Geometrical aspects of lattice gauge equivariant convolutional neural networks

TL;DR

This paper extends lattice gauge equivariant CNNs to include global symmetries like rotations and reflections, providing a geometric formulation and linking convolutions to gauge equivariant neural networks on SU(N) bundles.

Contribution

It introduces a geometric framework for L-CNNs with global symmetry equivariance and connects convolutions to gauge equivariant neural networks on principal bundles.

Findings

L-CNNs can be equipped with global group equivariance.

Convolutions in L-CNNs are special cases of gauge equivariant neural networks.

Provides a geometric formulation linking L-CNNs to principal bundles.

Abstract

Lattice gauge equivariant convolutional neural networks (L-CNNs) are a framework for convolutional neural networks that can be applied to non-Abelian lattice gauge theories without violating gauge symmetry. We demonstrate how L-CNNs can be equipped with global group equivariance. This allows us to extend the formulation to be equivariant not just under translations but under global lattice symmetries such as rotations and reflections. Additionally, we provide a geometric formulation of L-CNNs and show how convolutions in L-CNNs arise as a special case of gauge equivariant neural networks on SU($N$) principal bundles.