# VolterraNet: A higher order convolutional network with group equivariance for homogeneous manifolds

arXiv: https://arxiv.org/abs/2106.15301 · HTML: https://mlsanity.com/paper/2106.15301/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 43 references that are in the map.
- First posted on arXiv 5 Jun 2021.

## Builds on (most cited first, 14 of 14)

- [FaceNet: A Unified Embedding for Face Recognition and Clustering](https://mlsanity.com/paper/1503.03832/) (2015)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [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)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Volumetric and Multi-View CNNs for Object Classification on 3D Data](https://mlsanity.com/paper/1604.03265/) (2016)
- [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)
- [Polar Transformer Networks](https://mlsanity.com/paper/1709.01889/) (2017)
- [Non-linear Convolution Filters for CNN-based Learning](https://mlsanity.com/paper/1708.07038/) (2017)
- [Convolutional Networks for Spherical Signals](https://mlsanity.com/paper/1709.04893/) (2017)
- [Large-Scale 3D Shape Reconstruction and Segmentation from ShapeNet Core55](https://mlsanity.com/paper/1710.06104/) (2017)
- [Local Group Invariant Representations via Orbit Embeddings](https://mlsanity.com/paper/1612.01988/) (2016)

## Built on by (most cited first, 11 of 11)

- [Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach](https://mlsanity.com/paper/2305.04560/) (2023)
- [Diffusion MRI with Machine Learning](https://mlsanity.com/paper/2402.00019/) (2024)
- [PDO-s3DCNNs: Partial Differential Operator Based Steerable 3D CNNs](https://mlsanity.com/paper/2208.03720/) (2022)
- [Higher Order Gauge Equivariant CNNs on Riemannian Manifolds and Applications](https://mlsanity.com/paper/2305.16657/) (2023)
- [E(3) × SO(3)-Equivariant Networks for Spherical Deconvolution in Diffusion MRI](https://mlsanity.com/paper/2304.06103/) (2023)
- [Equivariant spatio-hemispherical networks for diffusion MRI deconvolution](https://mlsanity.com/paper/2411.11819/) (2024)
- [Gradient Boosted Filters For Signal Processing](https://mlsanity.com/paper/2405.09305/) (2024)
- [Towards Differential Handling of Various Blur Regions for Accurate Image Deblurring](https://mlsanity.com/paper/2502.19677/) (2025)
- [FoGE: Fock Space inspired encoding for graph prompting](https://mlsanity.com/paper/2507.02937/) (2025)
- [KVNN: Learnable Multi-Kernel Volterra Neural Networks](https://mlsanity.com/paper/2604.15141/) (2026)
- [Learning with Volterra Neural Networks: A System Theoretic Perspective](https://mlsanity.com/paper/2609.01928/) (2026)

## Abstract (arXiv)

Convolutional neural networks have been highly successful in image-based learning tasks due to their translation equivariance property. Recent work has generalized the traditional convolutional layer of a convolutional neural network to non-Euclidean spaces and shown group equivariance of the generalized convolution operation. In this paper, we present a novel higher order Volterra convolutional neural network (VolterraNet) for data defined as samples of functions on Riemannian homogeneous spaces. Analagous to the result for traditional convolutions, we prove that the Volterra functional convolutions are equivariant to the action of the isometry group admitted by the Riemannian homogeneous spaces, and under some restrictions, any non-linear equivariant function can be expressed as our homogeneous space Volterra convolution, generalizing the non-linear shift equivariant characterization of Volterra expansions in Euclidean space. We also prove that second order functional convolution operations can be represented as cascaded convolutions which leads to an efficient implementation. Beyond this, we also propose a dilated VolterraNet model. These advances lead to large parameter reductions relative to baseline non-Euclidean CNNs. To demonstrate the efficacy of the VolterraNet performance, we present several real data experiments involving classification tasks on spherical-MNIST, atomic energy, Shrec17 data sets, and group testing on diffusion MRI data. Performance comparisons to the state-of-the-art are also presented.