Local invertibility and sensitivity of atomic structure-feature mappings [version 1; peer review: 2 approved]

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 conver...

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Autores principales:	Sergey N. Pozdnyakov, Liwei Zhang, Christoph Ortner, Gábor Csányi, Michele Ceriotti
Formato:	article
Lenguaje:	EN
Publicado:	F1000 Research Limited 2021
Materias:	Science Q Social Sciences H
Acceso en línea:	https://doaj.org/article/a3898f2acb6942dd87685f4e26aaf998
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spelling	oai:doaj.org-article:a3898f2acb6942dd87685f4e26aaf9982021-11-22T12:41:10ZLocal invertibility and sensitivity of atomic structure-feature mappings [version 1; peer review: 2 approved]2732-512110.12688/openreseurope.14156.1https://doaj.org/article/a3898f2acb6942dd87685f4e26aaf9982021-10-01T00:00:00Zhttps://open-research-europe.ec.europa.eu/articles/1-126/v1https://doaj.org/toc/2732-5121Background: 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.Sergey N. PozdnyakovLiwei ZhangChristoph OrtnerGábor CsányiMichele CeriottiF1000 Research LimitedarticleScienceQSocial SciencesHENOpen Research Europe, Vol 1 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Science Q Social Sciences H
spellingShingle	Science Q Social Sciences H Sergey N. Pozdnyakov Liwei Zhang Christoph Ortner Gábor Csányi Michele Ceriotti Local invertibility and sensitivity of atomic structure-feature mappings [version 1; peer review: 2 approved]
description	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.
format	article
author	Sergey N. Pozdnyakov Liwei Zhang Christoph Ortner Gábor Csányi Michele Ceriotti
author_facet	Sergey N. Pozdnyakov Liwei Zhang Christoph Ortner Gábor Csányi Michele Ceriotti
author_sort	Sergey N. Pozdnyakov
title	Local invertibility and sensitivity of atomic structure-feature mappings [version 1; peer review: 2 approved]
title_short	Local invertibility and sensitivity of atomic structure-feature mappings [version 1; peer review: 2 approved]
title_full	Local invertibility and sensitivity of atomic structure-feature mappings [version 1; peer review: 2 approved]
title_fullStr	Local invertibility and sensitivity of atomic structure-feature mappings [version 1; peer review: 2 approved]
title_full_unstemmed	Local invertibility and sensitivity of atomic structure-feature mappings [version 1; peer review: 2 approved]
title_sort	local invertibility and sensitivity of atomic structure-feature mappings [version 1; peer review: 2 approved]
publisher	F1000 Research Limited
publishDate	2021
url	https://doaj.org/article/a3898f2acb6942dd87685f4e26aaf998
work_keys_str_mv	AT sergeynpozdnyakov localinvertibilityandsensitivityofatomicstructurefeaturemappingsversion1peerreview2approved AT liweizhang localinvertibilityandsensitivityofatomicstructurefeaturemappingsversion1peerreview2approved AT christophortner localinvertibilityandsensitivityofatomicstructurefeaturemappingsversion1peerreview2approved AT gaborcsanyi localinvertibilityandsensitivityofatomicstructurefeaturemappingsversion1peerreview2approved AT micheleceriotti localinvertibilityandsensitivityofatomicstructurefeaturemappingsversion1peerreview2approved
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Local invertibility and sensitivity of atomic structure-feature mappings [version 1; peer review: 2 approved]

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