# Deformation Robust Roto-Scale-Translation Equivariant CNNs

arXiv: https://arxiv.org/abs/2111.10978 · HTML: https://mlsanity.com/paper/2111.10978/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 57 references that are in the map.
- First posted on arXiv 22 Nov 2021.

## Builds on (most cited first, 30 of 43)

- [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)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networks](https://mlsanity.com/paper/1506.01497/) (2015)
- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Improved Regularization of Convolutional Neural Networks with Cutout](https://mlsanity.com/paper/1708.04552/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [End to End Learning for Self-Driving Cars](https://mlsanity.com/paper/1604.07316/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (2019)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [Rotation equivariant vector field networks](https://mlsanity.com/paper/1612.09346/) (2016)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Polar Transformer Networks](https://mlsanity.com/paper/1709.01889/) (2017)
- [Equivariant Point Network for 3D Point Cloud Analysis](https://mlsanity.com/paper/2103.14147/) (2021)
- [Oriented Response Networks](https://mlsanity.com/paper/1701.01833/) (2017)
- [DeepSphere: a graph-based spherical CNN](https://mlsanity.com/paper/2012.15000/) (2020)

## Built on by (most cited first, 7 of 7)

- [SE(3)-Equivariant and Noise-Invariant 3D Rigid Motion Tracking in Brain MRI](https://mlsanity.com/paper/2312.13534/) (2023)
- [Empowering Networks With Scale and Rotation Equivariance Using A Similarity Convolution](https://mlsanity.com/paper/2303.00326/) (2023)
- [Rotation-Scale Equivariant Steerable Filters](https://mlsanity.com/paper/2304.04600/) (2023)
- [SO(2) and O(2) Equivariance in Image Recognition with Bessel-Convolutional Neural Networks](https://mlsanity.com/paper/2304.09214/) (2023)
- [Adaptive aggregation of Monte Carlo augmented decomposed filters for efficient group-equivariant convolutional neural network](https://mlsanity.com/paper/2305.10110/) (2023)
- [Efficient 3D affinely equivariant CNNs with adaptive fusion of augmented spherical Fourier-Bessel bases](https://mlsanity.com/paper/2402.16825/) (2024)
- [Stability Analysis of Equivariant Convolutional Representations Through The Lens of Equivariant Multi-layered CKNs](https://mlsanity.com/paper/2408.04277/) (2024)

## Abstract (arXiv)

Incorporating group symmetry directly into the learning process has proved to be an effective guideline for model design. By producing features that are guaranteed to transform covariantly to the group actions on the inputs, group-equivariant convolutional neural networks (G-CNNs) achieve significantly improved generalization performance in learning tasks with intrinsic symmetry. General theory and practical implementation of G-CNNs have been studied for planar images under either rotation or scaling transformation, but only individually. We present, in this paper, a roto-scale-translation equivariant CNN (RST-CNN), that is guaranteed to achieve equivariance jointly over these three groups via coupled group convolutions. Moreover, as symmetry transformations in reality are rarely perfect and typically subject to input deformation, we provide a stability analysis of the equivariance of representation to input distortion, which motivates the truncated expansion of the convolutional filters under (pre-fixed) low-frequency spatial modes. The resulting model provably achieves deformation-robust RST equivariance, i.e., the RST symmetry is still "approximately" preserved when the transformation is "contaminated" by a nuisance data deformation, a property that is especially important for out-of-distribution generalization. Numerical experiments on MNIST, Fashion-MNIST, and STL-10 demonstrate that the proposed model yields remarkable gains over prior arts, especially in the small data regime where both rotation and scaling variations are present within the data.