Representing spherical tensors with scalar-based machine-learning models

TL;DR

This paper introduces a new approach for representing spherical tensors in machine learning models that leverages scalar functions and tensor bases, aiming to balance computational efficiency and symmetry adherence.

Contribution

It proposes a novel method to express equivariant functions as scalar functions multiplied by tensor bases, simplifying implementation and improving practical accuracy.

Findings

The method is faster and simpler to implement than traditional equivariant models.

It maintains high accuracy in practical applications despite lacking universal approximation.

The approach offers a middle ground between fully equivariant and unconstrained models.

Abstract

Rotational symmetry plays a central role in physics, providing an elegant framework to describe how the properties of 3D objects -- from atoms to the macroscopic scale -- transform under the action of rigid rotations. Equivariant models of 3D point clouds are able to approximate structure-property relations in a way that is fully consistent with the structure of the rotation group, by combining intermediate representations that are themselves spherical tensors. The symmetry constraints however make this approach computationally demanding and cumbersome to implement, which motivates increasingly popular unconstrained architectures that learn approximate symmetries as part of the training process. In this work, we explore a third route to tackle this learning problem, where equivariant functions are expressed as the product of a scalar function of the point cloud coordinates and a small basis of tensors with the appropriate symmetry. We also propose approximations of the general expressions that, while lacking universal approximation properties, are fast, simple to implement, and accurate in practical settings.