The first two apply to any physical system, and we factor them out of the representation by transforming the set of Cartesian coordinates into the vector of all interatomic distances, R = {||r i -r j ||} i<j . Note how the dimensionality of this representation scales with the square of the number of atoms, n, but this is of little consequence, since all our samples will lie on the 3n dimensional manifold. Alternatively, one could work with the well-known internal coordinates of the z-matrix, and this choice would not increase the dimensionality.