# Optimal Equivariant Architectures from the Symmetries of Matrix-Element Likelihoods

arXiv: https://arxiv.org/abs/2410.18553 · HTML: https://mlsanity.com/paper/2410.18553/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 89 references that are in the map.
- First posted on arXiv 24 Oct 2024.

## Builds on (most cited first, 30 of 37)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (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)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [Particle Transformer for Jet Tagging](https://mlsanity.com/paper/2202.03772/) (2022)
- [Approximating Likelihood Ratios with Calibrated Discriminative Classifiers](https://mlsanity.com/paper/1506.02169/) (2015)
- [On the Limitations of Representing Functions on Sets](https://mlsanity.com/paper/1901.09006/) (2019)
- [Energy Flow Networks: Deep Sets for Particle Jets](https://mlsanity.com/paper/1810.05165/) (2018)
- [ParticleNet: Jet Tagging via Particle Clouds](https://mlsanity.com/paper/1902.08570/) (2019)
- [Machine Learning in the Search for New Fundamental Physics](https://mlsanity.com/paper/2112.03769/) (2021)
- [Geometric Algebra Transformer](https://mlsanity.com/paper/2305.18415/) (2023)
- [Classification without labels: Learning from mixed samples in high energy physics](https://mlsanity.com/paper/1708.02949/) (2017)
- [Jet-Images -- Deep Learning Edition](https://mlsanity.com/paper/1511.05190/) (2015)
- [Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physics](https://mlsanity.com/paper/2405.14806/) (2024)
- [MadMiner: Machine learning-based inference for particle physics](https://mlsanity.com/paper/1907.10621/) (2019)
- [A Guide to Constraining Effective Field Theories with Machine Learning](https://mlsanity.com/paper/1805.00020/) (2018)
- [Explainable Equivariant Neural Networks for Particle Physics: PELICAN](https://mlsanity.com/paper/2307.16506/) (2023)
- [Symmetry Group Equivariant Architectures for Physics](https://mlsanity.com/paper/2203.06153/) (2022)
- [Permutationless Many-Jet Event Reconstruction with Symmetry Preserving Attention Networks](https://mlsanity.com/paper/2010.09206/) (2020)

## Built on by (most cited first, 5 of 5)

- [A Lorentz-Equivariant Transformer for All of the LHC](https://mlsanity.com/paper/2411.00446/) (2024)
- [Forecasting Generative Amplification](https://mlsanity.com/paper/2509.08048/) (2025)
- [Extrapolating Jet Radiation with Autoregressive Transformers](https://mlsanity.com/paper/2412.12074/) (2024)
- [Stable and Interpretable Jet Physics with IRC-Safe Equivariant Feature Extraction](https://mlsanity.com/paper/2509.22059/) (2025)
- [BitHEP -- The Limits of Low-Precision ML in HEP](https://mlsanity.com/paper/2504.03387/) (2025)

## Abstract (arXiv)

The Matrix-Element Method (MEM) has long been a cornerstone of data analysis in high-energy physics. It leverages theoretical knowledge of parton-level processes and symmetries to evaluate the likelihood of observed events. In parallel, the advent of geometric deep learning has enabled neural network architectures that incorporate known symmetries directly into their design, leading to more efficient learning. This paper presents a novel approach that combines MEM-inspired symmetry considerations with equivariant neural network design for particle physics analysis. Even though Lorentz invariance and permutation invariance overall reconstructed objects are the largest and most natural symmetry in the input domain, we find that they are sub-optimal in most practical search scenarios. We propose a longitudinal boost-equivariant message-passing neural network architecture that preserves relevant discrete symmetries. We present numerical studies demonstrating MEM-inspired architectures achieve new state-of-the-art performance in distinguishing di-Higgs decays to four bottom quarks from the QCD background, with enhanced sample and parameter efficiencies. This synergy between MEM and equivariant deep learning opens new directions for physics-informed architecture design, promising more powerful tools for probing physics beyond the Standard Model.