  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.
