Reconstructing local environments from concise atomistic representations
Symmetry-based representations of local atomic structure, such as the power spectrum or bispectrum, are routinely used to characterize the structural diversity of datasets and as input features for atomistic machine learning. Although these descriptors systematically incorporate increasingly complex geometric correlations, it remains unclear if a given feature can be mapped back to a discrete point cloud, whether such a reconstruction is unique, and how changes in the descriptor are reflected in the underlying atomic geometry. The choice and discretization of the radial and angular bases, as well as the high dimensionality of the resulting feature vectors -- which may contain hundreds or thousands of components -- make this interpretation even more challenging. In this work, we investigate the inverse problem of recovering atomic structures from local invariant descriptors. We show that accurate reconstructions can be obtained from remarkably compact descriptors of different correlation orders, each comprising only a few tens of features. Even representations that are formally incomplete or locally ill-conditioned can be inverted to accurate geometric reconstructions of atomic environments across molecular and material datasets. Our reconstruction framework provides a general algorithmic means of identifying approximate degeneracies of invariant descriptors and recovering distinct atomic environments that cannot be distinguished by a given representation. Finally, by reconstructing atomic configurations from descriptors, we examine how perturbations in invariant descriptors of different correlation orders translate into structural distortions.