# Beyond Canonicalization: How Tensorial Messages Improve Equivariant Message Passing

arXiv: https://arxiv.org/abs/2405.15389 · HTML: https://mlsanity.com/paper/2405.15389/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 46 references that are in the map.
- First posted on arXiv 24 May 2024.

## Builds on (most cited first, 30 of 35)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [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)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Deep Learning Scaling is Predictable, Empirically](https://mlsanity.com/paper/1712.00409/) (2017)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Relation-Shape Convolutional Neural Network for Point Cloud Analysis](https://mlsanity.com/paper/1904.07601/) (2019)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations](https://mlsanity.com/paper/2306.12059/) (2023)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Forces are not Enough: Benchmark and Critical Evaluation for Machine Learning Force Fields with Molecular Simulations](https://mlsanity.com/paper/2210.07237/) (2022)
- [Rotation Invariant Convolutions for 3D Point Clouds Deep Learning](https://mlsanity.com/paper/1908.06297/) (2019)
- [SE(3) Equivariant Graph Neural Networks with Complete Local Frames](https://mlsanity.com/paper/2110.14811/) (2021)
- [A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systems](https://mlsanity.com/paper/2312.07511/) (2023)
- [A Rotation-Invariant Framework for Deep Point Cloud Analysis](https://mlsanity.com/paper/2003.07238/) (2020)
- [TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentials](https://mlsanity.com/paper/2306.06482/) (2023)
- [Spherical Channels for Modeling Atomic Interactions](https://mlsanity.com/paper/2206.14331/) (2022)
- [A new perspective on building efficient and expressive 3D equivariant graph neural networks](https://mlsanity.com/paper/2304.04757/) (2023)
- [Quaternion Equivariant Capsule Networks for 3D Point Clouds](https://mlsanity.com/paper/1912.12098/) (2019)

## Built on by (most cited first, 15 of 15)

- [Lorentz Local Canonicalization: How to Make Any Network Lorentz-Equivariant](https://mlsanity.com/paper/2505.20280/) (2025)
- [Scaling laws for amplitude surrogates](https://mlsanity.com/paper/2601.13308/) (2026)
- [Learning Inter-Atomic Potentials without Explicit Equivariance](https://mlsanity.com/paper/2510.00027/) (2025)
- [Adaptive Canonicalization with Application to Invariant Anisotropic Geometric Networks](https://mlsanity.com/paper/2509.24886/) (2025)
- [Frame-based Equivariant Diffusion Models for 3D Molecular Generation](https://mlsanity.com/paper/2509.19506/) (2025)
- [RoVE: Rotary Value Embeddings Attention for Relative Position-dependent Value Pathways](https://mlsanity.com/paper/2606.11275/) (2026)
- [Learning conformational ensembles of proteins based on backbone geometry](https://mlsanity.com/paper/2503.05738/) (2025)
- [Equivariance by Local Canonicalization: A Matter of Representation](https://mlsanity.com/paper/2509.26499/) (2025)
- [Direct Molecular Polarizability Prediction with SO(3) Equivariant Local Frame GNNs](https://mlsanity.com/paper/2511.07087/) (2025)
- [Stable and Accurate Orbital-Free DFT Powered by Machine Learning](https://mlsanity.com/paper/2503.00443/) (2025)
- [Local-Canonicalization Equivariant Graph Neural Networks for Sample-Efficient and Generalizable Swarm Robot Control](https://mlsanity.com/paper/2509.14431/) (2025)
- [Tensor Channel Equivariant Graph Neural Networks for Molecular Polarizability Prediction](https://mlsanity.com/paper/2605.16891/) (2026)
- [Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs](https://mlsanity.com/paper/2608.28853/) (2026)
- [Rotational Equivariance in Machine Learning: A Comprehensive Tutorial](https://mlsanity.com/paper/2608.31045/) (2026)
- [All-electron machine-learned orbital-free density functional for periodic CHNO systems](https://mlsanity.com/paper/2610.17975/) (2026)

## Abstract (arXiv)

In numerous applications of geometric deep learning, the studied systems exhibit spatial symmetries and it is desirable to enforce these. For the symmetry of global rotations and reflections, this means that the model should be equivariant with respect to the transformations that form the group of mathrm O(d). While many approaches for equivariant message passing require specialized architectures, including non-standard normalization layers or non-linearities, we here present a framework based on local reference frames ("local canonicalization") which can be integrated with any architecture without restrictions. We enhance equivariant message passing based on local canonicalization by introducing tensorial messages to communicate geometric information consistently between different local coordinate frames. Our framework applies to message passing on geometric data in Euclidean spaces of arbitrary dimension. We explicitly show how our approach can be adapted to make a popular existing point cloud architecture equivariant. We demonstrate the superiority of tensorial messages and achieve state-of-the-art results on normal vector regression and competitive results on other standard 3D point cloud tasks.