Symmetry Discovery for Different Data Types

TL;DR

This paper introduces LieSD, a novel method for discovering symmetries in data using trained neural networks, which does not require prior knowledge of data types or symmetries and works effectively on various tasks.

Contribution

LieSD is a new approach that characterizes continuous symmetries via Lie algebra directly from neural network inputs, outputs, and gradients, extending to multi-channel and tensor data.

Findings

LieSD accurately determines the number of Lie algebra bases.

It performs well on non-uniform datasets.

Outperforms GAN-based methods in symmetry discovery.

Abstract

Equivariant neural networks incorporate symmetries into their architecture, achieving higher generalization performance. However, constructing equivariant neural networks typically requires prior knowledge of data types and symmetries, which is difficult to achieve in most tasks. In this paper, we propose LieSD, a method for discovering symmetries via trained neural networks which approximate the input-output mappings of the tasks. It characterizes equivariance and invariance (a special case of equivariance) of continuous groups using Lie algebra and directly solves the Lie algebra space through the inputs, outputs, and gradients of the trained neural network. Then, we extend the method to make it applicable to multi-channel data and tensor data, respectively. We validate the performance of LieSD on tasks with symmetries such as the two-body problem, the moment of inertia matrix prediction, and top quark tagging. Compared with the baseline, LieSD can accurately determine the number of Lie algebra bases without the need for expensive group sampling. Furthermore, LieSD can perform well on non-uniform datasets, whereas methods based on GANs fail.