# Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency

arXiv: https://arxiv.org/abs/2406.07598 · HTML: https://mlsanity.com/paper/2406.07598/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 71 of its 89 references that are in the map.
- First posted on arXiv 11 Jun 2024.

## Builds on (most cited first, 30 of 71)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [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)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [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)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [The Open Catalyst 2020 (OC20) Dataset and Community Challenges](https://mlsanity.com/paper/2010.09990/) (2020)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Geometric Latent Diffusion Models for 3D Molecule Generation](https://mlsanity.com/paper/2305.01140/) (2023)
- [TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentials](https://mlsanity.com/paper/2202.02541/) (2022)
- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [EquiBind: Geometric Deep Learning for Drug Binding Structure Prediction](https://mlsanity.com/paper/2202.05146/) (2022)
- [GOOD: A Graph Out-of-Distribution Benchmark](https://mlsanity.com/paper/2206.08452/) (2022)

## Built on by (most cited first, 19 of 19)

- [Probing the effects of broken symmetries in machine learning](https://mlsanity.com/paper/2406.17747/) (2024)
- [Relaxed Equivariance via Multitask Learning](https://mlsanity.com/paper/2410.17878/) (2024)
- [Lorentz Local Canonicalization: How to Make Any Network Lorentz-Equivariant](https://mlsanity.com/paper/2505.20280/) (2025)
- [Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetry](https://mlsanity.com/paper/2512.11855/) (2025)
- [Efficient Prediction of SO(3)-Equivariant Hamiltonian Matrices via SO(2) Local Frames](https://mlsanity.com/paper/2506.09398/) (2025)
- [Learning Inter-Atomic Potentials without Explicit Equivariance](https://mlsanity.com/paper/2510.00027/) (2025)
- [Equivariant Neural Networks for General Linear Symmetries on Lie Algebras](https://mlsanity.com/paper/2510.22984/) (2025)
- [Adaptive Canonicalization with Application to Invariant Anisotropic Geometric Networks](https://mlsanity.com/paper/2509.24886/) (2025)
- [Adaptive Symmetry Discovery for Dynamical System Identification](https://mlsanity.com/paper/2608.08091/) (2026)
- [A Geometry-Aware Message Passing Neural Network for Modeling Aerodynamics over Airfoils](https://mlsanity.com/paper/2412.09399/) (2024)
- [Edge Cluster Expansion with Radial Rotary Attention for Interatomic Potentials](https://mlsanity.com/paper/2607.10664/) (2026)
- [NeurIPS 2024 ML4CFD Competition: Results and Retrospective Analysis](https://mlsanity.com/paper/2506.08516/) (2025)
- [Tensor Decomposition Networks for Fast Machine Learning Interatomic Potential Computations](https://mlsanity.com/paper/2507.01131/) (2025)
- [Quantitative Approximation Rates for Group Equivariant Learning](https://mlsanity.com/paper/2602.20370/) (2026)
- [Sparse Data Augmentation for Optimization with Provable Guarantees](https://mlsanity.com/paper/2609.08133/) (2026)
- [Adaptive Sampling for Continuous Group Equivariant Neural Networks](https://mlsanity.com/paper/2409.08741/) (2024)
- [CrystalFramer: Rethinking the Role of Frames for SE(3)-Invariant Crystal Structure Modeling](https://mlsanity.com/paper/2503.02209/) (2025)
- [When and How to Canonize: A Generalization Perspective](https://mlsanity.com/paper/2605.11008/) (2026)
- [Dimension-Free Scaling Laws for Size Generalization: An Exponential Separation](https://mlsanity.com/paper/2610.19279/) (2026)

## Abstract (arXiv)

We consider achieving equivariance in machine learning systems via frame averaging. Current frame averaging methods involve a costly sum over large frames or rely on sampling-based approaches that only yield approximate equivariance. Here, we propose Minimal Frame Averaging (MFA), a mathematical framework for constructing provably minimal frames that are exactly equivariant. The general foundations of MFA also allow us to extend frame averaging to more groups than previously considered, including the Lorentz group for describing symmetries in space-time, and the unitary group for complex-valued domains. Results demonstrate the efficiency and effectiveness of encoding symmetries via MFA across a diverse range of tasks, including n-body simulation, top tagging in collider physics, and relaxed energy prediction. Our code is available at https://github.com/divelab/MFA.