  Many atomic descriptors are currently limited by their unfavourable scaling
with the number of chemical elements $S$ e.g. the length of body-ordered
descriptors, such as the Smooth Overlap of Atomic Positions (SOAP) power
spectrum (3-body) and the Atomic Cluster Expansion (ACE) (multiple
body-orders), scales as $(NS)^\nu$ where $\nu+1$ is the body-order and $N$ is
the number of radial basis functions used in the density expansion. We
introduce two distinct approaches which can be used to overcome this scaling
for the SOAP power spectrum. Firstly, we show that the power spectrum is
amenable to lossless compression with respect to both $S$ and $N$, so that the
descriptor length can be reduced from $\mathcal{O}(N^2S^2)$ to
$\mathcal{O}\left(NS\right)$. Secondly, we introduce a generalized SOAP kernel,
where compression is achieved through the use of the total, element agnostic
density, in combination with radial projection. The ideas used in the
generalized kernel are equally applicably to any other body-ordered descriptors
and we demonstrate this for the Atom Centered Symmetry Functions (ACSF).
Finally, both compression approaches are shown to offer comparable performance
to the original descriptor across a variety of numerical tests.
