# Rotation equivariant vector field networks

arXiv: https://arxiv.org/abs/1612.09346 · HTML: https://mlsanity.com/paper/1612.09346/

## Key facts

- Cited by 99 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Dec 2016.

## Built on by (most cited first, 30 of 99)

- [ReDet: A Rotation-equivariant Detector for Aerial Object Detection](https://mlsanity.com/paper/2103.07733/) (2021)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [EqMotion: Equivariant Multi-agent Motion Prediction with Invariant Interaction Reasoning](https://mlsanity.com/paper/2303.10876/) (2023)
- [EQ-VAE: Equivariance Regularized Latent Space for Improved Generative Image Modeling](https://mlsanity.com/paper/2502.09509/) (2025)
- [Equivariant Point Network for 3D Point Cloud Analysis](https://mlsanity.com/paper/2103.14147/) (2021)
- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [The Lie Derivative for Measuring Learned Equivariance](https://mlsanity.com/paper/2210.02984/) (2022)
- [DEVIANT: Depth EquiVarIAnt NeTwork for Monocular 3D Object Detection](https://mlsanity.com/paper/2207.10758/) (2022)
- [Constrained Learning with Non-Convex Losses](https://mlsanity.com/paper/2103.05134/) (2021)
- [Segment anything, from space?](https://mlsanity.com/paper/2304.13000/) (2023)
- [Sample Efficiency of Data Augmentation Consistency Regularization](https://mlsanity.com/paper/2202.12230/) (2022)
- [VN-Transformer: Rotation-Equivariant Attention for Vector Neurons](https://mlsanity.com/paper/2206.04176/) (2022)
- [E(2)-Equivariant Vision Transformer](https://mlsanity.com/paper/2306.06722/) (2023)
- [Fourier Series Expansion Based Filter Parametrization for Equivariant Convolutions](https://mlsanity.com/paper/2107.14519/) (2021)
- [Self-Supervised Equivariant Learning for Oriented Keypoint Detection](https://mlsanity.com/paper/2204.08613/) (2022)
- [Learning Rotation-Equivariant Features for Visual Correspondence](https://mlsanity.com/paper/2303.15472/) (2023)
- [Exploiting Redundancy: Separable Group Convolutional Networks on Lie Groups](https://mlsanity.com/paper/2110.13059/) (2021)
- [RIC-CNN: Rotation-Invariant Coordinate Convolutional Neural Network](https://mlsanity.com/paper/2211.11812/) (2022)
- [Scale-invariant scale-channel networks: Deep networks that generalise to previously unseen scales](https://mlsanity.com/paper/2106.06418/) (2021)
- [Rotation Equivariant Proximal Operator for Deep Unfolding Methods in Image Restoration](https://mlsanity.com/paper/2312.15701/) (2023)
- [Equivariant Networks for Pixelized Spheres](https://mlsanity.com/paper/2106.06662/) (2021)
- [Learning Invariant Weights in Neural Networks](https://mlsanity.com/paper/2202.12439/) (2022)
- [Learning Instance-Specific Augmentations by Capturing Local Invariances](https://mlsanity.com/paper/2206.00051/) (2022)
- [Reflection and Rotation Symmetry Detection via Equivariant Learning](https://mlsanity.com/paper/2203.16787/) (2022)
- [RiEMann: Near Real-Time SE(3)-Equivariant Robot Manipulation without Point Cloud Segmentation](https://mlsanity.com/paper/2403.19460/) (2024)
- [Rotation Equivariant Feature Image Pyramid Network for Object Detection in Optical Remote Sensing Imagery](https://mlsanity.com/paper/2106.00880/) (2021)
- [Normalization-Equivariant Neural Networks with Application to Image Denoising](https://mlsanity.com/paper/2306.05037/) (2023)
- [Approximately Equivariant Quantum Neural Network for p4m Group Symmetries in Images](https://mlsanity.com/paper/2310.02323/) (2023)
- [Training or Architecture? How to Incorporate Invariance in Neural Networks](https://mlsanity.com/paper/2106.10044/) (2021)

## Abstract (arXiv)

In many computer vision tasks, we expect a particular behavior of the output with respect to rotations of the input image. If this relationship is explicitly encoded, instead of treated as any other variation, the complexity of the problem is decreased, leading to a reduction in the size of the required model. In this paper, we propose the Rotation Equivariant Vector Field Networks (RotEqNet), a Convolutional Neural Network (CNN) architecture encoding rotation equivariance, invariance and covariance. Each convolutional filter is applied at multiple orientations and returns a vector field representing magnitude and angle of the highest scoring orientation at every spatial location. We develop a modified convolution operator relying on this representation to obtain deep architectures. We test RotEqNet on several problems requiring different responses with respect to the inputs' rotation: image classification, biomedical image segmentation, orientation estimation and patch matching. In all cases, we show that RotEqNet offers extremely compact models in terms of number of parameters and provides results in line to those of networks orders of magnitude larger.