Optimal radial basis for density-based atomic representations

Abstract

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 build 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 working with representations that correspond to high-body order correlations. We present examples that involve both molecular and condensed-phase machine-learning models.

Through the eyes of a descriptor: Constructing complete, invertible descriptions of atomic environments

“…It may even be possible to use such techniques to find a reduced set of invariants that work well for all signals that are sums of atomic feature functions (with a corresponding loss of discriminative power for other square-integrable functions). As this work is focused on universal descriptors we will not detail any particular approach here but we refer the reader to [47,63] for two examples of such techniques.…”

“…For invariants based on spherical harmonics we use Zernike functions as the radial basis, however other choices have been proposed including Bessel [64] and Chebyshev [44] functions which may have different numerical characteristics. (See Goscinski et al [63] for an excellent review of the performance of various radial basis functions.) The 3D Zernike polynomials are defined as…”

Compressing local atomic neighbourhood descriptors

“…-Optimal Radial Basis [26] with mixed-species basis p uu l = m c * ulm c u lm 1 2 umax(umax + 1)(L + 1)…”

Local invertibility and sensitivity of atomic structure-feature mappings

“…Only the latter term depend on the discretization of |A i 〉, while in the complete basis set limit the left singular vectors U k and the singular values s k can be converged with respect to n max and l max . Here we use an optimized radial basis [21] with n max = 20 components, built using the implementation in librascal [22] as the principal components computed based on 1000 structures from the random CH 4 dataset, and starting from 100 DVR features, a large angular cutoff l max = 20 and a very sharp density smearing σ a = 0.05 Å to approach the complete basis limit of the density correlation features. In order to reduce the dependence of the sensitivity analysis on the discretization of the features we do not plot an arbitrary component of 〈q|A i 〉, but write the Cartesian displacement around the reference structure in terms of distortions ∆ k projected along the left singular vectors…”

“…Mean condition number for the Jacobian of |ρ i 〉 features, computed from random CH 4 configurations[23], for C-centered environments. Left: mean CN as a function of n max and l max , for an optimal radial basis[21] and a density smearing σ a = 0.1Å. Center: mean CN as a function of n max , for two selected values of l max and comparing three different choices of basis, for σ a = 0.1Å.…”