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纸质出版日期： 2024-01-16 ，
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柳世禹， 戴文睿， 李成林， 熊红凯. 2024. 从图卷积网络到图散射网络：回顾与展望. 中国图象图形学报， 29(01):0045-0064
Liu Shiyu， Dai Wenrui， Li Chenglin， Xiong Hongkai. 2024. From graph convolution networks to graph scattering networks：a survey. Journal of Image and Graphics， 29(01):0045-0064
在图像与图形处理中，非欧氏空间数据与传统欧氏空间数据共同构成了数据的不同表达形式。随着面向图像、音频等传统信号的处理技术已经发展了数十年并趋于成熟，诸如图等非欧氏空间数据的兴起，对非欧氏空间的数据处理提取提出了更高的要求。图卷积网络的出现将面向传统信号的深度学习网络模型和卷积操作拓展到了图上，在一定程度上解决了学术界和工业界对图信号处理的需求。然而，空域特征聚合的图卷积网络容易产生过平滑问题。本文回顾了从图卷积网络到图散射网络的发展进程，分别梳理空域图卷积网络和谱域图卷积网络；并以图卷积网络为桥梁引出了图散射网络，比较和总结了图散射网络的前沿的理论和方法。传统的谱域图卷积网络虽然可以通过滤波器设计避免过平滑问题，但由于可训练参数较少、输出特征比较单一，往往存在表达能力不足的问题。图散射网络的提出很好地解决了图卷积网络中存在的问题。一方面，图散射变换将面向传统信号的散射变换操作拓展到图信号处理上，通过多尺度小波分解提取图信号的多分辨率特征，在保证网络稳定性的前提下解决了空域图卷积网络的特征过平滑问题；另一方面，相较于传统的谱域图卷积网络，图散射网络输出能够提取多尺度带通特征，增强模型的表达能力，提高了图分类等任务的结果。最后分析了现有图散射技术和理论的局限性，并提出了未来图散射网络可能的研究方向。
In image processing and computer graphics， non-Euclidean data such as graphs have gained increasing attention in recent years because Euclidean data such as images and videos fail to represent data with structure information. Compared with traditional Euclidean data， the scale of a graph can be arbitrary large. The structure of a graph usually contains information such as the relation between vertices， the logical sequences， and the properties of graph itself. While images can be easily converted into graphs based on the Euclidean position of pixels， graphs （especially for irregular graphs） can merely be converted into images. Therefore， graphs require a higher level of representation learning compared with traditional Euclidean data. However， in the era of deep learning， traditional convolution neural networks （CNNs） fail to learn representations for graphs due to permutation covariance for nodewise features and permutation invariance for outputs such as classification labels. The performance of CNNs in graph representation learning is still limited even if inputs are augmented by arbitrary permutation during training to learn permutation covariance. The development of graph neural networks and graph convolution operations achieves milestone in representation learning of non-Euclidean data such as graphs. Commonly， graph convolution neural networks （GCNs） can be divided into two categories： spatial GCNs and spectral GCNs. Spatial GCNs focus on the establishment of neighborhood and update with aggregation functions that combine the features of the center vertex and its neighbors. Though GCNs based on neighborhood feature aggregation encourage the propagation of nodewise features， deep GCNs usually suffer from oversmoothness issue， and the features of vertices become indistinguishable. Therefore， later works consider introducing skip connections in deep GCNs or constructing shallow GCNs with multiscale neighborhood considered within each convolution to alleviate this issue. Spectral GCNs focus on the graph spectral theorem and update their parameters by signal filtering in spectral domain with designed filters. However， eigen-decomposition of graph shift operator is costly for large graphs because its computation complexity is
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. Therefore， spectral GCNs usually apply K-order polynomials （i.e.， Chebyshev polynomials） to approximate the target filters and avoid eigen-decomposition. Though spectral GCNs may avoid oversmoothness issue with graph filter design， the limited number of learnable parameters and filter responses of spectral GCNs usually limit their expression ability. Spatial GCNs and spectral GCNs are not necessarily independent from one another. For example， 1-order Chebyshev polynomials with diffusion matrix are equivalent to feature aggregation within 1-hop neighborhood. Therefore， spatial GCNs based on feature aggregation with diffusion Laplacian matrix or lazy random walk matrix usually have the spectral form， which bridges the spatial GCNs and spectral GCNs. The rapid development in graph representation learning gives rise to the demand for survey and review that summarize existing works and serve as guidance for beginners. Currently， graph neural networks such as graph convolution neural networks， graph embeddings， and graph autoencoders have been reviewed. However， current surveys and reviews lack one domain in graph representation learning： graph scattering transforms （GSTs） and graph scattering networks （GSNs）. GSNs are non-trainable spectral GCNs based on wavelet decomposition. With the benefit of multiscale wavelets and the structure of networks， GSNs generate diverse features with nearly nonoverlapping frequency responses in the spectral domain. As one of the newly developed graph representation learning methods， GSNs are used in tasks such as graph classification and node classification. Recent works employed graph scattering transform to spatial GCNs to overcome the oversmoothness issue. Compared with spectral GCNs， GSNs generate diverse features that strengthen the expressive capability of model without introducing the oversmoothness issue. However， the nontrainable property of GSNs may limit the flexibility in graph representation learning on different graph datasets with different distributions of spectrum. GSNs suffer from the exponential growth of diffusion paths with the increase of scattering layers， which limit the depth of GSNs in practice. In this paper， a survey comprehensively reviews the designs from GCNs to GSNs. First， GCNs are divided into two categories： spatial GCNs and spectral GCNs. Spatial GCNs are categorized into the following types： 1） diffusion-based GCNs， 2） GCNs on large graphs with neighbor sampling or subgraph sampling， 3） GCNs with attention mechanism， and 4） GCNs with dynamic neighborhood construction. Spectral GCNs are reviewed according to different filters （filter kernels）： Chebyshev polynomials， Cayley polynomials， and K-order polynomials. After addressing the drawbacks of spatial GCNs and spectral GCNs， the definition of GSTs， the structure of classical GSNs， and the advantages of GSNs compared with GCNs are introduced. The current arts of GSNs are elaborated from the perspectives of network design along with application and stability in theory. The networks and application of graph scattering transform are reviewed in the following categories： 1） classical GSNs， 2） graph scattering transforms in GCNs to solve the over-smoothness issue， 3） graph attention networks with GSTs， 4） graph scattering transform on spatial-temporal graphs， 5） reducing scattering paths of GSNs via pruning to increase the efficiency of graph scattering transform， 6） GSNs with multiresolution graphs， and 7） trainable GSNs with wavelet scale selection and learnable spectral filters. In theory， the frame theorem and the stability theorem under signal and topology perturbation， respectively， are concluded. The limitations of GSNs （GSTs） in current works are analyzed， and possible directions for the development of graph scattering technics in the future are proposed.
深度学习图卷积网络（GCN）图散射网络（GSN）表征学习稳定性信号扰动拓扑扰动
deep learninggraph convolution network （GCN）graph scattering network （GSN）representation learningstabilitysignal perturbationtopology perturbation
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