https://github.com/Chen-Cai-OSU/awesome-equivariant-network

Paper list for equivariant neural network
https://github.com/Chen-Cai-OSU/awesome-equivariant-network

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Paper list for equivariant neural network

• Host: GitHub
• URL: https://github.com/Chen-Cai-OSU/awesome-equivariant-network
• Owner: Chen-Cai-OSU
• Created: 2021-01-13T18:17:01.000Z (almost 4 years ago)
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  • Readme: README.md

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README

        
# awesome-equivariant-network

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

Feel free to suggest relevant papers in the following format. 

```markdown

**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](#equivariance-and-Group-convolution)

- [Theory](#theory)

- [Equivariant Density Estimation and Sampling](equivariant-density-estimation-and-sampling)

- [Application](#application)

- [Permutation Equivariance](#permutation-equivariance)

- [Talk and Tutorial](#talk-and-tutorial)

- [TO READ](#to-read)

### [Equivariance and Group convolution](#content)

1. **Group Equivariant Convolutional Networks**  

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

   Note: first paper; discrete group; 

2. **Steerable CNNs**  

  Taco S. Cohen, Max Welling ICLR 2017 [paper](https://arxiv.org/abs/1612.08498)

3. **Harmonic Networks: Deep Translation and Rotation Equivariance**  

  Daniel E. Worrall, Stephan J. Garbin, Daniyar Turmukhambetov, Gabriel J. Brostow CVPR 2017 [paper](https://arxiv.org/abs/1612.04642)   

4. **Spherical CNNs**  

  Taco S. Cohen, Mario Geiger, Jonas Koehler, Max Welling ICLR 2018 best paper  [paper](https://arxiv.org/abs/1801.10130)  

  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](https://arxiv.org/abs/1806.09231)  

  Note: perform equivariant nonlinearity in Fourier space; 

6. **General E(2)-Equivariant Steerable CNNs**  

  Maurice Weiler, Gabriele Cesa NeurIPS 2019 [paper](https://arxiv.org/abs/1911.08251)  

  Note: nice benchmark on different reprsentations

7. **Learning Steerable Filters for Rotation Equivariant CNNs**  

   Maurice Weiler, Fred A. Hamprecht, Martin Storath CVPR 2018 [paper](https://arxiv.org/abs/1711.07289)   

   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](https://openaccess.thecvf.com/content_ECCV_2018/html/Carlos_Esteves_Learning_SO3_Equivariant_ECCV_2018_paper.html)  

   Note: SO(3) equivariance; zonal filter

9. **Polar Transformer Networks**  

  Carlos Esteves, Christine Allen-Blanchette, Xiaowei Zhou, Kostas Daniilidis ICLR 2018 [paper](https://arxiv.org/abs/1709.01889)  

10. **3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data**  

  Maurice Weiler, Mario Geiger, Max Welling, Wouter Boomsma, Taco Cohen  NeurIPS 2018 [paper](https://arxiv.org/abs/1807.02547)  

  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](https://arxiv.org/abs/1802.08219)  

   Note: SE(3) equivariance for point clouds

12. **Equivariant Multi-View Networks**  

   Carlos Esteves, Yinshuang Xu, Christine Allen-Blanchette, Kostas Daniilidis  ICCV 2019 [paper](https://arxiv.org/abs/1904.00993)   

13. **Gauge Equivariant Convolutional Networks and the Icosahedral CNN**  

   Taco S. Cohen, Maurice Weiler, Berkay Kicanaoglu, Max Welling ICML 2019 [paper](https://arxiv.org/abs/1902.04615), [talk](https://slideslive.com/38915809/gauge-equivariant-convolutional-networks?locale=de)  

   Note: gauge equivariance on general manifold

14. **Cormorant: Covariant Molecular Neural Networks**  

   Brandon Anderson, Truong-Son Hy, Risi Kondor NeurIPS 2019 [paper](https://arxiv.org/abs/1906.04015)

15. **Deep Scale-spaces: Equivariance Over Scale**  

   Daniel Worrall, Max Welling NeurIPS 2019 [paper](https://papers.nips.cc/paper/2019/hash/f04cd7399b2b0128970efb6d20b5c551-Abstract.html)

16. **Scale-Equivariant Steerable Networks**  

   Ivan Sosnovik, Michał Szmaja, Arnold Smeulders ICLR 2020 [paper](https://openreview.net/forum?id=HJgpugrKPS)

17. **B-Spline CNNs on Lie Groups**  

   Erik J Bekkers ICLR 2020 [paper](https://openreview.net/forum?id=H1gBhkBFDH)    

18. **SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks**  

   Fabian B. Fuchs, Daniel E. Worrall, Volker Fischer, Max Welling NeurIPS 2020  [paper](https://arxiv.org/abs/2006.10503), [blog](https://fabianfuchsml.github.io/se3transformer/)  

   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](https://arxiv.org/abs/2003.05425)  

   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](https://arxiv.org/abs/2006.04780)  

   Note: SO(1, 3) equivariance

21. **CNNs on Surfaces using Rotation-Equivariant Features**  

      Ruben Wiersma, Elmar Eisemann, Klaus Hildebrandt SIGGRAPH 2020 [paper](https://dl.acm.org/doi/pdf/10.1145/3386569.3392437), [code](https://github.com/rubenwiersma/hsn)  

22. **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](https://arxiv.org/abs/2002.12880)  

   Note: fairly generic architecture; use Monte Carlo sampling to achieve equivariance in expectation; 

23. **Spin-Weighted Spherical CNNs**  

   Carlos Esteves, Ameesh Makadia, Kostas Daniilidis NeurIPS 2020 [paper](https://arxiv.org/abs/2006.10731)  

   Note: anisotropic filter for vector field on sphere

24. **Learning Invariances in Neural Networks**  

   Gregory Benton, Marc Finzi, Pavel Izmailov, Andrew Gordon Wilson NeurIPS 2020 [paper](https://arxiv.org/abs/2010.11882)   

   Note: very interesting approch; enfore "soft" invariance via learning over both model parameters and distributions over augmentations

25. **LieTransformer: Equivariant self-attention for Lie Groups**  

   Michael Hutchinson, Charline Le Lan, Sheheryar Zaidi, Emilien Dupont, Yee Whye Teh, Hyunjik Kim [paper](https://arxiv.org/abs/2012.10885)  

   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](https://arxiv.org/abs/1911.07849)

27. **Attentive Group Equivariant Convolutional Networks**  

   David W. Romero, Erik J. Bekkers, Jakub M. Tomczak, Mark Hoogendoorn ICML 2020 [paper](https://arxiv.org/abs/2002.03830)

28. **Wavelet Networks: Scale Equivariant Learning From Raw Waveforms**  

   David W. Romero, Erik J. Bekkers, Jakub M. Tomczak, Mark Hoogendoorn [paper](https://arxiv.org/abs/2006.05259)

29. **Group Equivariant Stand-Alone Self-Attention For Vision**  

   David W. Romero, Jean-Baptiste Cordonnier ICLR 2021 [paper](https://arxiv.org/abs/2010.00977)

30. **Incorporating Symmetry into Deep Dynamics Models for Improved Generalization**  

   Rui Wang, Robin Walters, Rose Yu ICLR 2021 [paper](https://arxiv.org/abs/2002.03061)

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](https://arxiv.org/abs/2006.16908)

32. **Isometric Transformation Invariant and Equivariant Graph Convolutional Networks**  

   Masanobu Horie, Naoki Morita, Toshiaki Hishinuma, Yu Ihara, Naoto Mitsume ICLR 2021 [paper](https://arxiv.org/abs/2005.06316)

33. **E(n) Equivariant Graph Neural Networks**  

   Victor Garcia Satorras, Emiel Hoogeboom, Max Welling ICML 2021 [paper](https://arxiv.org/abs/2102.09844)  

   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](https://arxiv.org/abs/2104.12229)  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](https://arxiv.org/abs/2102.03150)  

36. **Field Convolutions For Surface CNNs**  

   Thomas W. Mitchel, Vladimir G. Kim, Michael Kazhdan ICCV 2021 (Oral) [paper](https://openaccess.thecvf.com/content/ICCV2021/html/Mitchel_Field_Convolutions_for_Surface_CNNs_ICCV_2021_paper.html)   

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](https://arxiv.org/abs/2106.06610)

38. **Efficient Equivariant Network**  

   Lingshen He, Yuxuan Chen, Zhengyang shen, Yiming Dong, Yisen Wang, Zhouchen Lin NeruIPS 2021 [paper](https://proceedings.neurips.cc/paper_files/paper/2021/hash/2a79ea27c279e471f4d180b08d62b00a-Abstract.html), [code](https://github.com/LingshenHe/Efficient-Equivariant-Network)

39. **GemNet: Universal Directional Graph Neural Networks for Molecules**  

   Johannes Klicpera, Florian Becker, Stephan Günnemann NeurIPS 2021 [paper](https://proceedings.neurips.cc/paper/2021/hash/35cf8659cfcb13224cbd47863a34fc58-Abstract.html)

40. **Automatic Symmetry Discovery with Lie Algebra Convolutional Network**  

   Nima Dehmamy, Robin Walters, Yanchen Liu, Dashun Wang, Rose Yu NeurIPS 2021 [paper](https://openreview.net/pdf?id=NPOWF_ZLfC5)

41. **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](https://arxiv.org/abs/2110.02905), [code](https://github.com/RobDHess/Steerable-E3-GNN)

42. **Frame Averaging for Invariant and Equivariant Network Design**  

   Omri Puny, Matan Atzmon, Heli Ben-Hamu, Ishan Misra, Aditya Grover, Edward J. Smith, Yaron Lipman [paper](https://arxiv.org/abs/2110.03336) ICLR 2022

43. **Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics**

   Albert Musaelian, Simon Batzner, Anders Johansson, Lixin Sun, Cameron J. Owen, Mordechai Kornbluth, Boris Kozinsky [paper](https://arxiv.org/abs/2204.05249) 

44. **Möbius Convolutions for Spherical CNNs**  

   Thomas W. Mitchel, Noam Aigerman, Vladimir G. Kim, Michael Kazhdan SIGGRAPH 2022 [paper](https://arxiv.org/abs/2201.12212)  

   (Note: Equivariance to the action of SL(2, C) on the sphere. To our knowledge this is the first *conformally* equivariant convolutional surface network)

45. **DeltaConv: Anisotropic Operators for Geometric Deep Learning on Point Clouds**

   Ruben Wiersma, Ahmad Nasikun, Elmar Eisemann, Klaus Hildebrandt SIGGRAPH 2022 [paper](https://arxiv.org/abs/2111.08799), [code](https://github.com/rubenwiersma/deltaconv)

   Rotation equivariance by using differential operators.

46. **Neural ePDOs: Spatially Adaptive Equivariant Partial Differential Operator Based Networks**  

 Lingshen He*, Yuxuan Chen*, Zhengyang shen, Yibo Yang, Zhouchen Lin  ICLR 2023 (spotlight) [paper](https://openreview.net/forum?id=D1Iqfm7WTkk), [code](https://github.com/YuxuanChen21/Neural_ePDOs)

47. **Steerable Partial Differential Operators for Equivariant Neural Networks**

Erik Jenner, Maurice Weiler ICLR 2022 [paper](https://arxiv.org/abs/2106.10163)

48. **A Program to build E(N)-Equivariant Steerable CNNs**

Gabriele Cesa, Leon Lang, Maurice Weiler ICLR 2022 [paper](https://openreview.net/pdf?id=WE4qe9xlnQw)

49. **Clifford-Steerable Convolutional Neural Networks**

Maksim Zhdanov, David Ruhe, Maurice Weiler, Ana Lucic, Johannes Brandstetter, Patrick Forré ICML 2024 [paper](https://arxiv.org/abs/2402.14730)

### [Theory](#content)

1. **On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups**  

  Risi Kondor, Shubhendu Trivedi ICML 2018 [paper](https://arxiv.org/abs/1802.03690)  

  Note: convolution is all you need (for scalar fields)

  

3. **A General Theory of Equivariant CNNs on Homogeneous Spaces**  

  Taco Cohen, Mario Geiger, Maurice Weiler NeurIPS 2019 [paper](https://arxiv.org/abs/1811.02017)  

  Note: convolution is all you need (for general fields)

  

4. **Equivariance Through Parameter-Sharing**  

  Siamak Ravanbakhsh, Jeff Schneider, Barnabas Poczos ICML 2017 [paper](https://arxiv.org/abs/1702.08389)

  

5. **Universal approximations of invariant maps by neural networks**  

  Dmitry Yarotsky [paper](https://arxiv.org/abs/1804.10306)

  

6. **A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels**  

  Leon Lang, Maurice Weiler ICLR 2021 [paper](https://arxiv.org/abs/2010.10952)  

  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.

  

7. **On the Universality of Rotation Equivariant Point Cloud Networks**  

  Nadav Dym, Haggai Maron ICLR 2021 [paper](https://arxiv.org/abs/2010.02449),   

  Note: universality for TFN and se3-transformer 

  

8. **Universal Equivariant Multilayer Perceptrons**  

Siamak Ravanbakhsh [paper](https://arxiv.org/abs/2002.02912)

9. **Provably Strict Generalisation Benefit for Equivariant Models**  

Bryn Elesedy, Sheheryar Zaidi ICML 2021 [paper](https://arxiv.org/abs/2102.10333)

10. **Implicit Bias of Linear Equivariant Networks**  

Hannah Lawrence, Kristian Georgiev, Andrew Dienes, Bobak T. Kiani ICML 2022 [paper](https://arxiv.org/abs/2110.06084)

11. **On the Expressive Power of Geometric Graph Neural Networks**  

Chaitanya K. Joshi, Cristian Bodnar, Simon V. Mathis, Taco Cohen, Pietro Liò ICML 2023 [paper](https://arxiv.org/abs/2301.09308)

12. **Equivariant and Coordinate Independent Convolutional Networks - A Gauge Field Theory of Neural Networks**

Maurice Weiler ,  Patrick Forré ,  Erik Verlinde ,  Max Welling [book](https://maurice-weiler.gitlab.io/cnn_book/EquivariantAndCoordinateIndependentCNNs.pdf)

### [Equivariant Density Estimation and Sampling](#content)

1. **Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities**    

  Jonas Köhler, Leon Klein, Frank Noé ICML 2020 [paper](https://arxiv.org/abs/2006.02425)  

  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](https://arxiv.org/abs/1909.13739)  

  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](https://arxiv.org/abs/2008.05456)    

  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](https://arxiv.org/abs/2008.02676)  

  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](https://arxiv.org/abs/2010.03242)  

  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](https://arxiv.org/abs/2006.01910)  

  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](https://arxiv.org/abs/2002.04913)  

  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](https://arxiv.org/abs/2012.00429)  

  Note: normalizing flows in phase space with equivariance with respect to changes in temperature.

9. **Equivariant Manifold Flows**  

  Isay Katsman, Aaron Lou, Derek Lim, Qingxuan Jiang, Ser-Nam Lim, Christopher De Sa NeurIPS 2021 [paper](https://arxiv.org/pdf/2107.08596.pdf)

  Note: normalizing flows that allow for learning over any Riemannian manifold with respect to any symmetry (isometry subgroup action invariance).

### [Application](#content)

1. **Trajectory Prediction using Equivariant Continuous Convolution**  

  Robin Walters, Jinxi Li, Rose Yu ICLR 2021 [paper](https://arxiv.org/abs/2010.11344)

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](https://arxiv.org/abs/2101.03164)

4. **Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks**  

  Tess E. Smidt, Mario Geiger, Benjamin Kurt Miller [paper](https://arxiv.org/abs/2007.02005)

5. **Group Equivariant Generative Adversarial Networks**  

  Neel Dey, Antong Chen, Soheil Ghafurian ICLR 2021  [paper](https://arxiv.org/abs/2005.01683)   

6. **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](https://arxiv.org/abs/1909.02487)  

7. **Symmetry-Aware Actor-Critic for 3D Molecular Design**    

  Gregor N. C. Simm, Robert Pinsler, Gábor Csányi, José Miguel Hernández-Lobato ICLR 2021 [paper](https://arxiv.org/abs/2011.12747)

8. **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](https://arxiv.org/abs/2002.08725)

9. **Scale Equivariance Improves Siamese Tracking**  

  Ivan Sosnovik\*, Artem Moskalev\*, Arnold Smeulders WACV 2021 [paper](https://arxiv.org/abs/2007.09115)

10. **3D G-CNNs for Pulmonary Nodule Detection**

  Marysia Winkels, Taco S. Cohen [paper](https://arxiv.org/abs/1804.04656) 

  International Conference on Medical Imaging with Deep Learning (MIDL), 2018.

11. **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](https://arxiv.org/abs/1804.03393)

12. **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](https://arxiv.org/abs/2102.09462)

13. **Rotation-Equivariant Deep Learning for Diffusion MRI**  

  Philip Müller, Vladimir Golkov, Valentina Tomassini, Daniel Cremers [paper](https://arxiv.org/abs/2102.06942)

14. **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](https://arxiv.org/abs/2104.05608)  

Note: equivariant nets + Morse graph for permeability tensor prediction

15. **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](https://onlinelibrary.wiley.com/doi/10.1002/advs.202004214) [arXiv](https://arxiv.org/abs/2009.05163)

16. **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](https://arxiv.org/abs/2106.02347)

17. **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](https://arxiv.org/abs/2111.07786)

18. **Roto-translated Local Coordinate Frames For Interacting Dynamical Systems**

Miltiadis Kofinas, Naveen Shankar Nagaraja, Efstratios Gavves NeurIPS 2021 [paper](https://arxiv.org/abs/2110.14961)

19. **MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields**


Ilyes Batatia, Dávid Péter Kovács, Gregor N. C. Simm, Christoph Ortner, Gábor Csányi, under review, 2022 [paper](https://arxiv.org/abs/2206.07697), [code](https://github.com/ACEsuit/mace)

20. **Equivariant Q Learning in Spatial Action Spaces**

Dian Wang, Robin Walters, Xupeng Zhu, Robert Platt, CoRL 2021 [paper](https://arxiv.org/pdf/2110.15443.pdf)

21. **SO(2)-Equivariant Reinforcement Learning**

Dian Wang, Robin Walters, Robert Platt, ICLR 2022 [paper](https://arxiv.org/pdf/2203.04439.pdf)

22. **Sample Efficient Grasp Learning Using Equivariant Models**

Xupeng Zhu, Dian Wang, Ondrej Biza, Guanang Su, Robin Walters, Robert Platt, RSS 2022 [paper](https://arxiv.org/pdf/2202.09468.pdf)

23. **Equivariant Transporter Network**

Haojie Huang, Dian Wang, Robin Walters, Robert Platt, RSS 2022 [paper](https://arxiv.org/pdf/2202.09400.pdf)

24. **On-Robot Learning With Equivariant Models**

Dian Wang, Mingxi Jia, Xupeng Zhu, Robin Walters, Robert Platt, CoRL 2022 [paper](https://arxiv.org/pdf/2203.04923.pdf)

25. **Edge Grasp Network: Graph-Based SE(3)-invariant Approach to Grasp Detection**

Haojie Huang, Dian Wang, Xupeng Zhu, Robin Walters, Robert Platt, Under Review [paper](https://arxiv.org/pdf/2211.00191.pdf)

26. **SEIL: Simulation-augmented Equivariant Imitation Learning**

Mingxi Jia, Dian Wang, Guanang Su, David Klee, Xupeng Zhu, Robin Walters, Robert Platt, Under Review [paper](https://arxiv.org/pdf/2211.00194.pdf)

27. **The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetry**

Dian Wang, Jung Yeon Park, Neel Sortur, Lawson L.S. Wong, Robin Walters, Robert Platt, Under Review [paper](https://arxiv.org/pdf/2211.09231.pdf)

### [Permutation Equivariance](#content)

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](https://arxiv.org/abs/1612.00593) 

2. **Deep Sets**  

Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Ruslan Salakhutdinov, Alexander Smola NeurIPS 2017   [paper](https://arxiv.org/abs/1703.06114)

3. **Invariant and Equivariant Graph Networks**  

  Haggai Maron, Heli Ben-Hamu, Nadav Shamir, Yaron Lipman ICLR 2019 [paper](https://arxiv.org/abs/1812.09902)  

4. **Provably Powerful Graph Networks**  

  Haggai Maron, Heli Ben-Hamu, Hadar Serviansky, Yaron Lipman NeurIPS 2019 [paper](https://arxiv.org/abs/1905.11136)  

5. **Universal Invariant and Equivariant Graph Neural Networks**  

  Nicolas Keriven, Gabriel Peyré NeurIPS 2019 [paper](https://papers.nips.cc/paper/2019/hash/ea9268cb43f55d1d12380fb6ea5bf572-Abstract.html)

6. **On Learning Sets of Symmetric Elements**  

  Haggai Maron, Or Litany, Gal Chechik, Ethan Fetaya [ICML 2020 best paper](https://arxiv.org/abs/2002.08599)

7. **On the Universality of Invariant Networks**  

  Haggai Maron, Ethan Fetaya, Nimrod Segol, Yaron Lipman [paper](https://arxiv.org/abs/1901.09342)

8. **Transformers Generalize DeepSets and Can be Extended to Graphs and Hypergraphs**

  Jinwoo Kim, Saeyoon Oh, Seunghoon Hong [paper](https://arxiv.org/abs/2110.14416)

### [Talk and Tutorial](#content)

IAS: [Graph Nets: The Next Generation - Max Welling - YouTube](https://www.youtube.com/watch?v=Wx8J-Kw3fTA&t=3602s)

[Equivariance and Data Augmentation workshop](https://sites.google.com/view/equiv-data-aug/home): many nice talks

IPAM: [Tess Smidt: "Euclidean Neural Networks for Emulating Ab Initio Calculations and Generating Atomi..." - YouTube](https://www.youtube.com/watch?v=8CF8Grb_brE)

IPAM: [E(3) Equivariant Neural Network Tutorial ](https://blondegeek.github.io/e3nn_tutorial/)

IPAM: [Risi Kondor: "Fourier space neural networks" ](https://www.youtube.com/watch?v=-PVyi0Keiec)

[NeurIPS 2020 tutorial: Equivariant Networks](https://nips.cc/virtual/2020/public/tutorial_3e267ff3c8b6621e5ad4d0f26142892b.html)

[Yaron Lipman - Deep Learning of Irregular and Geometric Data - YouTube](https://www.youtube.com/watch?v=fveyx5zKReo&feature=youtu.be)

Math-ML: [Erik J Bekkers: Group Equivariant CNNs beyond Roto-Translations: B-Spline CNNs on Lie Groups](https://youtu.be/rakcnrgX4oo)

Kostas Daniilidis: [Geometry-aware deep learning: A brief history of equivariant representations and recent results](https://mathinstitutes.org/videos/videos/view/15146)

Andrew White: [ Deep Learning for Molecules and Materials.](https://whitead.github.io/dmol-book/dl/Equivariant.html)

Erik Bekkers: [An Introduction to Group Equivariant Deep Learning](https://uvagedl.github.io) A course offered at UvA

Michael M. Bronstein, Joan Bruna, Taco Cohen, Petar Veličković: [Geometric Deep Learning Course](https://geometricdeeplearning.com/lectures/)

### [Background](#content)

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](https://www.amazon.com/Visual-Group-Theory-Problem-Book/dp/088385757X)   

  Note: very basic intro to group theory

2. [Theoretical Aspects of Group Equivariant Neural Networks](https://arxiv.org/abs/2004.05154)  

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](http://www.math.tau.ac.il/~borovoi/courses/ReprFG/Hatzagot.pdf)   

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](https://sv.20file.org/up1/1415_0.pdf)   

Note: covers representations of locally compact groups; induced representations.

5. [David Gurarie. Symmetries and Laplacians: Introduction to Harmonic Analysis, Group Representations and Applications.](https://www.amazon.com/Symmetries-Laplacians-Introduction-Representations-Applications/dp/0486462889)  

6. [Mark Hamilton. Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics](https://www.amazon.com/Mathematical-Gauge-Theory-Applications-Universitext/dp/3319684388)   

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)

2. **Extending Convolution Through Spatially Adaptive Alignment**  

Thomas W. Mitchel, PhD Thesis, Johns Hopkins University, 2022 [pdf](https://www.mitchel.computer/doc/thesis.pdf)  

   *Presents a novel, unified theoretical framework for transformation-equivariant convolutions on arbitrary homogenous spaces and 2D Riemannian manifolds. Can handle high-dimensional, non-compact transformation groups.*

### [TO READ](#content)

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

1. **Making Convolutional Networks Shift-Invariant Again**   

  Richard Zhang ICML 2019 [paper](https://arxiv.org/abs/1904.11486)

2. **Probabilistic symmetries and invariant neural networks**  

Benjamin Bloem-Reddy, Yee Whye Teh JMLR [paper](https://arxiv.org/abs/1901.06082)

3. **On Representing (Anti)Symmetric Functions**  

Marcus Hutter [paper](https://arxiv.org/abs/2007.15298)

4. **PDE-based Group Equivariant Convolutional Neural Networks**  

  Bart M.N. Smets, Jim Portegies, Erik J. Bekkers, Remco Duits [paper](https://arxiv.org/abs/2001.09046)