# Lorentz Local Canonicalization: How to Make Any Network Lorentz-Equivariant

arXiv: https://arxiv.org/abs/2505.20280 · HTML: https://mlsanity.com/paper/2505.20280/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 52 references that are in the map.
- First posted on arXiv 26 May 2025.

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Going deeper with Image Transformers](https://mlsanity.com/paper/2103.17239/) (2021)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Energy Flow Networks: Deep Sets for Particle Jets](https://mlsanity.com/paper/1810.05165/) (2018)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [Does equivariance matter at scale?](https://mlsanity.com/paper/2410.23179/) (2024)
- [A new perspective on building efficient and expressive 3D equivariant graph neural networks](https://mlsanity.com/paper/2304.04757/) (2023)
- [Equivariant Frames and the Impossibility of Continuous Canonicalization](https://mlsanity.com/paper/2402.16077/) (2024)
- [Quaternion Equivariant Capsule Networks for 3D Point Clouds](https://mlsanity.com/paper/1912.12098/) (2019)
- [Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physics](https://mlsanity.com/paper/2405.14806/) (2024)
- [Equivariant Point Cloud Analysis via Learning Orientations for Message Passing](https://mlsanity.com/paper/2203.14486/) (2022)
- [A Lorentz-Equivariant Transformer for All of the LHC](https://mlsanity.com/paper/2411.00446/) (2024)
- [PELICAN: Permutation Equivariant and Lorentz Invariant or Covariant Aggregator Network for Particle Physics](https://mlsanity.com/paper/2211.00454/) (2022)
- [Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency](https://mlsanity.com/paper/2406.07598/) (2024)
- [Beyond Canonicalization: How Tensorial Messages Improve Equivariant Message Passing](https://mlsanity.com/paper/2405.15389/) (2024)
- [Graph Neural Network with Local Frame for Molecular Potential Energy Surface](https://mlsanity.com/paper/2208.00716/) (2022)
- [Probing the effects of broken symmetries in machine learning](https://mlsanity.com/paper/2406.17747/) (2024)
- [Optimising simulations for diphoton production at hadron colliders using amplitude neural networks](https://mlsanity.com/paper/2106.09474/) (2021)

## Built on by (most cited first, 12 of 12)

- [Scaling laws for amplitude surrogates](https://mlsanity.com/paper/2601.13308/) (2026)
- [Forecasting Generative Amplification](https://mlsanity.com/paper/2509.08048/) (2025)
- [JetParticle-JEPA: An Efficient Self-Supervised Representation Learning method for Jet Tagging in High-Energy Physics](https://mlsanity.com/paper/2606.14813/) (2026)
- [Stable and Interpretable Jet Physics with IRC-Safe Equivariant Feature Extraction](https://mlsanity.com/paper/2509.22059/) (2025)
- [Statistical Properties of Training & Generalization](https://mlsanity.com/paper/2606.20299/) (2026)
- [Prompting Particle Physics: Tokenized Multi-modal Foundation Models for Combinatorially Many Tasks](https://mlsanity.com/paper/2609.31862/) (2026)
- [Equivariance by Local Canonicalization: A Matter of Representation](https://mlsanity.com/paper/2509.26499/) (2025)
- [Latent Representation Learning in Heavy-Ion Collisions with MaskPoint Transformer](https://mlsanity.com/paper/2510.06691/) (2025)
- [Generative models on phase space](https://mlsanity.com/paper/2604.02415/) (2026)
- [What Do Lorentz-Equivariant Jet Taggers Learn?](https://mlsanity.com/paper/2606.21790/) (2026)
- [Rotational Equivariance in Machine Learning: A Comprehensive Tutorial](https://mlsanity.com/paper/2608.31045/) (2026)
- [Machine learning for the LHC physics program: a 2025-2026 stocktake](https://mlsanity.com/paper/2609.32874/) (2026)

## Abstract (arXiv)

Lorentz-equivariant neural networks are becoming the leading architectures for high-energy physics. Current implementations rely on specialized layers, limiting architectural choices. We introduce Lorentz Local Canonicalization (LLoCa), a general framework that renders any backbone network exactly Lorentz-equivariant. Using equivariantly predicted local reference frames, we construct LLoCa-transformers and graph networks. We adapt a recent approach for geometric message passing to the non-compact Lorentz group, allowing propagation of space-time tensorial features. Data augmentation emerges from LLoCa as a special choice of reference frame. Our models achieve competitive and state-of-the-art accuracy on relevant particle physics tasks, while being 4× faster and using 10× fewer FLOPs.