Spectral graph convolution, an important tool of data filtering on graphs,
relies on two essential decisions; selecting spectral bases for signal
transformation and parameterizing the kernel for frequency analysis. While
recent techniques mainly focus on standard Fourier transform and vector-valued
spectral functions, they fall short in flexibility to describe specific signal
distribution for each node, and expressivity of spectral function. In this
paper, we present a novel wavelet-based graph convolution network, namely
WaveGC, which integrates multi-resolution spectral bases and a matrix-valued
filter kernel. Theoretically, we establish that WaveGC can effectively capture
and decouple short-range and long-range information, providing superior
filtering flexibility, surpassing existing graph convolutional networks and
graph Transformers (GTs). To instantiate WaveGC, we introduce a novel technique
for learning general graph wavelets by separately combining odd and even terms
of Chebyshev polynomials. This approach strictly satisfies wavelet
admissibility criteria. Our numerical experiments showcase the capabilities of
the new network. By replacing the Transformer part in existing architectures
with WaveGC, we consistently observe improvements in both short-range and
long-range tasks. This underscores the effectiveness of the proposed model in
handling different scenarios. Our code is available at
this https URL

本文提出了一种基于小波的图卷积网络 WaveGC，通过集成多分辨率的谱基和矩阵值滤波器核心，实现有效捕获和分离短程和长程信息，提供了更高的灵活性和表达能力，并在短程和长程任务中相比现有模型取得了改进。

通过一般的谱小波推进图卷积网络

Advancing Graph Convolutional Networks via General Spectral Wavelets

Although spatio-temporal graph neural networks have achieved great empirical
success in handling multiple correlated time series, they may be impractical in
some real-world scenarios due to a lack of sufficient high-quality training
data. Furthermore, spatio-temporal graph neural networks lack theoretical
interpretation. To address these issues, we put forth a novel mathematically
designed framework to analyze spatio-temporal data. Our proposed
spatio-temporal graph scattering transform (ST-GST) extends traditional
scattering transforms to the spatio-temporal domain. It performs iterative
applications of spatio-temporal graph wavelets and nonlinear activation
functions, which can be viewed as a forward pass of spatio-temporal graph
convolutional networks without training. Since all the filter coefficients in
ST-GST are mathematically designed, it is promising for the real-world
scenarios with limited training data, and also allows for a theoretical
analysis, which shows that the proposed ST-GST is stable to small perturbations
of input signals and structures. Finally, our experiments show that i) ST-GST
outperforms spatio-temporal graph convolutional networks by an increase of 35%
in accuracy for MSR Action3D dataset; ii) it is better and computationally more
efficient to design the transform based on separable spatio-temporal graphs
than the joint ones; and iii) the nonlinearity in ST-GST is critical to
empirical performance.

提出一种基于数学设计的空时图散射转换（ST-GST），将传统散射变换扩展到空时领域，通过使用空时图小波和非线性激活函数的迭代应用，对空时数据进行分析，不需要训练即可视为空时图卷积网络的正向传递，在限制的训练数据情况下表现出稳定性，实验表明其性能优于空时图卷积网络。

时空图散射变换

Spatio-Temporal Graph Scattering Transform

Scattering transforms are non-trainable deep convolutional architectures that
exploit the multi-scale resolution of a wavelet filter bank to obtain an
appropriate representation of data. More importantly, they are proven invariant
to translations, and stable to perturbations that are close to translations.
This stability property dons the scattering transform with a robustness to
small changes in the metric domain of the data. When considering network data,
regular convolutions do not hold since the data domain presents an irregular
structure given by the network topology.
In this work, we extend scattering transforms to network data by using
multiresolution graph wavelets, whose computation can be obtained by means of
graph convolutions. Furthermore, we prove that the resulting graph scattering
transforms are stable to metric perturbations of the underlying network. This
renders graph scattering transforms robust to changes on the network topology,
making it particularly useful for cases of transfer learning, topology
estimation or time-varying graphs.

本文介绍了在网络数据中使用多分辨率图形小波的散射变换，并证明了生成的散射变换对底层网络度量摄动的稳定性。这使得图形散射变换变得稳健，特别适用于转移学习、拓扑估计或时变图的情况。