A Unified Framework for Discovering Discrete Symmetries

TL;DR

This paper introduces a unified framework for discovering discrete symmetries in functions, utilizing a novel architecture and optimization techniques to identify symmetries like cyclic, dihedral, and local symmetries effectively.

Contribution

The paper presents a new architecture and a unified approach for symmetry discovery across various subgroup types, combining bandit algorithms and gradient descent for efficient learning.

Findings

Effective symmetry discovery demonstrated on image-digit sum tasks.

Successful polynomial regression experiments validate the framework.

Architecture's matrix-valued functions are essential for capturing symmetries.

Abstract

We consider the problem of learning a function respecting a symmetry from among a class of symmetries. We develop a unified framework that enables symmetry discovery across a broad range of subgroups including locally symmetric, dihedral and cyclic subgroups. At the core of the framework is a novel architecture composed of linear, matrix-valued and non-linear functions that expresses functions invariant to these subgroups in a principled manner. The structure of the architecture enables us to leverage multi-armed bandit algorithms and gradient descent to efficiently optimize over the linear and the non-linear functions, respectively, and to infer the symmetry that is ultimately learnt. We also discuss the necessity of the matrix-valued functions in the architecture. Experiments on image-digit sum and polynomial regression tasks demonstrate the effectiveness of our approach.