# Higher Order Gauge Equivariant CNNs on Riemannian Manifolds and Applications

arXiv: https://arxiv.org/abs/2305.16657 · HTML: https://mlsanity.com/paper/2305.16657/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 42 references that are in the map.
- First posted on arXiv 26 May 2023.

## Builds on (most cited first, 21 of 21)

- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Roto-Translation Covariant Convolutional Networks for Medical Image Analysis](https://mlsanity.com/paper/1804.03393/) (2018)
- [Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs](https://mlsanity.com/paper/2003.05425/) (2020)
- [Polar Transformer Networks](https://mlsanity.com/paper/1709.01889/) (2017)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [DeepSphere: a graph-based spherical CNN](https://mlsanity.com/paper/2012.15000/) (2020)
- [Spin-Weighted Spherical CNNs](https://mlsanity.com/paper/2006.10731/) (2020)
- [Non-linear Convolution Filters for CNN-based Learning](https://mlsanity.com/paper/1708.07038/) (2017)
- [Efficient Generalized Spherical CNNs](https://mlsanity.com/paper/2010.11661/) (2020)
- [End-to-End Kernel Learning with Supervised Convolutional Kernel Networks](https://mlsanity.com/paper/1605.06265/) (2016)
- [Convolutional Networks for Spherical Signals](https://mlsanity.com/paper/1709.04893/) (2017)
- [VolterraNet: A higher order convolutional network with group equivariance for homogeneous manifolds](https://mlsanity.com/paper/2106.15301/) (2021)

## Built on by (most cited first, 2 of 2)

- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [Adjusted Cup-Product Neural Layer](https://mlsanity.com/paper/2606.13568/) (2026)

## Abstract (arXiv)

With the advent of group equivariant convolutions in deep networks literature, spherical CNNs with mathsfSO(3)-equivariant layers have been developed to cope with data that are samples of signals on the sphere S2. One can implicitly obtain mathsfSO(3)-equivariant convolutions on S2 with significant efficiency gains by explicitly requiring gauge equivariance w.r.t. mathsfSO(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.