Graph Neural Networks (GNNs) Wu et al. (2019c) aim at imitating the expressive capability of deep neural networks from grid-like data (e.g. images and sequences) to graph structures. The fruitful progress of GNNs in the past decade has made them a crucial kind of tools for a variety of applications, from social networks Li et al. (2019)et al. (2018); Zeng et al. (2019), text classification Yao et al. (2019), to chemistry Liao et al. (2019).
Graph Attention Network () Veličković et al. (2018), as one central type of GNNs introduces the attention mechanism to further refine the convolution process in generic GCNs Kipf and Welling (2017). Specifically, during the node aggregation, assigns a self-attention weight to each edge, which can capture the local similarity among neighborhoods, and further boost the expressing power of GNNs as the weight itself is learnable. Many variants have been proposed since GAT Morris et al. (2019); Wang et al. (2019); Wu et al. (2019b).
, along with its variants, considers the attention in a straightforward way: learning the edge attentions in the spatial domain. In this sense, this attention can capture the local structure of graphs, i.e., the information from neighbors. However, it is unable to explicitly encode the global structure of graphs. Furthermore, computing the attention weights for every edge in the graph is inefficient, especially for large graphs.
In computer vision, a natural image can be decomposed into a low spatial frequency component containing the smoothly changing structure, e.g., background, and a high spatial frequency component describing the rapidly changing fine details, e.g., outlines. Figure 1(a) ~ 1(c) depict the example of low- and high-frequency components on a panda image. Obviously, the contribution of different frequency components varies with different downstream tasks. To accommodate this phenomenon, Chen et al. (2019b) proposed Octave Convolution () to factorize convolutional feature maps into two groups of different spatial frequencies and process them with different convolutions at their corresponding frequency.
In graph representational learning, this decomposing of low- and high-frequency can be observed more naturally, since graph signal processing (GSP) provides us a way to directly divide the low- and high-frequency components based on the ascending ordered eigenvalues of Laplacian in graphs. The eigenvectors associated with small eigenvalues carry smoothly varying signals, encouraging neighbor nodes to share similar values. In contrast, the eigenvectors associated with large eigenvalues carry sharply varying signals across edgesDonnat et al. (2018). As demonstrated in Figure 1(d) ~ 1(f), a barbell graph tends to retain the information inside the clusters when it is reconstructed with only low-frequency components, but reserve knowledge between the clusters when constructed with only high-frequency ones. As pointed out by Wu et al. (2019a); Maehara (2019), the low- and high-frequency components in the spectral domain may reflect the local and global structural information of graphs in the spatial domain respectively. Moreover, recent works Donnat et al. (2018); Maehara (2019) reveal the different importance of low- and high-frequency components of graphs that contributes to the learning of modern GNNs.
Inspired by recent works, we propose to extend the attention mechanism to the spectral domain of graph to explicitly encode the structural information of graphs from a global perspective. Accordingly, we present Spectral Graph Attention Network(). In
, we choose the graph wavelets as the spectral bases and decompose them into low- and high-frequency components with respect to their indices. Then we construct two distinct convolutional kernels according to the low- and high-frequency components and apply the attention mechanism on these two kernels to capture the their importance respectively. Finally, the pooling function, as well as the activation function, are applied to produce the output. Figure2 provides an overview of the design of . Furthermore, we employ the Chebyshev Polynomial approximation to compute the spectral wavelets of graphs and propose the variant - which is more efficient on large graphs. We thoroughly validate the performance of and - on five challenging benchmarks with eleven competitive baselines. and - achieve state-of-the-art results on most datasets. The contributions of this paper are summarized as follows:
To better exploit the local and global structural information of graphs, we propose to extend the attention to the spectral domain rather than the spatial domain and design . To the best of our knowledge, is the first attempt to adopt the attention mechanism to the spectral domain of graphs.
Compared with traditional , which needs to compute the attentions for each edge, only employs the attention operation on low- and high-frequency components in the spectral domain. We show that has the same parameter complexity as .
To accelerate the computation of the spectral wavelets, we propose the Chebyshev Polynomial approximation which reduces the computation complexity and achieves at most 7.9x acceleration in benchmark datasets.
Extensive experiments show the superiority of and demonstrate the rationale behind the attention on the spectral domain.
We denote as an undirected graph, where is the set of nodes, and is the set of edges, where . The adjacency matrix is defined as a symmetric matrix , where indicates an edge . We denote as the node degrees matrix, where represents the node degree of node .
The original version of is developed by Kipf and Welling (2017). The feed-forward GCN layer is defined as:
are the output of hidden vectors from the layer withas the input features. refers to the normalized adjacency matrix, where is the corresponding degree matrix of .
refers to the activation function, such as ReLu.refers to the learning parameters of the layer, where and refers to the feature dimension of input and output respectively.
From the spatial perspective, is viewed as the feature aggregation among the neighbors of nodes in the spatial domain of graphs. Therefore, we rewrite Eq. (1) to a more general form:
where refers to the neighborhood set of node in graph111Usually, we include in .; refers to the aggregation weight of neighbor for node ; and refers to the aggregation function that aggregates the output of each neighbor, such as and . Usually, can be viewed as the special case of Eq. (2) where , and .
Based on Eq. (2), Veličković et al. (2018) introduces the attention mechanism in graphs and proposes Graph Attention Network (). Concretely, instead of employing the (normalized) adjacency matrix as the aggregation weight, proposes to compute the weight by a self-attention strategy, namely:
where . refers to the attention weight. On one hand, compared with , it is expensive for to compute the attention weight for every edge in spatial domain because the parameter complexity of -head is , while that of is . On the other hand, the self-attention strategy in only consider the local structural information in graphs, i.e., the neighborhoods. It ignores the global structural information in graphs.
3 Spectral Graph Attention Network
Other than the neighbor aggregation in the spatial domain, from Kipf and Welling (2017); Jin et al. (2019); Chang et al. (2020), can also be understood as the Graph Signal Processing in the spectral domain:
where is a signal on every node. are the spectral bases extracted from graphs. is a diagonal filter parameterized by . Given Eq.( 4),
can be viewed as the spectral graph convolution based on the Fourier transform on graphs with first-order Chebyshev polynomial approximationsKipf and Welling (2017). Further, we can separate the spectral graph convolution into two stages Xu et al. (2019):
In Eq. (5), is the diagonal matrix for graph convolution kernel. For instance, the graph convolution kernel for is , is the eigenvalues of the normalized graph Laplacian matrix in ascending order, while the spectral bases for is the corresponding eigenvectors.
3.1 The Construction of Layer
In this section, we start to describe the construction of layer. From the Graph Signal Processing perspective, the diagonal values on can be treated as the frequencies on the graph. We denote the diagonal values with small / large indices as the low / high frequencies respectively. Meanwhile, the corresponding spectral bases in are low- and high-frequency components. As discussed in Section 1, the low- and high-frequency components carry different structural information in graphs. In this vein, we first split the spectral bases into two groups and re-write Eq. (5) as follows:
where and are the low- and high-frequency components, respectively. Here is a hyper-parameter that decides the splitting boundary of low- and high-frequency. When , Eq. (6) is equivalent to the graph convolution stage in Eq. (5).
In Eq. (6), can be viewed as the importance of the low- and high-frequency. Therefore, we introduce the learnable attention weights by exploiting the re-parameterization trick:
In Eq. (7), and are the two diagonal matrices parameterized by two learnable attention and , respectively. To ensure and are positive and comparable, we normalize them by the function:
Theoretically, there are many approaches to re-parameterize and , such as self-attention w.r.t the spectral bases
. However, these kinds of re-parameterization can not reflect the nature of low- and high- frequency components. On the other hand, they may introduce too many additional learnable parameters, especially for large graphs. These parameters might prohibit the efficient training due to the limited amount of training data in graphs, such as under graph-based semi-supervised learning setting. Meanwhile, we validate that such re-parameterization is simple but efficient and effective in practice.
3.2 Choice of Spectral Bases
Another important issue of is the choice of the spectral bases. While the Fourier bases have become the common choice in construction of spectral graph convolution, recent works Donnat et al. (2018); Xu et al. (2019) observed the advantages by utilizing spectral wavelets as bases in graph embedding techniques over traditional Fourier ones. Instead of Fourier bases, we choose the graph wavelets as spectral bases in .
Formally, the spectral graph wavelet is defined as the signal resulting from the modulation in the spectral domain of a signal centered around the associated node Hammond et al. (2011); Shuman et al. (2013). Then, given the graph , the graph wavelet transform is conducted by employing a set of wavelets as bases. Concretely, the spectral graph wavelet transformation is given as:
where is the eigenvectors of normalized graph Laplacian matrix ,
is a scaling matrix with heat kernel scaled by hyperparameter. The inverse of graph wavelets is obtained by simply replacing the in with corresponding to the heat kernel Donnat et al. (2018). Smaller indices in graph wavelets correspond to low-frequency components and vice versa.
The benefits that spectral graph wavelet bases have over Fourier bases mainly fall into three aspects: 1. Given the sparse real-world networks, the graph wavelet bases are usually much more sparse than Fourier bases, e.g., the density of is comparing with of Xu et al. (2019). The sparseness of graph wavelets makes them more computationally efficient for use. 2. In spectral graph wavelets, the signal resulting from heat kernel filter is typically localized on the graph and in the spectral domain Shuman et al. (2013). By adjusting the scaling parameter , one can easily constrain the range of localized neighborhood. Smaller values of generally associate with smaller neighborhoods. 3. Since the information of eigenvalue is implicitly contained in wavelets from the process of construction of wavelets, we would not suffer the information loss when do re-parameterization.
Therefore, the architecture of layer with graph wavelet bases can be written as:
In Eq. (9), aiming to further reduce the parameter complexity, we share the parameters in feature transformation stage for and , i.e, . In this way, we reduce the parameter complexity from to , which is nearly the same as . The parameter complexity of is much less than that of with -head attention. Comparing with GAT, which captures the local structure of graph from spatial domain, our proposed could better tackle global information by combining the low- and high-frequency features explicitly from spectral domain.
4 Fast Approximation of Spectral Wavelets via Chebyshev Polynomials
In , directly computing the transformation according to Eq.( 8) is intensive for large graphs, since diagonalizing Laplacian commonly requires computational complexity. Fortunately, we can employ the Chebyshev polynomials to fast approximate the spectral graph wavelet without eigen-decompositionHammond et al. (2011).
Let be the fixed scaling parameter in the heat filter kernel and be the degree of the Chebyshev polynomial approximations for the scaled wavelet (Larger value of yields more accurate approximations but higher computational cost in opposite), the graph wavelet is given by
where , is the order Chebyshev polynomial approximation, and is the Bessel function of the first kind.
Theorem 1 can be deviated from Section 6 in Hammond et al. (2011). To further accelerate the computation, we build a look-up table for the Bessel function to avoid addtional integral operations. With this Chebyshev polynomial approximation, the computational cost of spectral graph wavelets is decreased to . Due the real world graphs are usually sparse, this computational difference can be very significant. We denote with Chebyshev polynomial approximation as -.
5 Related Works
Spectral convolutional networks on graphs. Existing methods of defining a convolutional operation on graphs can be broadly divided into two categories: spectral based and spatial based methods Zhang et al. (2018). We focus on the spectral graph convolutions in this paper. Spectral CNN Bruna et al. (2014) first attempts to generalize CNNs to graphs based on the spectrum of the graph Laplacian and defines the convolutional kernel in the spectral domain. Boscaini et al. (2015) further employs windowed Fourier transformation to define a local spectral CNN approach. ChebyNet Defferrard et al. (2016) introduces a fast localized convolutional filter on graphs via Chebyshev polynomial approximation. Vanilla GCN Kipf and Welling (2017) further extends the spectral graph convolutions considering networks of significantly larger scale by several simplifications. Khasanova and Frossard (2017) learns graph-based features on images that are inherently invariant to isometric transformations. Cayleynets Levie et al. (2018) alternatively introduce Cayley polynomials allowing to efficiently compute spectral filters on graphs. FastGCN Chen et al. (2018) and ASGCN Huang et al. (2018) further accelerate the training of Vanilla GCN via sampling approaches. Lanczos algorithm is utilized in LanczosNet Liao et al. (2019) to construct low-rank approximations of the graph Laplacian for convolution. SGC Wu et al. (2019a) further reduces the complexity of Vanilla GCN by successively removing the non-linearities and collapsing weights between consecutive layers. Despite their effective performance, all these convolution theorem based methods lack the strategy to explicitly treat low- and high-frequency components with different importance.
Spectral graph wavelets. Theoretically, the lifting scheme is proposed for the construction of wavelets that can be adapted to irregular graphs in Sweldens (1998). Hammond et al. (2011) defines wavelet transforms appropriate for graphs and describes a fast algorithm for computation via fast Chebyshev polynomial approximation. For applications, Tremblay and Borgnat (2014) utilizes graph wavelets for multi-scale community mining and obtains a local view of the graph from each node. Donnat et al. (2018) introduces the property of graph wavelets that describes information diffusion and learns structural node embeddings accordingly. Xu et al. (2019) first attempts to construct graph neural networks with graph wavelets. These works emphasize the local and sparse property of graph wavelets for graph signal processing both theoretically and practically.
Space/spectrum-aware feature representation. In computer vision, Chen et al. (2019b) first defines space-aware feature representations based on scale-space theory and reduces spatial redundancy of vanilla CNN models by proposing the Octave Convolution () model. Durall et al. (2019)
further leverages octave convolutions for designing stabilizing GANs. To our knowledge, this is the first time that spectrum-aware feature representations are considered in irregular graph domain and established with graph convolutional neural networks.
Joining the practice of previous works, we focus on five node classification benchmark datasets under semi-supervised setting with different graph size and feature type. (1) Three citation networks: Citeseer, Cora and Pubmed Sen et al. (2008)
, which aims to classify the research topics of papers. (2) A coauthor network: Coauthor CS which aims to predict the most active fields of study for each author from the KDD Cup 2016 challenge222https://kddcup2016.azurewebsites.net. (3) A co-purchase network: Amazon Photo McAuley et al. (2015) which aims to predict the category of products from Amazon. For the citation networks, we follow the public split setting provided by Yang et al. (2016), that is, 20 labeled nodes per class in each dataset for training and 500 / 1000 labeled samples for valiation / test respectively. For the other two datasets, we follow the splitting setting from Shchur et al. (2018); Chen et al. (2019a). Statistical overview of all datasets is given in Table 1. Label rate denotes the ratio of labeled nodes fetched in training process.
|Model||Citeseer||Cora||Pubmed||Coauthor CS||Amazon Photo|
|Perozzi et al. (2014)|
|Yang et al. (2016)|
|Li et al. (2016)|
|Defferrard et al. (2016)|
|Kipf and Welling (2017)|
|Hamilton et al. (2017b)|
|Verma et al. (2018)|
|Veličković et al. (2018)|
|Bai et al. (2019)|
|Xu et al. (2019)|
|Bianchi et al. (2019)|
|Morris et al. (2019)|
We thoroughly evaluate the performance of with 11 representative baselines. Among them, Perozzi et al. (2014) and Yang et al. (2016) are the traditional graph embedding methods. Defferrard et al. (2016), Kipf and Welling (2017), Xu et al. (2019), Bianchi et al. (2019) are the spectral-based GNNs. Li et al. (2016), Hamilton et al. (2017a), Veličković et al. (2018), Verma et al. (2018), Bai et al. (2019) and Morris et al. (2019) are the spatial-based GNNs. In addition, we also implement the the variant of with Chebyshev Polynomial approximation, which is denoted as -.
6.3 Experimental setup
For all experiments, a 2-layer network of our model is constructed using TensorFlowAbadi et al. (2015) with 64 hidden units. We train our model utilizing the Adam optimizer Kingma and Ba (2014) with an initial learning rate . We train the model using early stopping with a window size of 100. Most training process are stopped in less than 200 steps as expected. We initialize the weights matrix following Glorot and Bengio (2010), employ L2 regularization on weights and dropout input and hidden layers to prevent overfitting Srivastava et al. (2014).
For constructing wavelets , we follow the suggestion from Xu et al. (2019) for each dataset, i.e., for Citeseer, for Pubmed and for Cora, Coauthor CS and Amazon Photo. In addition, we employ the grid search to determine the best of low-frequency components in and the impact of this parameter would be discussed in Section 6.5.1. Without specification, we use the aggregation function in .
6.4 Performance on Semi-supervised Node Classification
Table 2 summaries the results on all datasets. For all baselines, we reuse result the reported in literatures Veličković et al. (2018); Chen et al. (2019a). From Table 2, we have these findings. (1) Clearly, the attention-based GNNs (, and -) achieve the best performance among all datasets. It validates that the attention mechanism can capture the important pattern from either spatial and spectral perspective. (2) Specifically, achieves best performance on four datasets; particularly on Pubmed, the best accuracy by - is 80.5% and it is better than the previous best(79.0%), which is regarded as a remarkable boost considering the challenge on this benchmark. Meanwhile, compared with baselines, - can also achieve the better performance on three datasets and even gain the best performance on two of them. (4). Compared with aggregation, aggregation seems a better choice for . This may due to aggregation can preserve the significant signals learned by . (5) It’s worth to note that to achieve such results, both and - only employ the attention on low- and high-frequency filter of graphs in spectral domain, while needs to learn the attention weights on every edge in spatial domain. It verifies that is more efficient than since the spectral domain contains the meaningful signals and can capture the global information of graphs.
6.5 Ablation Studies
6.5.1 The impact of low-frequency components proportion
To evaluate the impact of the hyperparameter , we fix the other hyperparameters and vary from to linearly to run on Citeseer, Cora and Pubmed, respectively. Figure 4
depicts the mean (in bold line) and variance (in light area) of everyon three datasets. As shown in Figure 4, the mean value curve of three datasets exhibits the similar pattern, that is, the best performance are achieved when is small. The best proportion of low-frequency components are , , and for Citeseer, Cora and Pubmed, respectively. In the other words, consistently, only a small fraction of components needs to be treated as the low-frequency components in .
6.5.2 The ablation study on low- and high-frequency Components
To further elaborate the importance of low- and high-frequency components in , we conduct the ablation study on the classification results by testing only with low- or high-frequency components w.r.t the best proportion in Section 6.5.1. Specially, we manually set or to 0 during testing stage to observe how the learned low- and high-frequency components in graphs affect the classification accuracy. As shown in Table 3, both low- and high-frequency components are essential for the model. Meanwhile, we can find that with very small proportion (5% - 15%) of low-frequency components can achieve the comparable results to those obtained by full . It reads that the low-frequency components contain more information that can contribute to the feature representation learned from the model.
6.5.3 The learned attention on low- and high-frequency components
To investigate the results in Section 6.5.2, we further show the learned attentions of w.r.t the best proportion for Cietseer, Cora and Pubmed which are demonstrated in Table 4. Interestingly, despite the small proportion, the attention weight of low-frequency components learned by is much larger than that of high-frequency components in each layer consistently. Hence, is successfully to capture the importance of low- and high-frequency components of graphs in the spectral domain. Moreover, as pointed out by Donnat et al. (2018); Maehara (2019), the low-frequency components in graphs usually indicate smooth varying signals which can reflect the locality property in graphs. It implies that the local structural information is important for these datasets. This may explain why GAT also gains good performance on these datasets.
|Methods||Citeseer ()||Cora ()||Pubmed ()|
6.6 Time Efficiency of Chebyshev Polynomials Approximation
As discussed in Section 4, we propose the fast approximation of spectral wavelets according to Chebyshev polynomials. To elaborate its efficiency, we compare the time cost of calculating between via eigen-decomposition () and fast approximation (-). We report the mean time cost of and - with second-order Chebyshev Polynomials after 10 runs for Core, Citeseer and Pubmed respectively. As shown in Table 5, we can find that this fast approximation can greatly accelerate the training process. Specifically, - run 7.9x times faster than that of . It validate the efficiency of the proposed fast approximation approaches.
|Dataset||Citeseer ()||Cora ()||Pubmed ()|
|Attention filter weights|
|Learned value (first layer)||0.838||0.162||0.722||0.278||0.860||0.140|
|Learned value (second layer)||0.935||0.065||0.929||0.071||0.928||0.072|
6.7 t-SNE Visualization of Learned Embeddings
To evalute the effectiveness of the learned features of qualitatively, we depict the t-SNE visualization Maaten and Hinton (2008) of learned embeddings of on Citeseer and Pubmed in Figure 4 comparing with , and . The representation exhibits discernible clustering in the projected 2D space. In Figure 4, the color indicates the class label in datasets. Compared with the other methods, the intersections of different classes in are more separated. It verifies the discriminative power of across the classes.
In this paper, we propose , a novel spectral-based graph convolutional neural network to learn the representation of graph with respect to different frequency components in the spectral domain. By introduce the distinct trainable attention weight for low- and high-frequency component, can effectively capture both local and global information in graphs and enhance the performance of GNNs. Furthermore, a fast Chebyshev polynomial approximation is proposed to accelerate the spectral wavelet calculation. To the best of our knowledge, this is the first attempt to adopt the attention mechanism to the spectral domain of graphs. It is expected that will shed light on building more efficient architectures for learning with graphs.
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