# Scale-Rotation-Equivariant Lie Group Convolution Neural Networks (Lie Group-CNNs)

arXiv: https://arxiv.org/abs/2306.06934 · HTML: https://mlsanity.com/paper/2306.06934/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 42 references that are in the map.
- First posted on arXiv 12 Jun 2023.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [DeepLab: Semantic Image Segmentation with Deep Convolutional Nets, Atrous Convolution, and Fully Connected CRFs](https://mlsanity.com/paper/1606.00915/) (2016)
- [MLP-Mixer: An all-MLP Architecture for Vision](https://mlsanity.com/paper/2105.01601/) (2021)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Multi-Scale Context Aggregation by Dilated Convolutions](https://mlsanity.com/paper/1511.07122/) (2015)
- [Spatial Pyramid Pooling in Deep Convolutional Networks for Visual Recognition](https://mlsanity.com/paper/1406.4729/) (2014)
- [Res2Net: A New Multi-scale Backbone Architecture](https://mlsanity.com/paper/1904.01169/) (2019)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Rotation Equivariant CNNs for Digital Pathology](https://mlsanity.com/paper/1806.03962/) (2018)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Understanding image representations by measuring their equivariance and equivalence](https://mlsanity.com/paper/1411.5908/) (2014)
- [Why do deep convolutional networks generalize so poorly to small image transformations?](https://mlsanity.com/paper/1805.12177/) (2018)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [Convolutional Networks with Dense Connectivity](https://mlsanity.com/paper/2001.02394/) (2020)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Rotation-invariant convolutional neural networks for galaxy morphology prediction](https://mlsanity.com/paper/1503.07077/) (2015)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Scale-Invariant Convolutional Neural Networks](https://mlsanity.com/paper/1411.6369/) (2014)
- [Fourier Series Expansion Based Filter Parametrization for Equivariant Convolutions](https://mlsanity.com/paper/2107.14519/) (2021)
- [Learning Generalized Transformation Equivariant Representations via Autoencoding Transformations](https://mlsanity.com/paper/1906.08628/) (2019)
- [Equivariant Wavelets: Fast Rotation and Translation Invariant Wavelet Scattering Transforms](https://mlsanity.com/paper/2104.11244/) (2021)

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

- [Relaxed Equivariance via Multitask Learning](https://mlsanity.com/paper/2410.17878/) (2024)
- [Unified Alignment Protocol: Making Sense of the Unlabeled Data in New Domains](https://mlsanity.com/paper/2505.21010/) (2025)

## Abstract (arXiv)

The weight-sharing mechanism of convolutional kernels ensures translation-equivariance of convolution neural networks (CNNs). Recently, rotation-equivariance has been investigated. However, research on scale-equivariance or simultaneous scale-rotation-equivariance is insufficient. This study proposes a Lie group-CNN, which can keep scale-rotation-equivariance for image classification tasks. The Lie group-CNN includes a lifting module, a series of group convolution modules, a global pooling layer, and a classification layer. The lifting module transfers the input image from Euclidean space to Lie group space, and the group convolution is parameterized through a fully connected network using Lie-algebra of Lie-group elements as inputs to achieve scale-rotation-equivariance. The Lie group SIM(2) is utilized to establish the Lie group-CNN with scale-rotation-equivariance. Scale-rotation-equivariance of Lie group-CNN is verified and achieves the best recognition accuracy on the blood cell dataset (97.50%) and the HAM10000 dataset (77.90%) superior to Lie algebra convolution network, dilation convolution, spatial transformer network, and scale-equivariant steerable network. In addition, the generalization ability of the Lie group-CNN on SIM(2) on rotation-equivariance is verified on rotated-MNIST and rotated-CIFAR10, and the robustness of the network is verified on SO(2) and SE(2). Therefore, the Lie group-CNN can successfully extract geometric features and performs equivariant recognition on images with rotation and scale transformations.