With the advent of group equivariant convolutions in deep networks
literature, spherical CNNs with $\mathsf{SO}(3)$-equivariant layers have been
developed to cope with data that are samples of signals on the sphere $S^2$.
One can implicitly obtain $\mathsf{SO}(3)$-equivariant convolutions on $S^2$
with significant efficiency gains by explicitly requiring gauge equivariance
w.r.t. $\mathsf{SO}(2)$. In this paper, we build on this fact by introducing a
higher order generalization of the gauge equivariant convolution, whose
implementation is dubbed a gauge equivariant Volterra network (GEVNet). This
allows us to model spatially extended nonlinear interactions within a given
receptive field while still maintaining equivariance to global isometries. We
prove theoretical results regarding the equivariance and construction of higher
order gauge equivariant convolutions. Then, we empirically demonstrate the
parameter efficiency of our model, first on computer vision benchmark data
(e.g. spherical MNIST), and then in combination with a convolutional kernel
network (CKN) on neuroimaging data. In the neuroimaging data experiments, the
resulting two-part architecture (CKN + GEVNet) is used to automatically
discriminate between patients with Lewy Body Disease (DLB), Alzheimer's Disease
(AD) and Parkinson's Disease (PD) from diffusion magnetic resonance images
(dMRI). The GEVNet extracts micro-architectural features within each voxel,
while the CKN extracts macro-architectural features across voxels. This
compound architecture is uniquely poised to exploit the intra- and inter-voxel
information contained in the dMRI data, leading to improved performance over
the classification results obtained from either of the individual components.

本文介绍一个高阶推广的标准等变卷积的实现方式，即等变 Volterra 网络，使得在给定的接受范围内可以建模空间扩展的非线性交互，同时保持全局同构等变性，最后将它应用到神经影像数据的分类中。

在黎曼流形上进行高阶规范等变 CNN 和应用

Higher Order Gauge Equivariant CNNs on Riemannian Manifolds and  Applications

We tackle classification based on brain connectivity derived from diffusion
magnetic resonance images. We propose a machine-learning model inspired by
graph convolutional networks (GCNs), which takes a brain connectivity input
graph and processes the data separately through a parallel GCN mechanism with
multiple heads. The proposed network is a simple design that employs different
heads involving graph convolutions focused on edges and nodes, capturing
representations from the input data thoroughly. To test the ability of our
model to extract complementary and representative features from brain
connectivity data, we chose the task of sex classification. This quantifies the
degree to which the connectome varies depending on the sex, which is important
for improving our understanding of health and disease in both sexes. We show
experiments on two publicly available datasets: PREVENT-AD (347 subjects) and
OASIS3 (771 subjects). The proposed model demonstrates the highest performance
compared to the existing machine-learning algorithms we tested, including
classical methods and (graph and non-graph) deep learning. We provide a
detailed analysis of each component of our model.

本文提出了一种受图卷积网络启发的机器学习模型，针对基于扩散磁共振图像的脑连接性进行分类。该模型通过并行地使用具有多个头的图形卷积机制对脑连接输入图进行处理，并使用不同的头涉及图卷积来捕获输入数据的表示，证明了在两个公开数据集上比现有的机器学习算法展现出更高的性能，包括经典方法和 (图形和非图形) 深度学习。