Local invertibility and sensitivity of atomic structure-feature mappings

Abstract

Background: 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. Methods: Here, we analyze the sensitivity of the mapping to atomic displacements, using a singular value decomposition of the Jacobian of the transformation to quantify the sensitivity for different configurations, choice of representations and implementation details. Results: We show 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. Conclusions: These results exemplify the subtle issues that arise when constructing symmetric representations of atomic structures, and provide conceptual and numerical tools to identify and investigate them in both benchmark and realistic applications.

Compressing local atomic neighbourhood descriptors

“…Perhaps more surprising is that using ν; μ ¼ 1; 1-projecting one atom onto the unit sphere-did not reduce the sensitivity more drastically. Of course, these tests are not exhaustive and it is probable that special environments, likely closely related to any additional degeneracies, exist where this is not the case 50 . However, we find these results promising and leave further investigation to future work.…”

Incompleteness of graph neural networks for points clouds in three dimensions

“…This systematic study has revealed that -at least for low values of ν -atom-centered representations are incomplete, i.e. there are pairs of structures that are distinct, but which contain environments that are indistinguishable based on the unordered list of distances and angles 11 , which also affects the ability of the representa-tion to resolve local deformations of certain structures 13 . This issue is also closely related to classical problems in invariant theory 14 , that aim to determine under which conditions two point clouds can be unequivocally identified by the unordered list of distances, or distances and angles.…”

Comment on "Manifolds of quasi-constant SOAP and ACSF fingerprints and the resulting failure to machine learn four body interactions"

“…Here we show that: (1) Low-sensitivity regions are a direct manifestation of the degenerate manifolds discussed in Ref. 2, which lead to configurations where the Jacobian is exactly singular [3]; (2) The extent to which these singular configurations induce a region of low sensitivity depends on the numerical details of the implementation; (3) Although low sensitivity affects the extrapolative power of the model, it does not altogether prevent model fitting in the interpolative regime.…”

“…2, one can find that matching pairs are arranged along two continuous manifolds, A ± : at their intersection one finds structures A 0 for which J has a "spurious" singular value -the features change as the square of the atomic displacement away from A 0 , even if A 0 is not symmetric. [3] Both "zero-order" degenerate pairs A ± and "first-order" singular configurations A 0 are intrinsic problems of any 3-body representation. The "quasi-constant" manifolds observed by Parsaeifard and Goedecker are not a distinct, unrelated pathology, but a direct consequence of the structures being geometrically close to a first-order singularity, the spurious zero-sensitivity structure at the intersection of the manifolds of degenerate pairs.…”