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Nonlinear gradient denoising : Finding accurate extrema from inaccurate functional derivatives

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2015

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Snyder, John C.
Müller, Klaus-Robert
Burke, Kieron

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International Journal of Quantum Chemistry. Wiley-Blackwell. 2015, 115(16), pp. 1102-1114. ISSN 0020-7608. eISSN 1097-461X. Available under: doi: 10.1002/qua.24937

Zusammenfassung

A method for nonlinear optimization with machine learning (ML) models, called nonlinear gradient denoising (NLGD), is developed, and applied with ML approximations to the kinetic energy density functional in an orbital‐free density functional theory. Due to systematically inaccurate gradients of ML models, in particular when the data is very high‐dimensional, the optimization must be constrained to the data manifold. We use nonlinear kernel principal component analysis (PCA) to locally reconstruct the manifold, enabling a projected gradient descent along it. A thorough analysis of the method is given via a simple model, designed to clarify the concepts presented. Additionally, NLGD is compared with the local PCA method used in previous work. Our method is shown to be superior in cases when the data manifold is highly nonlinear and high dimensional. Further applications of the method in both density functional theory and ML are discussed.

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density functional theory, machine learning, nonlinear gradient denoising, orbital‐free density functional theory

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ISO 690SNYDER, John C., Matthias RUPP, Klaus-Robert MÜLLER, Kieron BURKE, 2015. Nonlinear gradient denoising : Finding accurate extrema from inaccurate functional derivatives. In: International Journal of Quantum Chemistry. Wiley-Blackwell. 2015, 115(16), pp. 1102-1114. ISSN 0020-7608. eISSN 1097-461X. Available under: doi: 10.1002/qua.24937
BibTex
@article{Snyder2015Nonli-52135,
  year={2015},
  doi={10.1002/qua.24937},
  title={Nonlinear gradient denoising : Finding accurate extrema from inaccurate functional derivatives},
  number={16},
  volume={115},
  issn={0020-7608},
  journal={International Journal of Quantum Chemistry},
  pages={1102--1114},
  author={Snyder, John C. and Rupp, Matthias and Müller, Klaus-Robert and Burke, Kieron}
}
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    <dcterms:abstract xml:lang="eng">A method for nonlinear optimization with machine learning (ML) models, called nonlinear gradient denoising (NLGD), is developed, and applied with ML approximations to the kinetic energy density functional in an orbital‐free density functional theory. Due to systematically inaccurate gradients of ML models, in particular when the data is very high‐dimensional, the optimization must be constrained to the data manifold. We use nonlinear kernel principal component analysis (PCA) to locally reconstruct the manifold, enabling a projected gradient descent along it. A thorough analysis of the method is given via a simple model, designed to clarify the concepts presented. Additionally, NLGD is compared with the local PCA method used in previous work. Our method is shown to be superior in cases when the data manifold is highly nonlinear and high dimensional. Further applications of the method in both density functional theory and ML are discussed.</dcterms:abstract>
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