Group equivariant convolutional networks (GCNNs) endow classical
convolutional networks with additional symmetry priors, which can lead to a
considerably improved performance. Recent advances in the theoretical
description of GCNNs revealed that such models can generally be understood as
performing convolutions with G-steerable kernels, that is, kernels that satisfy
an equivariance constraint themselves. While the G-steerability constraint has
been derived, it has to date only been solved for specific use cases - a
general characterization of G-steerable kernel spaces is still missing. This
work provides such a characterization for the practically relevant case of G
being any compact group. Our investigation is motivated by a striking analogy
between the constraints underlying steerable kernels on the one hand and
spherical tensor operators from quantum mechanics on the other hand. By
generalizing the famous Wigner-Eckart theorem for spherical tensor operators,
we prove that steerable kernel spaces are fully understood and parameterized in
terms of 1) generalized reduced matrix elements, 2) Clebsch-Gordan
coefficients, and 3) harmonic basis functions on homogeneous spaces.

本篇论文对 G 任意紧致群下可转向核空间进行了全面的研究，通过将可转向核空间和量子力学中的球张量算符进行类比，广义化了著名的 Wigner-Eckart 定理，证明了可转向核空间可由广义约化矩阵元，Clebsch-Gordan 系数和单群空间上的谐函数完全理解和参数化。

一种针对群等变卷积核的 Wigner-Eckart 定理

A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels

Driven by the outstanding performance of neural networks in the structured
Euclidean domain, recent years have seen a surge of interest in developing
neural networks for graphs and data supported on graphs. The graph is leveraged
at each layer of the neural network as a parameterization to capture detail at
the node level with a reduced number of parameters and computational
complexity. Following this rationale, this paper puts forth a general framework
that unifies state-of-the-art graph neural networks (GNNs) through the concept
of EdgeNet. An EdgeNet is a GNN architecture that allows different nodes to use
different parameters to weigh the information of different neighbors. By
extrapolating this strategy to more iterations between neighboring nodes, the
EdgeNet learns edge- and neighbor-dependent weights to capture local detail.
This is a general linear and local operation that a node can perform and
encompasses under one formulation all existing graph convolutional neural
networks (GCNNs) as well as graph attention networks (GATs). In writing
different GNN architectures with a common language, EdgeNets highlight specific
architecture advantages and limitations, while providing guidelines to improve
their capacity without compromising their local implementation. An interesting
conclusion is the unification of GCNNs and GATs -- approaches that have been so
far perceived as separate. In particular, we show that GATs are GCNNs on a
graph that is learned from the features. This particularization opens the doors
to develop alternative attention mechanisms for improving discriminatory power.

本文提出 EdgeNet 框架，将现有的图卷积神经网络（GCNNs）和图注意力网络（GATs）统一起来，使得同一问题可以使用不同的 GNN 结构来解决，并且指导和优化各种 GNNs 的性能和实现。

EdgeNets: 边缘变化的图神经网络

EdgeNets:Edge Varying Graph Neural Networks

Graph Convolutional Neural Networks (GCNNs) are the most recent exciting
advancement in deep learning field and their applications are quickly spreading
in multi-cross-domains including bioinformatics, chemoinformatics, social
networks, natural language processing and computer vision. In this paper, we
expose and tackle some of the basic weaknesses of a GCNN model with a capsule
idea presented in \cite{hinton2011transforming} and propose our Graph Capsule
Network (GCAPS-CNN) model. In addition, we design our GCAPS-CNN model to solve
especially graph classification problem which current GCNN models find
challenging. Through extensive experiments, we show that our proposed Graph
Capsule Network can significantly outperforms both the existing state-of-art
deep learning methods and graph kernels on graph classification benchmark
datasets.

本文提出了一种名为 Graph Capsule Network (GCAPS-CNN) 的模型，通过使用胶囊网络的思想对 GCNN 模型的一些基本缺陷进行了改进，并将其应用于解决当前 GCNN 模型在图分类问题上面临的挑战，实验结果表明，GCAPS-CNN 模型可以显著地提高图分类基准数据集上的性能。