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With the advent of group equivariant convolutions in deep networksliterature, spherical CNNs with $\mathsf{SO}(3)$-equivariant layers have beendeveloped 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 equivariancew.r.t. $\mathsf{SO}(2)$. In this paper, we build on this fact by introducing ahigher order generalization of the gauge equivariant convolution, whoseimplementation is dubbed a gauge equivariant Volterra network (GEVNet). Thisallows us to model spatially extended nonlinear interactions within a givenreceptive field while still maintaining equivariance to global isometries. Weprove theoretical results regarding the equivariance and construction of higherorder gauge equivariant convolutions. Then, we empirically demonstrate theparameter efficiency of our model, first on computer vision benchmark data(e.g. spherical MNIST), and then in combination with a convolutional kernelnetwork (CKN) on neuroimaging data. In the neuroimaging data experiments, theresulting two-part architecture (CKN + GEVNet) is used to automaticallydiscriminate 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. Thiscompound architecture is uniquely poised to exploit the intra- and inter-voxelinformation contained in the dMRI data, leading to improved performance overthe classification results obtained from either of the individual components.

Higher Order Gauge Equivariant CNNs on Riemannian Manifolds and Applications