Accelerated Discovery of Machine-Learned Symmetries: Deriving the Exceptional Lie Groups G2, F4 and E6

TL;DR

This paper introduces faster algorithms for discovering continuous symmetries in data using deep learning, successfully deriving generators for complex Lie groups including G2, F4, and E6, demonstrating broad applicability.

Contribution

The paper presents two improved algorithms that accelerate the discovery of symmetry transformations and a post-processing method for sparse generator representation, applied to complex Lie groups.

Findings

Significant speed-up over standard methods

Successful derivation of generators for G2, F4, E6

Applicable to various labeled datasets

Abstract

Recent work has applied supervised deep learning to derive continuous symmetry transformations that preserve the data labels and to obtain the corresponding algebras of symmetry generators. This letter introduces two improved algorithms that significantly speed up the discovery of these symmetry transformations. The new methods are demonstrated by deriving the complete set of generators for the unitary groups U(n) and the exceptional Lie groups $G_2$, $F_4$, and $E_6$. A third post-processing algorithm renders the found generators in sparse form. We benchmark the performance improvement of the new algorithms relative to the standard approach. Given the significant complexity of the exceptional Lie groups, our results demonstrate that this machine-learning method for discovering symmetries is completely general and can be applied to a wide variety of labeled datasets.