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。