  Machine-learning of atomic-scale properties amounts to extracting
correlations between structure, composition and the quantity that one wants to
predict. Representing the input structure in a way that best reflects such
correlations makes it possible to improve the accuracy of the model for a given
amount of reference data. When using a description of the structures that is
transparent and well-principled, optimizing the representation might reveal
insights into the chemistry of the data set. Here we show how one can
generalize the SOAP kernel to introduce a distance-dependent weight that
accounts for the multi-scale nature of the interactions, and a description of
correlations between chemical species. We show that this improves substantially
the performance of ML models of molecular and materials stability, while making
it easier to work with complex, multi-component systems and to extend SOAP to
coarse-grained intermolecular potentials. The element correlations that give
the best performing model show striking similarities with the conventional
periodic table of the elements, providing an inspiring example of how machine
learning can rediscover, and generalize, intuitive concepts that constitute the
foundations of chemistry.
