Continuous Symmetry Discovery and Enforcement Using Infinitesimal Generators of Multi-parameter Group Actions

TL;DR

This paper presents a computational framework for discovering and enforcing continuous symmetries in machine learning models, extending beyond affine transformations to include infinitesimal isometries and improving generalization.

Contribution

It introduces a method to automatically discover multi-parameter infinitesimal generators and enforce symmetries in neural networks, including non-Euclidean settings.

Findings

Framework efficiently discovers non-affine infinitesimal generators.

Enforcing symmetries improves model generalization.

Extends symmetry discovery to neural networks and Riemannian metrics.

Abstract

Symmetry-informed machine learning can exhibit advantages over machine learning which fails to account for symmetry. In the context of continuous symmetry detection, current state of the art experiments are largely limited to detecting affine transformations. Herein, we outline a computationally efficient framework for discovering infinitesimal generators of multi-parameter group actions which are not generally affine transformations. This framework accommodates the automatic discovery of the number of linearly independent infinitesimal generators. We build upon recent work in continuous symmetry discovery by extending to neural networks and by restricting the symmetry search space to infinitesimal isometries. We also introduce symmetry enforcement of smooth models using vector field regularization, thereby improving model generalization. The notion of vector field similarity is also generalized for non-Euclidean Riemannian metric tensors.