# SymmetryGAN: Symmetry Discovery with Deep Learning

**Authors:** Krish Desai, Benjamin Nachman, Jesse Thaler  
**arXiv:** [2112.05722](https://arxiv.org/abs/2112.05722)  
**Date:** 2021-12-10  
**Summary page:** https://www.summarizepaper.com/en/paper/2112.05722v1/

## Key points

- Authors explore the concept of symmetries in datasets
- Symmetries of individual data elements can be characterized by their invariance under different transformations
- Symmetries of an ensemble of data elements become ambiguous due to Jacobian factors introduced during coordinate changes
- Authors propose a statistical definition of dataset symmetries involving inertial reference densities, drawing an analogy to inertial frames in classical mechanics
- Introduce SymmetryGAN as a novel approach for discovering symmetries using deep learning techniques based on generative adversarial networks (GANs)
- Performance of SymmetryGAN evaluated on Gaussian examples and demonstrates excellent empirical results consistent with expectations from the analytic loss landscape
- Applied to simulated dijet events from the Large Hadron Collider (LHC) to showcase potential utility in high-energy collider physics applications
- Procedures explored to infer underlying symmetry group from empirical data
- Contributes to advancing understanding of dataset symmetries and provides a powerful tool for discovering and analyzing these symmetries using deep learning methods
- Findings have implications for various fields where understanding and leveraging dataset symmetries are crucial for further research and applications.

## Summary

In the paper "SymmetryGAN: Symmetry Discovery with Deep Learning," authors Krish Desai, Benjamin Nachman, and Jesse Thaler explore the concept of symmetries in datasets. While the symmetries of individual data elements can be characterized by their invariance under different transformations, the symmetries of an ensemble of data elements become ambiguous due to Jacobian factors introduced during coordinate changes. To address this issue, the authors propose a rigorous statistical definition of dataset symmetries that involves inertial reference densities, drawing an analogy to inertial frames in classical mechanics. They introduce SymmetryGAN as a novel approach to automatically discover symmetries using deep learning techniques based on generative adversarial networks (GANs). The performance of SymmetryGAN is evaluated on Gaussian examples and demonstrates excellent empirical results consistent with expectations from the analytic loss landscape. Furthermore, it is applied to simulated dijet events from the Large Hadron Collider (LHC) to showcase its potential utility in high-energy collider physics applications. Beyond symmetry discovery, the paper explores procedures to infer the underlying symmetry group from empirical data. This work contributes to advancing our understanding of dataset symmetries and provides a powerful tool for discovering and analyzing these symmetries using deep learning methods. Overall, this paper presents a comprehensive investigation into dataset symmetries and introduces an innovative approach for their discovery and analysis. The findings have implications for various fields where understanding and leveraging dataset symmetries are crucial for further research and applications.