awesome-equivariant-network

Paper list for equivariant neural network. Work-in-progress.

Feel free to suggest relevant papers in the following format.

**Group Equivariant Convolutional Networks**  
Taco S. Cohen, Max Welling ICML 2016 [paper](https://arxiv.org/pdf/1602.07576.pdf)   

Acknowledgement: I would like to thank Maurice Weiler, Fabian Fuchs, Tess Smidt, Rui Wang, David Pfau, Jonas Köhler, Taco Cohen, Gregor Simm, Erik J Bekkers, Jean-Baptiste Cordonnier, David W. Romero, Ivan Sosnovik, Kostas Daniilidis for paper suggestions! Thank Weihao Xia for helping out typesetting!

Table of Contents

• Equivariance and Group convolution
• Theory
• Equivariant Density Estimation and Sampling
• Application
• Permutation Equivariance
• Talk and Tutorial
• TO READ

Equivariance and Group convolution

1. Group Equivariant Convolutional Networks
   Taco S. Cohen, Max Welling ICML 2016 paper
   Note: first paper; discrete group;
2. Steerable CNNs
   Taco S. Cohen, Max Welling ICLR 2017 paper
3. Harmonic Networks: Deep Translation and Rotation Equivariance
   Daniel E. Worrall, Stephan J. Garbin, Daniyar Turmukhambetov, Gabriel J. Brostow CVPR 2017 paper
4. Spherical CNNs
   Taco S. Cohen, Mario Geiger, Jonas Koehler, Max Welling ICLR 2018 best paper paper
   Note: use generalized FFT to speed up convolution on $S^2$ and $SO(3)$
5. Clebsch–Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network
   Risi Kondor, Zhen Lin, Shubhendu Trivedi NeurIPS 2018 paper
   Note: perform equivariant nonlinearity in Fourier space;
6. General E(2)-Equivariant Steerable CNNs
   Maurice Weiler, Gabriele Cesa NeurIPS 2019 paper
   Note: nice benchmark on different reprsentations
7. Learning Steerable Filters for Rotation Equivariant CNNs
   Maurice Weiler, Fred A. Hamprecht, Martin Storath CVPR 2018 paper
   Note: group convolutions, kernels parameterized in circular harmonic basis (steerable filters);
8. Learning SO(3) Equivariant Representations with Spherical CNNs
   Carlos Esteves, Christine Allen-Blanchette, Ameesh Makadia, Kostas Daniilidis ECCV 2018 paper
   Note: SO(3) equivariance; zonal filter
9. Polar Transformer Networks
   Carlos Esteves, Christine Allen-Blanchette, Xiaowei Zhou, Kostas Daniilidis ICLR 2018 paper
10. 3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data
    Maurice Weiler, Mario Geiger, Max Welling, Wouter Boomsma, Taco Cohen NeurIPS 2018 paper
    Note: SE(3) equivariance; characterize the basis of steerable kernel
11. Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds
    Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, Patrick Riley paper
    Note: SE(3) equivariance for point clouds
12. Equivariant Multi-View Networks
    Carlos Esteves, Yinshuang Xu, Christine Allen-Blanchette, Kostas Daniilidis ICCV 2019 paper
13. Gauge Equivariant Convolutional Networks and the Icosahedral CNN
    Taco S. Cohen, Maurice Weiler, Berkay Kicanaoglu, Max Welling ICML 2019 paper, talk
    Note: gauge equivariance on general manifold
14. Cormorant: Covariant Molecular Neural Networks
    Brandon Anderson, Truong-Son Hy, Risi Kondor NeurIPS 2019 paper
15. Deep Scale-spaces: Equivariance Over Scale
    Daniel Worrall, Max Welling NeurIPS 2019 paper
16. Scale-Equivariant Steerable Networks
    Ivan Sosnovik, Michał Szmaja, Arnold Smeulders ICLR 2020 paper
17. B-Spline CNNs on Lie Groups
    Erik J Bekkers ICLR 2020 paper
18. SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks
    Fabian B. Fuchs, Daniel E. Worrall, Volker Fischer, Max Welling NeurIPS 2020 paper, blog
    Note: TFN + equivariant self-attention; improved spherical harmonics computation
19. Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs
    Pim de Haan, Maurice Weiler, Taco Cohen, Max Welling ICLR 2021 paper
    Note: anisotropic gauge equivariant kernels + message passing by parallel transporting features over mesh edges
20. Lorentz Group Equivariant Neural Network for Particle Physics
    Alexander Bogatskiy, Brandon Anderson, Jan T. Offermann, Marwah Roussi, David W. Miller, Risi Kondor ICML 2020 paper
    Note: SO(1, 3) equivariance
21. Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data
    Marc Finzi, Samuel Stanton, Pavel Izmailov, Andrew Gordon Wilson ICML 2020 paper
    Note: fairly generic architecture; use Monte Carlo sampling to achieve equivariance in expectation;
22. Spin-Weighted Spherical CNNs
    Carlos Esteves, Ameesh Makadia, Kostas Daniilidis NeurIPS 2020 paper
    Note: anisotropic filter for vector field on sphere
23. Learning Invariances in Neural Networks
    Gregory Benton, Marc Finzi, Pavel Izmailov, Andrew Gordon Wilson NeurIPS 2020 paper
    Note: very interesting approch; enfore "soft" invariance via learning over both model parameters and distributions over augmentations
24. Lie Algebra Convolutional Neural Networks with Automatic Symmetry Extraction
    Nima Dehmamy, Yanchen Liu, Robin Walters, Rose Yu paper
    Note: very interesting paper; It’s unfortunate that it is rejected by ICLR 2021
25. LieTransformer: Equivariant self-attention for Lie Groups
    Michael Hutchinson, Charline Le Lan, Sheheryar Zaidi, Emilien Dupont, Yee Whye Teh, Hyunjik Kim paper
    Note: equivariant self attention to arbitrary Lie groups and their discrete subgroups
26. Co-Attentive Equivariant Neural Networks: Focusing Equivariance On Transformations Co-Occurring In Data
    David W. Romero, Mark Hoogendoorn ICLR 2020 paper
27. Attentive Group Equivariant Convolutional Networks
    David W. Romero, Erik J. Bekkers, Jakub M. Tomczak, Mark Hoogendoorn ICML 2020 paper
28. Wavelet Networks: Scale Equivariant Learning From Raw Waveforms
    David W. Romero, Erik J. Bekkers, Jakub M. Tomczak, Mark Hoogendoorn paper
29. Group Equivariant Stand-Alone Self-Attention For Vision
    David W. Romero, Jean-Baptiste Cordonnier ICLR 2021 paper
30. Incorporating Symmetry into Deep Dynamics Models for Improved Generalization
    Rui Wang, Robin Walters, Rose Yu ICLR 2021 paper
31. MDP Homomorphic Networks: Group Symmetries in Reinforcement Learning
    Elise van der Pol, Daniel E. Worrall, Herke van Hoof, Frans A. Oliehoek, Max Welling NeurIPS 2020 paper
32. Isometric Transformation Invariant and Equivariant Graph Convolutional Networks
    Masanobu Horie, Naoki Morita, Toshiaki Hishinuma, Yu Ihara, Naoto Mitsume ICLR 2021 paper
33. E(n) Equivariant Graph Neural Networks
    Victor Garcia Satorras, Emiel Hoogeboom, Max Welling ICML 2021 paper
    Note: a simple alternative that achieves E(n) equivariance
34. Vector Neurons: A General Framework for SO(3)-Equivariant Networks
    Congyue Deng, Or Litany, Yueqi Duan, Adrien Poulenard, Andrea Tagliasacchi, Leonidas Guibas paper
    Note: a simple MLP for type-1 features
35. Equivariant message passing for the prediction of tensorial properties and molecular spectra
    Kristof T. Schütt, Oliver T. Unke, Michael Gastegger ICML 2021 paper
36. Field Convolutions For Surface CNNs Thomas W. Mitchel, Vladimir G. Kim, Michael Kazhdan ICCV 2021 (Oral) paper
37. Scalars are universal: Equivariant machine learning, structured like classical physics Soledad Villar, David W.Hogg, Kate Storey-Fisher, Weichi Yao, Ben Blum-Smith NeruIPS 2021 paper
38. GemNet: Universal Directional Graph Neural Networks for Molecules Johannes Klicpera, Florian Becker, Stephan Günnemann NeurIPS 2021 paper
39. Geometric and Physical Quantities improve E(3) Equivariant Message Passing Johannes Brandstetter and Rob Hesselink and Elise van der Pol and Erik J Bekkers and Max Welling ICLR 2022 (spotlight) paper, code

Theory

1. On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups
   Risi Kondor, Shubhendu Trivedi ICML 2018 paper
   Note: convolution is all you need (for scalar fields)

2. A General Theory of Equivariant CNNs on Homogeneous Spaces
   Taco Cohen, Mario Geiger, Maurice Weiler NeurIPS 2019 paper
   Note: convolution is all you need (for general fields)

3. Equivariance Through Parameter-Sharing
   Siamak Ravanbakhsh, Jeff Schneider, Barnabas Poczos ICML 2017 paper

4. Universal approximations of invariant maps by neural networks
   Dmitry Yarotsky paper

5. A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels
   Leon Lang, Maurice Weiler ICLR 2021 paper
   Note: steerable kernel spaces are fully understood and parameterized in terms of 1) generalized reduced matrix elements, 2) Clebsch-Gordan coefficients, and 3) harmonic basis functions on homogeneous spaces.

6. On the Universality of Rotation Equivariant Point Cloud Networks
   Nadav Dym, Haggai Maron ICLR 2021 paper,
   Note: universality for TFN and se3-transformer

7. Universal Equivariant Multilayer Perceptrons
   Siamak Ravanbakhsh paper

8. Provably Strict Generalisation Benefit for Equivariant Models
   Bryn Elesedy, Sheheryar Zaidi ICML 2021 paper

Equivariant Density Estimation and Sampling

1. Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities
   Jonas Köhler, Leon Klein, Frank Noé ICML 2020 paper
   Note: general framework for constructing equivariant normalizing flows on euclidean spaces. Instantiation for particle systems/point clouds = simultanoues SE(3) and permutation equivariance.
2. Equivariant Hamiltonian Flows
   Danilo Jimenez Rezende, Sébastien Racanière, Irina Higgins, Peter Toth NeurIPS 2019 ML4Phys workshop paper
   Note: general framework for constructing equivariant normalizing flows in phase space utilizing Hamiltonian dynamics. Instantiation for SE(2) equivariance.
3. Sampling using SU(N) gauge equivariant flows
   Denis Boyda, Gurtej Kanwar, Sébastien Racanière, Danilo Jimenez Rezende, Michael S. Albergo, Kyle Cranmer, Daniel C. Hackett, Phiala E. Shanahan paper
   Note: normalizing flows for lattice gauge theory. Instantiation for SU(2)/SU(3) equivariance.
4. Exchangeable neural ode for set modeling
   Yang Li, Haidong Yi, Christopher M. Bender, Siyuan Shan, Junier B. Oliva NeurIPS 2020 paper
   Note: framework for permutation equivariant flows for set data. Instantiation for permutation equivariance.
5. Equivariant Normalizing Flows for Point Processes and Sets
   Marin Biloš, Stephan Günnemann NeurIPS 2020 paper
   Note: framework for permutation equivariant flows for set data. Instantiation for permutation equivariance.
6. The Convolution Exponential and Generalized Sylvester Flows
   Emiel Hoogeboom, Victor Garcia Satorras, Jakub M. Tomczak, Max Welling NeurIPS 2020 paper
   Note: invertible convolution operators. Instantiation for permutation equivariance.
7. Targeted free energy estimation via learned mappings
   Peter Wirnsberger, Andrew J. Ballard, George Papamakarios, Stuart Abercrombie, Sébastien Racanière, Alexander Pritzel, Danilo Jimenez Rezende, Charles Blundell J Chem Phys. 2020 Oct 14;153(14):144112. paper
   Note: normalizing flows for particle systems on a torus. Instantiation for permutation equivariance.
8. Temperature-steerable flows
   Manuel Dibak, Leon Klein, Frank Noé NeurIPS 2020 ML4Phys workshops paper
   Note: normalizing flows in phase space with equivariance with respect to changes in temperature.

Application

1. Trajectory Prediction using Equivariant Continuous Convolution
   Robin Walters, Jinxi Li, Rose Yu ICLR 2021 paper
2. SE(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials
   Simon Batzner, Tess E. Smidt, Lixin Sun, Jonathan P. Mailoa, Mordechai Kornbluth, Nicola Molinari, Boris Kozinsky paper
3. Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks
   Tess E. Smidt, Mario Geiger, Benjamin Kurt Miller paper
4. Group Equivariant Generative Adversarial Networks
   Neel Dey, Antong Chen, Soheil Ghafurian ICLR 2021 paper
5. Ab-Initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks
   David Pfau, James S. Spencer, Alexander G. de G. Matthews, W. M. C. Foulkes paper
6. Symmetry-Aware Actor-Critic for 3D Molecular Design
   Gregor N. C. Simm, Robert Pinsler, Gábor Csányi, José Miguel Hernández-Lobato ICLR 2021 paper
7. Roto-translation equivariant convolutional networks: Application to histopathology image analysis
   Maxime W. Lafarge, Erik J. Bekkers, Josien P.W. Pluim, Remco Duits, Mitko Veta MedIA paper
8. Scale Equivariance Improves Siamese Tracking
   Ivan Sosnovik*, Artem Moskalev*, Arnold Smeulders WACV 2021 paper
9. 3D G-CNNs for Pulmonary Nodule Detection Marysia Winkels, Taco S. Cohen paper International Conference on Medical Imaging with Deep Learning (MIDL), 2018.
10. Roto-translation covariant convolutional networks for medical image analysis
    Erik J. Bekkers, Maxime W. Lafarge, Mitko Veta, Koen A.J. Eppenhof, Josien P.W. Pluim, Remco Duits MICCAI 2018 Young Scientist Award paper
11. Equivariant Spherical Deconvolution: Learning Sparse Orientation Distribution Functions from Spherical Data
    Axel Elaldi*, Neel Dey*, Heejong Kim, Guido Gerig, Information Processing in Medical Imaging (IPMI) 2021 paper
12. Rotation-Equivariant Deep Learning for Diffusion MRI
    Philip Müller, Vladimir Golkov, Valentina Tomassini, Daniel Cremers paper
13. Equivariant geometric learning for digital rock physics: estimating formation factor and effective permeability tensors from Morse graph
    Chen Cai, Nikolaos Vlassis, Lucas Magee, Ran Ma, Zeyu Xiong, Bahador Bahmani, Teng-Fong Wong, Yusu Wang, WaiChing Sun paper
    Note: equivariant nets + Morse graph for permeability tensor prediction
14. Direct prediction of phonon density of states with Euclidean neural network Zhantao Chen, Nina Andrejevic, Tess Smidt, Zhiwei Ding, Yen-Ting Chi, Quynh T. Nguyen, Ahmet Alatas, Jing Kong, Mingda Li, Advanced Science (2021) paper arXiv
15. SE(3)-equivariant prediction of molecular wavefunctions and electronic densities Oliver T. Unke, Mihail Bogojeski, Michael Gastegger, Mario Geiger, Tess Smidt, Klaus-Robert Müller paper
16. Independent SE(3)-Equivariant Models for End-to-End Rigid Protein Docking Octavian-Eugen Ganea, Xinyuan Huang, Charlotte Bunne, Yatao Bian, Regina Barzilay, Tommi Jaakkola, Andreas Krause, under review, 2022 paper

Permutation Equivariance

There are many paper on this topics. I only added very few of them.

1. PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation
   Charles R. Qi, Hao Su, Kaichun Mo, Leonidas J. Guibas CVPR 2017 paper
2. Deep Sets
   Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Ruslan Salakhutdinov, Alexander Smola NeurIPS 2017 paper
3. Invariant and Equivariant Graph Networks
   Haggai Maron, Heli Ben-Hamu, Nadav Shamir, Yaron Lipman ICLR 2019 paper
4. Provably Powerful Graph Networks
   Haggai Maron, Heli Ben-Hamu, Hadar Serviansky, Yaron Lipman NeurIPS 2019 paper
5. Universal Invariant and Equivariant Graph Neural Networks
   Nicolas Keriven, Gabriel Peyré NeurIPS 2019 paper
6. On Learning Sets of Symmetric Elements
   Haggai Maron, Or Litany, Gal Chechik, Ethan Fetaya ICML 2020 best paper
7. On the Universality of Invariant Networks
   Haggai Maron, Ethan Fetaya, Nimrod Segol, Yaron Lipman paper
8. Transformers Generalize DeepSets and Can be Extended to Graphs and Hypergraphs Jinwoo Kim, Saeyoon Oh, Seunghoon Hong paper

Talk and Tutorial

IAS: Graph Nets: The Next Generation - Max Welling - YouTube

Equivariance and Data Augmentation workshop: many nice talks

IPAM: Tess Smidt: "Euclidean Neural Networks for Emulating Ab Initio Calculations and Generating Atomi..." - YouTube

IPAM: E(3) Equivariant Neural Network Tutorial

IPAM: Risi Kondor: "Fourier space neural networks"

NeurIPS 2020 tutorial: Equivariant Networks

Yaron Lipman - Deep Learning of Irregular and Geometric Data - YouTube

Math-ML: Erik J Bekkers: Group Equivariant CNNs beyond Roto-Translations: B-Spline CNNs on Lie Groups

Kostas Daniilidis: Geometry-aware deep learning: A brief history of equivariant representations and recent results

Andrew White: Deep Learning for Molecules and Materials.

Erik Bekkers: An Introduction to Group Equivariant Deep Learning A course offered at UvA

Michael M. Bronstein, Joan Bruna, Taco Cohen, Petar Veličković: Geometric Deep Learning Course

Background

I am by no means an expert in this field. Here are books and articles suggest by Taco Cohen when asked references to learn group theory and representation theory.

1. Carter, Visual Group Theory
   Note: very basic intro to group theory

2. Theoretical Aspects of Group Equivariant Neural Networks
   Carlos Esteves
   Note: covers all the math you need for equivariant nets in a fairly compact and accessible manner.

3. Serre, Linear Representations of Finite Groups
   Note: classic text on representations of finite groups. First few chapters are relevant to equivariant nets.

4. G B Folland. A Course in Abstract Harmonic Analysis
   Note: covers representations of locally compact groups; induced representations.

5. David Gurarie. Symmetries and Laplacians: Introduction to Harmonic Analysis, Group Representations and Applications.

6. Mark Hamilton. Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics
   Note: covers fiber bundles, useful for understanding homogeneous G-CNNs and Gauge CNNs.

Theses / Dissertations

1. Taco Cohen, Equivariant Convolutional Networks, PhD Thesis, University of Amsterdam, 2021 [pdf] (Note: Part II contains a lot of new material, not published before)

TO READ

There are many paper I haven't read carefully yet.

1. Making Convolutional Networks Shift-Invariant Again
   Richard Zhang ICML 2019 paper
2. Probabilistic symmetries and invariant neural networks
   Benjamin Bloem-Reddy, Yee Whye Teh JMLR paper
3. On Representing (Anti)Symmetric Functions
   Marcus Hutter paper
4. PDE-based Group Equivariant Convolutional Neural Networks
   Bart M.N. Smets, Jim Portegies, Erik J. Bekkers, Remco Duits paper