  Machine-learning models based on a point-cloud representation of a physical
object are ubiquitous in scientific applications and particularly well-suited
to the atomic-scale description of molecules and materials. Among the many
different approaches that have been pursued, the description of local atomic
environments in terms of their neighbor densities has been used widely and very
succesfully. We propose a novel density-based method which involves computing
``Wigner kernels''. These are fully equivariant and body-ordered kernels that
can be computed iteratively with a cost that is independent of the
radial-chemical basis and grows only linearly with the maximum body-order
considered. This is in marked contrast to feature-space models, which comprise
an exponentially-growing number of terms with increasing order of correlations.
We present several examples of the accuracy of models based on Wigner kernels
in chemical applications, for both scalar and tensorial targets, reaching
state-of-the-art accuracy on the popular QM9 benchmark dataset, and we discuss
the broader relevance of these ideas to equivariant geometric machine-learning.
