# Moving Frame Net: SE(3)-Equivariant Network for Volumes

arXiv: https://arxiv.org/abs/2211.03420 · HTML: https://mlsanity.com/paper/2211.03420/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 27 references that are in the map.
- First posted on arXiv 7 Nov 2022.

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

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [MedMNIST v2 -- A large-scale lightweight benchmark for 2D and 3D biomedical image classification](https://mlsanity.com/paper/2110.14795/) (2021)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [MedMNIST Classification Decathlon: A Lightweight AutoML Benchmark for Medical Image Analysis](https://mlsanity.com/paper/2010.14925/) (2020)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Equivariant Point Network for 3D Point Cloud Analysis](https://mlsanity.com/paper/2103.14147/) (2021)
- [CubeNet: Equivariance to 3D Rotation and Translation](https://mlsanity.com/paper/1804.04458/) (2018)
- [PDO-eConvs: Partial Differential Operator Based Equivariant Convolutions](https://mlsanity.com/paper/2007.10408/) (2020)
- [Structured Receptive Fields in CNNs](https://mlsanity.com/paper/1605.02971/) (2016)
- [3D Medical Point Transformer: Introducing Convolution to Attention Networks for Medical Point Cloud Analysis](https://mlsanity.com/paper/2112.04863/) (2021)
- [Steerable Partial Differential Operators for Equivariant Neural Networks](https://mlsanity.com/paper/2106.10163/) (2021)
- [Embed Me If You Can: A Geometric Perceptron](https://mlsanity.com/paper/2006.06507/) (2020)
- [Fully trainable Gaussian derivative convolutional layer](https://mlsanity.com/paper/2207.08424/) (2022)
- [Rotation-Invariant Point Convolution With Multiple Equivariant Alignments](https://mlsanity.com/paper/2012.04048/) (2020)
- [Differential invariants for SE(2)-equivariant networks](https://mlsanity.com/paper/2206.13279/) (2022)
- [PDO-s3DCNNs: Partial Differential Operator Based Steerable 3D CNNs](https://mlsanity.com/paper/2208.03720/) (2022)

## Built on by (most cited first, 6 of 6)

- [ILPO-NET: Network for the invariant recognition of arbitrary volumetric patterns in 3D](https://mlsanity.com/paper/2403.19612/) (2024)
- [On the Fourier analysis in the SO(3) space : EquiLoPO Network](https://mlsanity.com/paper/2404.15979/) (2024)
- [Equivariant Eikonal Neural Networks: Grid-Free, Scalable Travel-Time Prediction on Homogeneous Spaces](https://mlsanity.com/paper/2505.16035/) (2025)
- [Entropy Transformer Networks: A Learning Approach via Tangent Bundle Data Manifold](https://mlsanity.com/paper/2307.12517/) (2023)
- [Adaptive Sampling for Continuous Group Equivariant Neural Networks](https://mlsanity.com/paper/2409.08741/) (2024)
- [bispectrum: Selective G-Bispectra Made Practical](https://mlsanity.com/paper/2605.07270/) (2026)

## Abstract (arXiv)

Equivariance of neural networks to transformations helps to improve their performance and reduce generalization error in computer vision tasks, as they apply to datasets presenting symmetries (e.g. scalings, rotations, translations). The method of moving frames is classical for deriving operators invariant to the action of a Lie group in a manifold.Recently, a rotation and translation equivariant neural network for image data was proposed based on the moving frames approach. In this paper we significantly improve that approach by reducing the computation of moving frames to only one, at the input stage, instead of repeated computations at each layer. The equivariance of the resulting architecture is proved theoretically and we build a rotation and translation equivariant neural network to process volumes, i.e. signals on the 3D space. Our trained model overperforms the benchmarks in the medical volume classification of most of the tested datasets from MedMNIST3D.