# Transformation Coding: Simple Objectives for Equivariant Representations

arXiv: https://arxiv.org/abs/2202.10930 · HTML: https://mlsanity.com/paper/2202.10930/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 45 references that are in the map.
- First posted on arXiv 19 Feb 2022.

## Builds on (most cited first, 30 of 31)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Contrastive Multiview Coding](https://mlsanity.com/paper/1906.05849/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [A Point Set Generation Network for 3D Object Reconstruction from a Single Image](https://mlsanity.com/paper/1612.00603/) (2016)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Whitening for Self-Supervised Representation Learning](https://mlsanity.com/paper/2007.06346/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Contrastive Learning of Structured World Models](https://mlsanity.com/paper/1911.12247/) (2019)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Deep Convolutional Inverse Graphics Network](https://mlsanity.com/paper/1503.03167/) (2015)
- [Explorations in Homeomorphic Variational Auto-Encoding](https://mlsanity.com/paper/1807.04689/) (2018)
- [Symmetry-Based Disentangled Representation Learning requires Interaction with Environments](https://mlsanity.com/paper/1904.00243/) (2019)
- [Interpretable Transformations with Encoder-Decoder Networks](https://mlsanity.com/paper/1710.07307/) (2017)
- [Group Equivariant Capsule Networks](https://mlsanity.com/paper/1806.05086/) (2018)
- [Learning Covariant Feature Detectors](https://mlsanity.com/paper/1605.01224/) (2016)
- [Transformation Properties of Learned Visual Representations](https://mlsanity.com/paper/1412.7659/) (2014)
- [Linear Disentangled Representations and Unsupervised Action Estimation](https://mlsanity.com/paper/2008.07922/) (2020)

## Abstract (arXiv)

We present a simple non-generative approach to deep representation learning that seeks equivariant deep embedding through simple objectives. In contrast to existing equivariant networks, our transformation coding approach does not constrain the choice of the feed-forward layer or the architecture and allows for an unknown group action on the input space. We introduce several such transformation coding objectives for different Lie groups such as the Euclidean, Orthogonal and the Unitary groups. When using product groups, the representation is decomposed and disentangled. We show that the presence of additional information on different transformations improves disentanglement in transformation coding. We evaluate the representations learnt by transformation coding both qualitatively and quantitatively on downstream tasks, including reinforcement learning.