n\n|Method|Cora|Citeseer|Pubmed|Cora_ca|20News|MN40|Congress|Walmart|Senate|House|\n|:-|-|-|-|-|-|-|-|-|-|-|\n|HGNN|2.301|2.800|2.423|2.506|2.499|3.958|4.950|20.277|2.540|2.543|\n|HNHN|2.023|1.996|2.877|2.024|3.676|2.720|3.925|14.435|2.278|2.267|\n|HCHA|3.290|3.485|4.448|2.850|2.894|4.072|6.675|8.558|3.297|4.584|\n|UniGCN|4.553|1.932|6.693|2.511|6.825|4.315|8.497|22.121|2.342|1.920|\n|UniGCNII|4.193|3.812|2.069|4.826|4.025|3.831|7.472|24.452|2.277|2.419|\n|AllDeepSets|15.456|18.331|8.889|16.108|8.420|9.378|25.499|60.619|13.232|11.461|\n|AllSetTransformer|2.597|3.167|8.865|3.462|14.273|4.924|6.114|51.702|3.260|3.435|\n|ED-HNN|4.267|4.458|4.407|6.588|11.072|5.664|39.791|31.137|2.336|2.487|\n|ZEN (**ours**)|**0.266**|**0.736**|**0.807**|**0.280**|**0.237**|**0.197**|**0.093**|**0.672**|**0.003**|**0.007**|"}]

Author rebuttal:

Meta-review: This paper introduces ZEN, a parameter-free hypergraph neural network for few-shot node classification. ZEN is built on a unified linearized formulation of HNNs. Two key novelties include a tractable closed-form solution for weight computation and a redundancy-aware propagation mechanism to avoid self-information leakage. Extensive experiments across 11 real-world datasets show consistent improvements over strong baselines.

The main strengths lie in its originality and simplicity: the parameter-free formulation is theoretically grounded and empirically outperforms previous hypergraph neural networks. The paper is also clearly written, with thorough ablations and case studies.

The weaknesses are relatively minor. The method is currently limited to node classification; broader applicability to other hypergraph tasks remains unexplored. The evaluation setup only partially follows the conventional few-shot learning paradigm, which could be better justified. ZEN is restricted to 2-hop propagation, which may limit representation capacity, and some comparisons (e.g., with more few-shot methods or deeper hypergraph variants) are missing. Finally, further clarification on time complexity, error bounds, and reporting conventions would improve the presentation.

Overall, despite these weaknesses, the paper makes a solid and meaningful contribution to hypergraph learning. Its theoretical insights, empirical effectiveness, and efficiency justify acceptance. I recommend acceptance.
Decision: Accept (poster)