# A Simple Strategy to Provable Invariance via Orbit Mapping

arXiv: https://arxiv.org/abs/2209.11916 · HTML: https://mlsanity.com/paper/2209.11916/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 83 references that are in the map.
- First posted on arXiv 24 Sep 2022.

## Builds on (most cited first, 30 of 51)

- [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)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Unsupervised Representation Learning by Predicting Image Rotations](https://mlsanity.com/paper/1803.07728/) (2018)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Unsupervised Learning of Visual Representations by Solving Jigsaw Puzzles](https://mlsanity.com/paper/1603.09246/) (2016)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Normalized Object Coordinate Space for Category-Level 6D Object Pose and Size Estimation](https://mlsanity.com/paper/1901.02970/) (2019)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [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)
- [High-Performance Large-Scale Image Recognition Without Normalization](https://mlsanity.com/paper/2102.06171/) (2021)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Understanding image representations by measuring their equivariance and equivalence](https://mlsanity.com/paper/1411.5908/) (2014)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [PPF-FoldNet: Unsupervised Learning of Rotation Invariant 3D Local Descriptors](https://mlsanity.com/paper/1808.10322/) (2018)
- [Deep Functional Maps: Structured Prediction for Dense Shape Correspondence](https://mlsanity.com/paper/1704.08686/) (2017)
- [3D Point Capsule Networks](https://mlsanity.com/paper/1812.10775/) (2018)
- [Rotation equivariant vector field networks](https://mlsanity.com/paper/1612.09346/) (2016)
- [Polar Transformer Networks](https://mlsanity.com/paper/1709.01889/) (2017)
- [Exploring the Landscape of Spatial Robustness](https://mlsanity.com/paper/1712.02779/) (2017)
- [AET vs. AED: Unsupervised Representation Learning by Auto-Encoding Transformations rather than Data](https://mlsanity.com/paper/1901.04596/) (2019)
- [Deep Shells: Unsupervised Shape Correspondence with Optimal Transport](https://mlsanity.com/paper/2010.15261/) (2020)

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

- [How Much Data Are Augmentations Worth? An Investigation into Scaling Laws, Invariance, and Implicit Regularization](https://mlsanity.com/paper/2210.06441/) (2022)
- [Improving Equivariance in State-of-the-Art Supervised Depth and Normal Predictors](https://mlsanity.com/paper/2309.16646/) (2023)

## Abstract (arXiv)

Many applications require robustness, or ideally invariance, of neural networks to certain transformations of input data. Most commonly, this requirement is addressed by training data augmentation, using adversarial training, or defining network architectures that include the desired invariance by design. In this work, we propose a method to make network architectures provably invariant with respect to group actions by choosing one element from a (possibly continuous) orbit based on a fixed criterion. In a nutshell, we intend to 'undo' any possible transformation before feeding the data into the actual network. Further, we empirically analyze the properties of different approaches which incorporate invariance via training or architecture, and demonstrate the advantages of our method in terms of robustness and computational efficiency. In particular, we investigate the robustness with respect to rotations of images (which can hold up to discretization artifacts) as well as the provable orientation and scaling invariance of 3D point cloud classification.