  The input of almost every machine learning algorithm targeting the properties
of matter at the atomic scale involves a transformation of the list of
Cartesian atomic coordinates into a more symmetric representation. Many of the
most popular representations can be seen as an expansion of the symmetrized
correlations of the atom density, and differ mainly by the choice of basis.
Considerable effort has been dedicated to the optimization of the basis set,
typically driven by heuristic considerations on the behavior of the regression
target. Here we take a different, unsupervised viewpoint, aiming to determine
the basis that encodes in the most compact way possible the structural
information that is relevant for the dataset at hand. For each training dataset
and number of basis functions, one can determine a unique basis that is optimal
in this sense, and can be computed at no additional cost with respect to the
primitive basis by approximating it with splines. We demonstrate that this
construction yields representations that are accurate and computationally
efficient, particularly when constructing representations that correspond to
high-body order correlations. We present examples that involve both molecular
and condensed-phase machine-learning models.
