  The increasingly common applications of machine-learning schemes to
atomic-scale simulations have triggered efforts to better understand the
mathematical properties of the mapping between the Cartesian coordinates of the
atoms and the variety of representations that can be used to convert them into
a finite set of symmetric descriptors or features. Here, we analyze the
sensitivity of the mapping to atomic displacements, showing that the
combination of symmetry and smoothness leads to mappings that have singular
points at which the Jacobian has one or more null singular values (besides
those corresponding to infinitesimal translations and rotations). This is in
fact desirable, because it enforces physical symmetry constraints on the values
predicted by regression models constructed using such representations. However,
besides these symmetry-induced singularities, there are also spurious singular
points, that we find to be linked to the incompleteness of the mapping, i.e.
the fact that, for certain classes of representations, structurally distinct
configurations are not guaranteed to be mapped onto different feature vectors.
Additional singularities can be introduced by a too aggressive truncation of
the infinite basis set that is used to discretize the representations.
