Symmetry Discovery Beyond Affine Transformations

TL;DR

Under the manifold assumption, this work outlines a framework for discovering continuous symmetry in data beyond the affine transformation group and provides a similar framework for discovering discrete symmetry.

Abstract

Symmetry detection can improve various machine learning tasks. In the context of continuous symmetry detection, current state of the art experiments are limited to detecting affine transformations. Under the manifold assumption, we outline a framework for discovering continuous symmetry in data beyond the affine transformation group. We also provide a similar framework for discovering discrete symmetry. We experimentally compare our method to an existing method known as LieGAN and show that our method is competitive at detecting affine symmetries for large sample sizes and superior than LieGAN for small sample sizes. We also show our method is able to detect continuous symmetries beyond the affine group and is generally more computationally efficient than LieGAN.

Figures (11)

• Figure 1: The ROCKET-PHATE embedded Bear Lake Weather dataset, colored by invariant function value.
• Figure 2: A visual representation of the kernel density estimate of the circular dataset.
• Figure 3: A representation of the estimated invariant function of the symmetry of the circular probability distribution.
• Figure 4: The circular dataset, not uniformly distributed, projected onto the $(x,y)$ plane.
• Figure 5: A visual representation of the function values of the estimated invariant function of $X$ as given in \ref{['le3']}.