SymmetryLens: Unsupervised Symmetry Learning via Locality and Density Preservation

TL;DR

SymmetryLens introduces an unsupervised method to learn and explicitly represent underlying symmetries in data, including complex and hidden symmetries, using an information-theoretic loss that couples symmetry and locality.

Contribution

It provides a novel unsupervised approach to discover and explicitly represent symmetries in data, including those not visually apparent, by leveraging an information-theoretic loss and locality constraints.

Findings

Successfully learns pixel translation symmetry from data.

Capable of identifying various complex symmetries.

Produces stable and reproducible symmetry representations.

Abstract

We develop a new unsupervised symmetry learning method that starts with raw data and provides the minimal generator of an underlying Lie group of symmetries, together with a symmetry-equivariant representation of the data, which turns the hidden symmetry into an explicit one. The method is able to learn the pixel translation operator from a dataset with only an approximate translation symmetry and can learn quite different types of symmetries that are not apparent to the naked eye. The method is based on the formulation of an information-theoretic loss function that measures both the degree of symmetry of a dataset under a candidate symmetry generator and a proposed notion of locality of the samples, which is coupled to symmetry. We demonstrate that this coupling between symmetry and locality, together with an optimization technique developed for entropy estimation, results in a stable system that provides reproducible results.