# GMR-Conv: An Efficient Rotation and Reflection Equivariant Convolution Kernel Using Gaussian Mixture Rings

arXiv: https://arxiv.org/abs/2504.02819 · HTML: https://mlsanity.com/paper/2504.02819/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 52 references that are in the map.
- First posted on arXiv 3 Apr 2025.

## Builds on (most cited first, 30 of 33)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Feature Pyramid Networks for Object Detection](https://mlsanity.com/paper/1612.03144/) (2016)
- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [A Closer Look at Spatiotemporal Convolutions for Action Recognition](https://mlsanity.com/paper/1711.11248/) (2017)
- [Understanding the Effective Receptive Field in Deep Convolutional Neural Networks](https://mlsanity.com/paper/1701.04128/) (2017)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [ShellNet: Efficient Point Cloud Convolutional Neural Networks using Concentric Shells Statistics](https://mlsanity.com/paper/1908.06295/) (2019)
- [Exploiting Cyclic Symmetry in Convolutional Neural Networks](https://mlsanity.com/paper/1602.02660/) (2016)
- [Polar Transformer Networks](https://mlsanity.com/paper/1709.01889/) (2017)
- [Oriented Response Networks](https://mlsanity.com/paper/1701.01833/) (2017)
- [CryoNuSeg: A Dataset for Nuclei Instance Segmentation of Cryosectioned H&E-Stained Histological Images](https://mlsanity.com/paper/2101.00442/) (2021)
- [Learning Pixel Trajectories with Multiscale Contrastive Random Walks](https://mlsanity.com/paper/2201.08379/) (2022)
- [RIC-CNN: Rotation-Invariant Coordinate Convolutional Neural Network](https://mlsanity.com/paper/2211.11812/) (2022)
- [CyCNN: A Rotation Invariant CNN using Polar Mapping and Cylindrical Convolution Layers](https://mlsanity.com/paper/2007.10588/) (2020)
- [Rotation Equivariant Siamese Networks for Tracking](https://mlsanity.com/paper/2012.13078/) (2020)
- [Deep-HiTS: Rotation Invariant Convolutional Neural Network for Transient Detection](https://mlsanity.com/paper/1701.00458/) (2017)
- [Rotated Ring, Radial and Depth Wise Separable Radial Convolutions](https://mlsanity.com/paper/2010.00873/) (2020)
- [Sample Efficient Semantic Segmentation using Rotation Equivariant Convolutional Networks](https://mlsanity.com/paper/1807.00583/) (2018)
- [SPIN: Simplifying Polar Invariance for Neural networks Application to vision-based irradiance forecasting](https://mlsanity.com/paper/2111.14507/) (2021)
- [Use of symmetric kernels for convolutional neural networks](https://mlsanity.com/paper/1805.09421/) (2018)
- [SymNet: Symmetrical Filters in Convolutional Neural Networks](https://mlsanity.com/paper/1906.04252/) (2019)
- [SRE-CNN: A Spatiotemporal Rotation-Equivariant CNN for Cardiac Cine MR Imaging](https://mlsanity.com/paper/2409.08537/) (2024)

## Built on by (most cited first, 1 of 1)

- [Equi-ViT: Rotational Equivariant Vision Transformer for Robust Histopathology Analysis](https://mlsanity.com/paper/2601.09130/) (2026)

## Abstract (arXiv)

Symmetry, where certain features remain invariant under geometric transformations, can often serve as a powerful prior in designing convolutional neural networks (CNNs). While conventional CNNs inherently support translational equivariance, extending this property to rotation and reflection has proven challenging, often forcing a compromise between equivariance, efficiency, and information loss. In this work, we introduce Gaussian Mixture Ring Convolution (GMR-Conv), an efficient convolution kernel that smooths radial symmetry using a mixture of Gaussian-weighted rings. This design mitigates discretization errors of circular kernels, thereby preserving robust rotation and reflection equivariance without incurring computational overhead. We further optimize both the space and speed efficiency of GMR-Conv via a novel parameterization and computation strategy, allowing larger kernels at an acceptable cost. Extensive experiments on eight classification and one segmentation datasets demonstrate that GMR-Conv not only matches conventional CNNs' performance but can also surpass it in applications with orientation-less data. GMR-Conv is also proven to be more robust and efficient than the state-of-the-art equivariant learning methods. Our work provides inspiring empirical evidence that carefully applied radial symmetry can alleviate the challenges of information loss, marking a promising advance in equivariant network architectures. The code is available at https://github.com/XYPB/GMR-Conv.