  Statistical learning methods show great promise in providing an accurate
prediction of materials and molecular properties, while minimizing the need for
computationally demanding electronic structure calculations. The accuracy and
transferability of these models are increased significantly by encoding into
the learning procedure the fundamental symmetries of rotational and
permutational invariance of scalar properties. However, the prediction of
tensorial properties requires that the model respects the appropriate geometric
transformations, rather than invariance, when the reference frame is rotated.
We introduce a formalism that can be used to perform machine-learning of
tensorial properties of arbitrary rank for general molecular geometries. To
demonstrate it, we derive a tensor kernel adapted to rotational symmetry, which
is the natural generalization of the smooth overlap of atomic positions (SOAP)
kernel commonly used for the prediction of scalar properties at the atomic
scale. The performance and generality of the approach is demonstrated by
learning the instantaneous electrical response of water oligomers of increasing
complexity, from the isolated molecule to the condensed phase.
