# Equivariance-bridged SO(2)-Invariant Representation Learning using Graph Convolutional Network

arXiv: https://arxiv.org/abs/2106.09996 · HTML: https://mlsanity.com/paper/2106.09996/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 36 references that are in the map.
- First posted on arXiv 18 Jun 2021.

## Builds on (most cited first, 24 of 24)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [The Effectiveness of Data Augmentation in Image Classification using Deep Learning](https://mlsanity.com/paper/1712.04621/) (2017)
- [Multi-Label Image Recognition with Graph Convolutional Networks](https://mlsanity.com/paper/1904.03582/) (2019)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Deep Convolutional Networks on Graph-Structured Data](https://mlsanity.com/paper/1506.05163/) (2015)
- [Graph Convolutional Networks for Temporal Action Localization](https://mlsanity.com/paper/1909.03252/) (2019)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Rotation equivariant vector field networks](https://mlsanity.com/paper/1612.09346/) (2016)
- [TI-POOLING: transformation-invariant pooling for feature learning in Convolutional Neural Networks](https://mlsanity.com/paper/1604.06318/) (2016)
- [Graph-based Isometry Invariant Representation Learning](https://mlsanity.com/paper/1703.00356/) (2017)
- [Invariant Deep Compressible Covariance Pooling for Aerial Scene Categorization](https://mlsanity.com/paper/2011.05702/) (2020)

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

- [H-NeXt: The next step towards roto-translation invariant networks](https://mlsanity.com/paper/2311.01111/) (2023)
- [Harmformer: Harmonic Networks Meet Transformers for Continuous Roto-Translation Equivariance](https://mlsanity.com/paper/2411.03794/) (2024)

## Abstract (arXiv)

Training a Convolutional Neural Network (CNN) to be robust against rotation has mostly been done with data augmentation. In this paper, another progressive vision of research direction is highlighted to encourage less dependence on data augmentation by achieving structural rotational invariance of a network. The deep equivariance-bridged SO(2) invariant network is proposed to echo such vision. First, Self-Weighted Nearest Neighbors Graph Convolutional Network (SWN-GCN) is proposed to implement Graph Convolutional Network (GCN) on the graph representation of an image to acquire rotationally equivariant representation, as GCN is more suitable for constructing deeper network than spectral graph convolution-based approaches. Then, invariant representation is eventually obtained with Global Average Pooling (GAP), a permutation-invariant operation suitable for aggregating high-dimensional representations, over the equivariant set of vertices retrieved from SWN-GCN. Our method achieves the state-of-the-art image classification performance on rotated MNIST and CIFAR-10 images, where the models are trained with a non-augmented dataset only. Quantitative validations over invariance of the representations also demonstrate strong invariance of deep representations of SWN-GCN over rotations.