  Density based representations of atomic environments that are invariant under
Euclidean symmetries have become a widely used tool in the machine learning of
interatomic potentials, broader data-driven atomistic modelling and the
visualisation and analysis of materials datasets.The standard mechanism used to
incorporate chemical element information is to create separate densities for
each element and form tensor products between them. This leads to a steep
scaling in the size of the representation as the number of elements increases.
Graph neural networks, which do not explicitly use density representations,
escape this scaling by mapping the chemical element information into a fixed
dimensional space in a learnable way. We recast this approach as tensor
factorisation by exploiting the tensor structure of standard neighbour density
based descriptors. In doing so, we form compact tensor-reduced representations
whose size does not depend on the number of chemical elements, but remain
systematically convergeable and are therefore applicable to a wide range of
data analysis and regression tasks.
