# Lorentz group equivariant autoencoders

arXiv: https://arxiv.org/abs/2212.07347 · HTML: https://mlsanity.com/paper/2212.07347/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 101 references that are in the map.
- First posted on arXiv 14 Dec 2022.

## Builds on (most cited first, 30 of 46)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [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)
- [A Point Set Generation Network for 3D Object Reconstruction from a Single Image](https://mlsanity.com/paper/1612.00603/) (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)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Recent Advances in Autoencoder-Based Representation Learning](https://mlsanity.com/paper/1812.05069/) (2018)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Energy Flow Networks: Deep Sets for Particle Jets](https://mlsanity.com/paper/1810.05165/) (2018)
- [Explainable Artificial Intelligence: a Systematic Review](https://mlsanity.com/paper/2006.00093/) (2020)
- [CaloGAN: Simulating 3D High Energy Particle Showers in Multi-Layer Electromagnetic Calorimeters with Generative Adversarial Networks](https://mlsanity.com/paper/1712.10321/) (2017)
- [A Living Review of Machine Learning for Particle Physics](https://mlsanity.com/paper/2102.02770/) (2021)
- [ParticleNet: Jet Tagging via Particle Clouds](https://mlsanity.com/paper/1902.08570/) (2019)
- [Accelerating Science with Generative Adversarial Networks: An Application to 3D Particle Showers in Multi-Layer Calorimeters](https://mlsanity.com/paper/1705.02355/) (2017)
- [FSPool: Learning Set Representations with Featurewise Sort Pooling](https://mlsanity.com/paper/1906.02795/) (2019)
- [Learning Particle Physics by Example: Location-Aware Generative Adversarial Networks for Physics Synthesis](https://mlsanity.com/paper/1701.05927/) (2017)
- [Anomaly Detection With Conditional Variational Autoencoders](https://mlsanity.com/paper/2010.05531/) (2020)
- [Particle Cloud Generation with Message Passing Generative Adversarial Networks](https://mlsanity.com/paper/2106.11535/) (2021)
- [Calorimetry with Deep Learning: Particle Simulation and Reconstruction for Collider Physics](https://mlsanity.com/paper/1912.06794/) (2019)

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

- [Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles](https://mlsanity.com/paper/2301.05638/) (2023)
- [Anomalies, Representations, and Self-Supervision](https://mlsanity.com/paper/2301.04660/) (2023)
- [Fast Particle-based Anomaly Detection Algorithm with Variational Autoencoder](https://mlsanity.com/paper/2311.17162/) (2023)
- [Oracle-Preserving Latent Flows](https://mlsanity.com/paper/2302.00806/) (2023)
- [Optimal Equivariant Architectures from the Symmetries of Matrix-Element Likelihoods](https://mlsanity.com/paper/2410.18553/) (2024)
- [mathbbZ2× mathbbZ2 Equivariant Quantum Neural Networks: Benchmarking against Classical Neural Networks](https://mlsanity.com/paper/2311.18744/) (2023)
- [Enhancing anomaly detection with topology-aware autoencoders](https://mlsanity.com/paper/2502.10163/) (2025)
- [Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoders](https://mlsanity.com/paper/2505.13569/) (2025)
- [jBOT: Semantic Jet Representation Clustering Emerges from Self-Distillation](https://mlsanity.com/paper/2601.11719/) (2026)
- [Lorentz Equivariant Model for Knowledge-Enhanced Hyperbolic Collaborative Filtering](https://mlsanity.com/paper/2302.04545/) (2023)
- [Learning symmetries in datasets](https://mlsanity.com/paper/2504.05174/) (2025)
- [Artificial Intelligence and Symmetries: Learning, Encoding, and Discovering Structure in Physical Data](https://mlsanity.com/paper/2602.02351/) (2026)
- [Learnable cut flow for high energy physics](https://mlsanity.com/paper/2503.22498/) (2025)
- [Generative models on phase space](https://mlsanity.com/paper/2604.02415/) (2026)
- [Contrastive Learning for Interpretable Anomaly Detection at Collider Experiments](https://mlsanity.com/paper/2608.13652/) (2026)

## Abstract (arXiv)

There has been significant work recently in developing machine learning (ML) models in high energy physics (HEP) for tasks such as classification, simulation, and anomaly detection. Often these models are adapted from those designed for datasets in computer vision or natural language processing, which lack inductive biases suited to HEP data, such as equivariance to its inherent symmetries. Such biases have been shown to make models more performant and interpretable, and reduce the amount of training data needed. To that end, we develop the Lorentz group autoencoder (LGAE), an autoencoder model equivariant with respect to the proper, orthochronous Lorentz group SO^+(3,1), with a latent space living in the representations of the group. We present our architecture and several experimental results on jets at the LHC and find it outperforms graph and convolutional neural network baseline models on several compression, reconstruction, and anomaly detection metrics. We also demonstrate the advantage of such an equivariant model in analyzing the latent space of the autoencoder, which can improve the explainability of potential anomalies discovered by such ML models.