Representing spherical tensors with scalar-based machine-learning models

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. In particular, we show that it is always possible to separate the learning of an equivariant property into learnable scalars and fixed geometric terms built as the maximal coupling of interatomic vectors. We also propose approximations of the general expressions that, while lacking universal approximation properties, are fast, simple to implement, and accurate in practical settings.