Hypergraph Neural Networksgaoyue.org/paper/HGNN.pdf · Introduction Graph-based convolutional neural networks (Kipf and Welling 2017), (Defferrard, Bresson, and Vandergheynst 2016)

Hypergraph Neural Networks

Yifan Feng†, Haoxuan You‡, Zizhao Zhang‡, Rongrong Ji†, Yue Gao‡∗† Fujian Key Laboratory of Sensing and Computing for Smart City,

School of Information Science and Engineering, Xiamen University, 361005, China‡BNRist, KLISS, School of Software, Tsinghua University, 100084, China.

evanfeng97, [email protected], [email protected], zz-z14,[email protected]

Abstract

In this paper, we present a hypergraph neural networks(HGNN) framework for data representation learning, whichcan encode high-order data correlation in a hypergraph struc-ture. Confronting the challenges of learning representationfor complex data in real practice, we propose to incorpo-rate such data structure in a hypergraph, which is more flexi-ble on data modeling, especially when dealing with complexdata. In this method, a hyperedge convolution operation isdesigned to handle the data correlation during representationlearning. In this way, traditional hypergraph learning proce-dure can be conducted using hyperedge convolution opera-tions efficiently. HGNN is able to learn the hidden layer rep-resentation considering the high-order data structure, whichis a general framework considering the complex data correla-tions. We have conducted experiments on citation networkclassification and visual object recognition tasks and com-pared HGNN with graph convolutional networks and othertraditional methods. Experimental results demonstrate thatthe proposed HGNN method outperforms recent state-of-the-art methods. We can also reveal from the results that the pro-posed HGNN is superior when dealing with multi-modal datacompared with existing methods.

IntroductionGraph-based convolutional neural networks (Kipf andWelling 2017), (Defferrard, Bresson, and Vandergheynst2016) have attracted much attention in recent years. Dif-ferent from traditional convolutional neural networks, graphconvolution is able to encode the graph structure of differentinput data using a neural network model and it can be usedin the semi-supervised learning procedure. Graph convolu-tional neural networks have shown superiority on represen-tation learning compared with traditional neural networksdue to its ability of using data graph structure.

In traditional graph convolutional neural network meth-ods, the pairwise connections among data are employed. Itis noted that the data structure in real practice could be be-yond pairwise connections and even far more complicated.Confronting the scenarios with multi-modal data, the situa-tion for data correlation modelling could be more complex.

∗Corresponding author. This work was finished when YifanFeng visited Tsinghua University.Copyright c© 2019, Association for the Advancement of ArtificialIntelligence (www.aaai.org). All rights reserved.

Visual connections

Textconnections

Socialconnections

Tweets/Microblogs

Figure 1: Examples of complex connections on social me-dia data. Each color point represents a tweet or microblog,and there could be visual connections, text connections andsocial connections among them.

Figure 1 provides examples of complex connections on so-cial media data. On one hand, the data correlation can bemore complex than pairwise relationship, which is difficultto be modeled by a graph structure. On the other hand, thedata representation tends to be multi-modal, such as the vi-sual connections, text connections and social connections inthis example. Under such circumstances, traditional graphstructure has the limitation to formulate the data correlation,which limits the application of graph convolutional neuralnetworks. Under such circumstance, it is important and ur-gent to further investigate better and more general data struc-ture model to learn representation.

To tackle this challenging issue, in this paper, we proposea hypergraph neural networks (HGNN) framework, whichuses the hypergraph structure for data modeling. Comparedwith simple graph, on which the degree for all edges ismandatory 2, a hypergraph can encode high-order data cor-relation (beyond pairwise connections) using its degree-freehyperedges, as shown in Figure 2. In Figure 2, the graph isrepresented using the adjacency matrix, in which each edgeconnects just two vertices. On the contrary, a hypergraphis easy to be expanded for multi-modal and heterogeneous

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Figure 2: The comparison between graph and hypergraph.

data representation using its flexible hyperedges. For exam-ple, a hypergraph can jointly employ multi-modal data forhypergraph generation by combining the adjacency matrix,as illustrated in Figure 2. Therefore, hypergraph has beenemployed in many computer vision tasks such as classifi-cation and retrieval tasks (Gao et al. 2012). However, tra-ditional hypergraph learning methods (Zhou, Huang, andScholkopf 2007) suffer from their high computation com-plexity and storage cost, which limits the wide applicationof hypergraph learning methods.

In this paper, we propose a hypergraph neural networksframework (HGNN) for data representation learning. In thismethod, the complex data correlation is formulated in a hy-pergraph structure, and we design a hyperedge convolutionoperation to better exploit the high-order data correlationfor representation learning. More specifically, HGNN is ageneral framework which can incorporate with multi-modaldata and complicated data correlations. Traditional graphconvolutional neural networks can be regarded as a specialcase of HGNN. To evaluate the performance of the pro-posed HGNN framework, we have conducted experimentson citation network classification and visual object recog-nition tasks. The experimental results on four datasets andcomparisons with graph convolutional network (GCN) andother traditional methods have shown better performanceof HGNN. These results indicate that the proposed HGNNmethod is more effective on learning data representation us-ing high-order and complex correlations.

The main contributions of this paper are two-fold:1. We propose a hypergraph neural networks framework,

i.e., HGNN, for representation learning using hypergraphstructure. HGNN is able to formulate complex and high-order data correlation through its hypergraph structureand can be also efficient using hyperedge convolutionoperations. It is effective on dealing with multi-modaldata/features. Moreover, GCN (Kipf and Welling 2017)can be regarded as a special case of HGNN, for which theedges in simple graph can be regarded as 2-order hyper-edges which connect just two vertices.

2. We have conducted extensive experiments on citation net-work classification and visual object classification tasks.Comparisons with state-of-the-art methods demonstratethe effectiveness of the proposed HGNN framework. Ex-periments also indicate the better performance of the pro-posed method when dealing with multi-modal data.

Related WorkIn this section, we briefly review existing works of hyper-graph learning and neural networks on graph.

Hypergraph learningIn many computer vision tasks, the hypergraph structure hasbeen employed to model high-order correlation among data.Hypergraph learning is first introduced in (Zhou, Huang, andScholkopf 2007), as a propagation process on hypergraphstructure. The transductive inference on hypergraph aims tominimize the label difference among vertices with strongerconnections on hypergraph. In (Huang, Liu, and Metaxas2009), hypergraph learning is further employed in video ob-ject segmentation. (Huang et al. 2010) used the hypergraphstructure to model image relationship and conducted trans-ductive inference process for image ranking. To further im-prove the hypergraph structure, research attention has beenattracted for leaning the weights of hyperedges, which havegreat influence on modeling the correlation of data. In (Gaoet al. 2013), a l2 regularize on the weights is introduced tolearn optimal hyperedge weights. In (Hwang et al. 2008),the correlation among hyperedges is further explored by aassumption that highly correlated hyperedges should havesimilar weights. Regarding the multi-modal data, in (Gao etal. 2012), multi-hypergraph structure is introduced to assignweights for different sub-hypergraphs, which corresponds todifferent modalities.

Neural networks on graphSince many irregular data that do not own a grid-like struc-ture can only be represented in the form of graph, extending

Multimodal DataTraining Data Testing Data

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Hypergraph Neural Networks

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Figure 3: The proposed HGNN framework.

neural networks to graph structure has attracted great atten-tion from researchers. In (Gori, Monfardini, and Scarselli2005) and (Scarselli et al. 2009), the neural network ongraph is first introduced to apply recurrent neural networksto deal with graphs. For generalizing convolution network tograph, the methods are divided into spectral and non-spectralapproaches.

For spectral approaches, the convolution operation is for-mulated in spectral domain of graph. (Bruna et al. 2014) in-troduces the first graph CNN, which uses the graph Lapla-cian eigenbasis as an analogy of the Fourier transform.In (Henaff, Bruna, and LeCun 2015), the spectral filterscan be parameterized with smooth coefficients to makethem spatial-localized. In (Defferrard, Bresson, and Van-dergheynst 2016), a Chebyshev expansion of the graphLaplacian is further used to approximate the spectral filters.Then, in (Kipf and Welling 2017), the chebyshev polynomi-als are simplified into 1-order polynomials to form an effi-cient layer-wise propagation model.

For spatial approaches, the convolution operation is de-fined in groups of spatial close nodes. In (Atwood andTowsley 2016), the powers of a transition matrix is em-ployed to define the neighborhood of nodes. (Monti et al.2017) uses the local path operators in the form of Gaussianmixture models to generalize convolution in spatial domain.In (Velickovic et al. 2018), the attention mechanisms is in-troduced into the graph to build attention-based architectureto perform the node classification task on graph.

Hypergraph Neural NetworksIn this section, we introduce our proposed hypergraph neu-ral networks (HGNN). We first briefly introduce hypergraphlearning, and then the spectral convolution on hypergraphis provided. Following, we analyze the relations betweenHGNN and existing methods. In the last part of the section,some implementation details will be given.

Hypergraph learning statementWe first review the hypergraph analysis theory. Differentfrom simple graph, a hyperedge in a hypergraph connectstwo or more vertices. A hypergraph is defined as G =(V, E ,W), which includes a vertex set V , a hyperedge set E .Each hyperedge is assigned with a weight by W, a diagonal

matrix of edge weights. The hypergraph G can be denotedby a |V| × |E| incidence matrix H, with entries defined as

h(v, e) =

1, if v ∈ e0, if v 6∈ e, (1)

For a vertex v ∈ V , its degree is defined as d(v) =∑e∈E ω(e)h(v, e). For an edge e ∈ E , its degree is defined

as δ(e) =∑v∈V h(v, e). Further, De and Dv denote the di-

agonal matrices of the edge degrees and the vertex degrees,respectively.

Here let us consider the node(vertex) classification prob-lem on hypergraph, where the node labels should be smoothon the hypergraph structure. The task can be formulated asa regularization framework as introduced by (Zhou, Huang,and Scholkopf 2007):

arg minfRemp(f) + Ω(f), (2)

where Ω(f) is a regularize on hypergraph,Remp(f) denotesthe supervised empirical loss, f(·) is a classification func-tion. The regularize Ω(f) is defined as:

Ω(f) =1

2

∑e∈E

∑u,v∈V

w(e)h(u, e)h(v, e)

δ(e)( f(u)√d(u)

− f(v)√d(v)

)2,

(3)

We let θ = D−1/2v HWD−1e H>D

−1/2v and ∆ = I−Θ.

Then, the normalized Ω(f) can be written as

Ω(f) = f>∆, (4)

where ∆ is positive semi-definite, and usually called the hy-pergraph Laplacian.

Spectral convolution on hypergraphGiven a hypergraph G = (V, E ,∆) with n vertices,since the hypergraph Laplacian ∆ is a n × n positivesemi-definite matrix, the eigen decomposition ∆ = ΦΛΦ>

can be employed to get the orthonormal eigen vectorsΦ = diag(φ1, . . . , φn) and a diagonal matrix Λ =diag(λ1, . . . , λn) containing corresponding non-negative

Node Feature Transform

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Node Feature Aggregating N: Node Number

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Node Feature Hyperedge Feature Node Feature

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Figure 4: The illustration of the hyperedge convolution layer.

eigenvalues. Then, the Fourier transform for a signal x =(x1, . . . ,xn) in hypergraph is defined as x = Φ>x, wherethe eigen vectors are regarded as the Fourier bases and theeigenvalues are interpreted as frequencies. The spectral con-volution of signal x and filter g can be denoted as

g ? x = Φ((Φ>g) (Φ>x)) = Φg(Λ)Φ>x, (5)where denotes the element-wise Hadamard product andg(Λ) = diag(g(λ1), . . . ,g(λn)) is a function of the Fouriercoefficients. However, the computation cost in forward andinverse Fourier transform is O(n2). To solve the prob-lem, we can follow (Defferrard, Bresson, and Vandergheynst2016) to parametrize g(Λ) with K order polynomials. Fur-thermore, we use the truncated Chebyshev expansion asone such polynomial. Chebyshv polynomials Tk(x) is recur-sively computed by Tk(x) = 2xTk−1(x) − Tk−2(x), withT0(x) = 1 and T1(x) = x. Thus, the g(Λ) can be parame-tried as

g ? x ≈K∑k=0

θkTk(∆)x, (6)

where Tk(∆) is the Chebyshev polynomial of order k withscaled Laplacian ∆ = 2

λmax∆ − I. In Equation 6, the ex-

pansive computation of Laplacian Eigen vectors is excludedand only matrix powers, additions and multiplications areincluded, which brings further improvement in computationcomplexity. We can further let K = 1 to limit the orderof convolution operation due to that the Laplacian in hyper-graph can already well represent the high-order correlationbetween nodes. It is also suggested in (Kipf and Welling2017) that λmax ≈ 2 because of the scale adaptability ofneural networks. Then, the convolution operation can be fur-ther simplified to

g ? x ≈ θ0x− θ1D−1/2HWD−1e H>D−1/2v x, (7)where θ0 and θ1 is parameters of filters over all nodes. Wefurther use a single parameter θ to avoid the overfitting prob-lem, which is defined as

θ1 = − 12θ

θ0 = 12θD

−1/2v HD−1e H>D

−1/2v ,

(8)

Then, the convolution operation can be simplified to thefollowing expression

g ? x ≈ 1

2θD−1/2v H(W + I)D−1e H>D−1/2v x

≈ θD−1/2v HWD−1e H>D−1/2v x,

(9)

where (W + I) can be regarded as the weight of the hyper-edges. W is initialized as an identity matrix, which meansequal weights for all hyperedges.

When we have a hypergraph signal X ∈ Rn×C1 with nnodes and C1 dimensional features, our hyperedge convolu-tion can be formulated by

Y = D−1/2v HWD−1e H>D−1/2v XΘ, (10)

where W = diag(w1, . . . ,wn). Θ ∈ RC1×C2 is the param-eter to be learned during the training process. The filter Θis applied over the nodes in hypergraph to extract features.After convolution, we can obtain Y ∈ Rn×C2 , which can beused for classification.

Hypergraph neural networks analysisFigure 3 illustrates the details of the hypergraph neural net-works. Multi-modality datasets are divided into training dataand testing data, and each data contains several nodes withfeatures. Then multiple hyperedge structure groups are con-structed from the complex correlation of the multi-modalitydatasets. We concatenate the hyperedge groups to generatethe hypergraph adjacent matrix H. The hypergraph adjacentmatrix H and the node feature are fed into the HGNN to getthe node output labels. As introduced in the above section,we can build a hyperedge convolutional layer f(X,W,Θ)in the following formulation

X(l+1) = σ(D−1/2v HWD−1e H>D−1/2v X(l)Θ(l)), (11)

where X(1) ∈ RN×C is the signal of hypergraph at l layer,X(0) = X and σ denotes the nonlinear activation function.

The HGNN model is based on the spectral convolution onthe hypergraph. Here, we further investigate HGNN in the

property of exploiting high-order correlation among data. Asis shown in Figure 4, the HGNN layer can perform node-edge-node transform, which can better refine the featuresusing the hypergraph structure. More specifically, at first,the initial node feature X(1) is processed by learnable fil-ter matrix Θ(1) to extract C2-dimensional feature. Then, thenode feature is gathered according to the hyperedge to formthe hyperedge feature RE×C2 , which is implemented by themultiplication of H> ∈ RE×N. Finally the output node fea-ture is obtained by aggregating their related hyperedge fea-ture, which is achieved by multiplying matrix H. Denotethat Dv and De play a role of normalization in Equation 11.Thus, the HGNN layer can efficiently extract the high-ordercorrelation on hypergraph by the node-edge-node transform.

Relations to existing methods When the hyperedges onlyconnect two vertices, the hypergraph is simplified into a sim-ple graph and the Laplacian ∆ is also coincident with theLaplacian of simple graph up to a factor of 1

2 . Comparedwith the existing graph convolution methods, our HGNN cannaturally model high-order relationship among data, whichis effectively exploited and encoded in forming feature ex-traction. Compared with the traditional hypergraph method,our model is highly efficient in computation without the in-verse operation of Laplacian ∆. It should also be noted thatour HGNN has great expansibility toward multi-modal fea-ture with the flexibility of hyperedge generation.

ImplementationHypergraph construction In our visual object classifica-tion task, the features of N visual object data can be repre-sented as X = [x1, . . . ,xn]

>. We build the hypergraph ac-cording to the distance between two features. More specif-ically, Euclidean distance is used to calculate d(xi,xj). Inthe construction, each vertex represents one visual object,and each hyperedge is formed by connecting one vertex andits K nearest neighbors, which brings N hyperedges thatlinks K + 1 vertices. And thus, we get the incidence ma-trix H ∈ RN×N with N × (K + 1) entries equaling to 1while others equaling to 0. In the citation network classifi-cation, where the data are organized in graph structure, eachhyperedge is built by linking one vertex and their neighborsaccording to the adjacency relation on graph. So we also getN hyperedges and H ∈ RN×N.

Model for node classification In the problem of nodeclassification, we build the HGNN model as in Figure 3.The dataset is divided into training data and test data. Thenhypergraph is constructed as the section above, which gen-erates the incidence matrix H and corresponding De. Webuild a two-layer HGNN model to employ the powerful ca-pacity of HGNN layer. And the softmax function is used togenerate predicted labels. During training, the cross-entropyloss for the training data is back-propagated to update the pa-rameters Θ and in testing, the labels of test data is predictedfor evaluating the performance. When there are multi-modalinformation incorporate them by the construction of hyper-

edge groups and then various hyperedges are fused togetherto model the complex relationship on data.

ExperimentsIn this section, we evaluate our proposed hypergraph neu-ral networks on two task: citation network classification andvisual object recognition. We also compare the proposedmethod with graph convolutional networks and other state-of-the-art methods.

Dataset Cora Pumbed

Nodes 2708 19717Edges 5429 44338

Feature 1433 500Training node 140 60

Validation node 500 500Testing node 1000 1000

Classes 7 3

Table 1: Summary of the citation classification datasets.

Citation network classificationDatasets In this experiment, the task is to classify citationdata. Here, two widely used citation network datasets, i.e.,Cora and Pubmed (Sen et al. 2008) are employed. The ex-perimental setup follows the settings in (Yang, Cohen, andSalakhutdinov 2016). In both of those two datasets, the fea-ture for each data is the bag-of-words representation of doc-uments. The data connection, i.e., the graph structure, in-dicates the citations among those data. To generate the hy-pergraph structure for HGNN, each time one vertex in thegraph is selected as the centroid and its connected verticesare used to generate one hyperedge including the centroiditself. Through this we can obtain the same size incidencematrix compared with the original graph. It is noted that asthere are no more information for data relationship, the gen-erated hypergraph constructure is quite similar to the graph.The Cora dataset contains 2708 data and 5% are used as la-beled data for training. The Pubmed dataset contains 19717data, and only 0.3% are used for training. The detailed de-scription for the two datasets listed in Table 1.

Experimental settings In this experiment, a two-layerHGNN is applied. The feature dimension of the hidden layeris set as 16 and the dropout (Srivastava et al. 2014) is em-ployed to avoid overfitting with drop rate p = 0.5. Wechoose the ReLU as the nonlinear activation function. Dur-ing the training process, we use Adam optimizer (Kingmaand Ba 2014) to minimize our cross-entropy loss functionwith a learning rate of 0.001. We have also compared theproposed HGNN with recent methods in these experiments.

Results and discussion The results of the experimental re-sults and comparisons on the citation network dataset areshown in Table 2. For our HGNN model, we report the aver-age classification accuracy of 100 runs on Core and Pumbed,

which is 81.6% and 80.1%. As shown in the results, the pro-posed HGNN model can achieve the best or comparable per-formance compared with the state-of-the-art methods. Com-pared with GCN, the proposed HGNN method can achieve aslight improvement on the Cora dataset and 1.1% improve-ment on the Pubmed dataset. We note that the generated hy-pergraph structure is quite similar to the graph structure asthere is neither extra nor more complex information in thesedata. Therefore, the gain obtained by HGNN is not very sig-nificant.

Method Cora Pubmed

DeepWalk (Perozzi, Al-Rfou,and Skiena 2014)

67.2% 65.3%

ICA (Lu and Getoor 2003) 75.1% 73.9%Planetoid (Yang, Cohen, andSalakhutdinov 2016)

75.7% 77.2%

Chebyshev (Defferrard, Bres-son, and Vandergheynst 2016)

81.2% 74.4%

GCN (Kipf and Welling 2017) 81.5% 79.0%

HGNN 81.6% 80.1%

Table 2: Classification results on the Cora and Pubmeddatasets.

Visual object classificationDatasets and experimental settings In this experiment,the task is to classify visual objects. Two public benchmarksare employed here, including the Princeton ModelNet40dataset (Wu et al. 2015) and the National Taiwan University(NTU) 3D model dataset (Chen et al. 2003), as shown in Ta-ble 3. The ModelNet40 dataset consists of 12,311 objectsfrom 40 popular categories, and the same training/testingsplit is applied as introduced in (Wu et al. 2015), where9,843 objects are used for training and 2,468 objects areused for testing. The NTU dataset is composed of 2,0123D shapes from 67 categories, including car, chair, chess,chip, clock, cup, door, frame, pen, plant leaf and so on. Inthe NTU dataset, 80% data are used for training and theother 20% data are used for testing. In this experiment, each3D object is represented by the extracted features. Here,two recent state-of-the-art shape representation methods areemployed, including Multi-view Convolutional Neural Net-work (MVCNN) (Su et al. 2015) and Group-View Convolu-tional Neural Network (GVCNN) (Feng et al. 2018). Thesetwo methods are selected due to that they have shown sat-isfactory performance on 3D object representation. We fol-low the experimental settings of MVCNN and GVCNN togenerate multiple views of each 3D object. Here, 12 virtualcameras are employed to capture views with a interval angleof 30 degree, and then both the MVCNN and the GVCNNfeatures are extracted accordingly.

To compare with GCN method, it is noted that there is noavailable graph structure in the ModelNet40 dataset and theNTU dataset. Therefore, we construct a probability graphbased on the distance of nodes. Given the features of data,

the affinity matrix A is generated to represent the relation-ship among different vertices, and Aij can be calculated by:

Aij = exp(−2Dij2

∆) (12)

where Dij indicates the Euclidean distance between nodei and node j. ∆ is the average pairwise distance betweennodes. For the GCN experiment with two features con-structed simple graphs, we simply average the two modal-ity adjacency matrices to get the fused graph structure forcomparison.

Dataset ModelNet40 NTU

Objects 12311 2012MVCNN Feature 4096 4096GVCNN Feature 2048 2048

Training node 9843 1639Testing node 2468 373

Classes 40 67

Table 3: The detailed information of the ModelNet40 andthe NTU datasets.

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Figure 5: An example of hyperedge generation in the vi-sual object classification task. Left: For each node we ag-gregate its N neighbor nodes by Euclidean distance to gen-erate a hyperedge. Right: To generate the multi-modality hy-pergraph adjacent matrix we concatenate adjacent matrix oftwo modality.

Hypergraph structure construction on visual datasetsIn experiments on ModelNet40 and NTU datasets, two hy-pergraph construction methods are employed. The first oneis based on single modality feature and the other one is basedon multi-modality feature. In the first case, only one featureis used. Each time one object in the dataset is selected as thecentroid, and its 10 nearest neighbors in the selected featurespace are used to generate one hyperedge including the cen-troid itself, as shown in Figure 5. Then, a hypergraph G withN hyperedges can be constructed. In the second case, mul-tiple features are used to generate a hypergraph G modelingcomplex multi-modality correlation. Here, for the ith modal-ity data, a hypergraph adjacent matrix Hi is constructed ac-cordingly. After all the hypergraphs from different featureshave been generated, these adjacent matrices Hi can be con-catenated to build the multi-modality hypergraph adjacentmatrix H. In this way, the hypergraphs using single modalfeature and multi-modal features can be constructed.

FeatureFeatures for Structure

GVCNN MVCNN GVCNN+MVCNNGCN HGNN GCN HGNN GCN HGNN

GVCNN (Feng et al. 2018) 91.8% 92.6% 91.5% 91.8% 92.8% 96.6%MVCNN (Su et al. 2015) 92.5% 92.9% 86.7% 91.0% 92.3% 96.6%

GVCNN+MVCNN - - - - 94.4% 96.7%

Table 4: Comparison between GCN and HGNN on the ModelNet40 dataset.

FeatureFeatures for Structure

GVCNN MVCNN GVCNN+MVCNNGCN HGNN GCN HGNN GCN HGNN

GVCNN ((Feng et al. 2018)) 78.8% 82.5% 78.8% 79.1% 75.9% 84.2%MVCNN ((Su et al. 2015)) 74.0% 77.2% 71.3% 75.6% 73.2% 83.6%

GVCNN+MVCNN − − − − 76.1% 84.2%

Table 5: Comparison between GCN and HGNN on the NTU dataset.

Method ClassificationAccuracy

PointNet (Qi et al. 2017a) 89.2%PointNet++ (Qi et al. 2017b) 90.7%

PointCNN (Li et al. 2018) 91.8%SO-Net (Li, Chen, and Lee 2018) 93.4%

HGNN 96.7%

Table 6: Experimental comparison among recent classifica-tion methods on ModelNet40 dataset.

Results and discussions Experiments and comparisonson the visual object recognition task are shown in Table 4and Table 5, respectively. For the ModelNet40 dataset, wehave compared the proposed method using two features withrecent state-of-the-are methods in Table 6. As shown in theresults, we can have the following observations:

1. The proposed HGNN method outperforms the state-of-the-art object recognition methods in the ModelNet40dataset. More specifically, compared with PointCNN andSO-Net, the proposed HGNN method can achieve gainsof 4.8% and 3.2%, respectively. These results demonstratethe superior performance of the proposed HGNN methodon visual object recognition.

2. Compared with GCN, the proposed method achieves bet-ter performance in all experiments. As shown in Ta-ble 4 and Table 5, when only one feature is used forgraph/hypergraph structure generation, HGNN can ob-tain slightly improvement. For example, when GVCNNis used as the object feature and MVCNN is used forgraph/hypergraph structure generation, HGNN achievesgains of 0.3% and 2.0% compared with GCN on theModelNet40 and the NTU datasets, respectively. Whenmore features, i.e., both GVCNN and MVCNN, areused for graph/hypergraph structure generation, HGNNachieves much better performance compared with GCN.

For example, HGNN achieves gains of 8.3%, 10.4% and8.1% compared with GCN when GVCNN, MVCNN andGVCNN+MVCNN are used as the object features on theNTU dataset, respectively.The better performance can be dedicated to the employed

hypergraph structure. The hypergraph structure is able toconvey complex and high-order correlations among data,which can better represent the underneath data relation-ship compared with graph structure or the methods withoutgraph structure. Moreover, when multi-modal data/featuresare available, HGNN has the advantage of combining suchmulti-modal information in the same structure by its flexiblehyperedges. Compared with traditional hypergraph learningmethods, which may suffer from the high computationalcomplexity and storage cost, the proposed HGNN frame-work is much more efficient through the hyperedge convo-lution operation.

ConclusionIn this paper, we propose a framework of hypergraph neuralnetworks (HGNN). In this method, HGNN generalizes theconvolution operation to the hypergraph learning process.The convolution on spectral domain is conducted with hy-pergraph Laplacian and further approximated by truncatedchebyshev polynomials. HGNN is a more general frame-work which is able to handle the complex and high-ordercorrelations through the hypergraph structure for representa-tion learning compared with traditional graph. We have con-ducted experiments on citation network classification andvisual object recognition tasks to evaluate the performanceof the proposed HGNN method. Experimental results andcomparisons with the state-of-the-art methods demonstratebetter performance of the proposed HGNN model. HGNNis able to take complex data correlation into representationlearning and thus lead to potential wide applications in manytasks, such as visual recognition, retrieval and data classifi-cation.

AcknowledgementsThis work was supported by National Key R&D Pro-gram of China (Grant No. 2017YFC0113000, andNo.2016YFB1001503), and National Natural ScienceFunds of China (U1701262, U1705262, 61671267,61772443, 61572410), National Science and TechnologyMajor Project (No. 2016ZX01038101), MIIT IT funds(Research and application of TCN key technologies) ofChina, and The National Key Technology R and D Program(No. 2015BAG14B01-02), Post Doctoral Innovative TalentSupport Program under Grant BX201600094, China Post-Doctoral Science Foundation under Grant 2017M612134,Scientific Research Project of National Language Commit-tee of China (Grant No. YB135-49), and Nature ScienceFoundation of Fujian Province, China (No. 2017J01125 andNo. 2018J01106).

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Documents

Hypergraph Neural Networksgaoyue.org/paper/HGNN.pdf · Introduction Graph-based convolutional neural networks (Kipf and Welling 2017), (Defferrard, Bresson, and Vandergheynst 2016)