Latent Space Symmetry Discovery

TL;DR

This paper introduces LaLiGAN, a generative model that automatically discovers complex nonlinear symmetries in data by mapping them to a latent space where symmetries are linear, aiding understanding and downstream tasks.

Contribution

The paper presents LaLiGAN, a novel generative model capable of discovering nonlinear symmetries in data by learning a latent space where symmetries are linear, extending symmetry discovery beyond simple cases.

Findings

Successfully discovers intrinsic symmetries in high-dimensional systems

Produces a structured latent space useful for equation discovery

Handles nonlinear group actions effectively

Abstract

Equivariant neural networks require explicit knowledge of the symmetry group. Automatic symmetry discovery methods aim to relax this constraint and learn invariance and equivariance from data. However, existing symmetry discovery methods are limited to simple linear symmetries and cannot handle the complexity of real-world data. We propose a novel generative model, Latent LieGAN (LaLiGAN), which can discover symmetries of nonlinear group actions. It learns a mapping from the data space to a latent space where the symmetries become linear and simultaneously discovers symmetries in the latent space. Theoretically, we show that our model can express nonlinear symmetries under some conditions about the group action. Experimentally, we demonstrate that our method can accurately discover the intrinsic symmetry in high-dimensional dynamical systems. LaLiGAN also results in a well-structured latent space that is useful for downstream tasks including equation discovery and long-term forecasting.