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Mathematics of Computation

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Integration in reproducing kernel Hilbert spaces of Gaussian kernels


Authors: Toni Karvonen, Chris J. Oates and Mark Girolami
Journal: Math. Comp. 90 (2021), 2209-2233
MSC (2020): Primary 65D30, 46E22; Secondary 65D12, 41A25
DOI: https://doi.org/10.1090/mcom/3659
Published electronically: June 18, 2021
MathSciNet review: 4280298
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Abstract: The Gaussian kernel plays a central role in machine learning, uncertainty quantification and scattered data approximation, but has received relatively little attention from a numerical analysis standpoint. The basic problem of finding an algorithm for efficient numerical integration of functions reproduced by Gaussian kernels has not been fully solved. In this article we construct two classes of algorithms that use $N$ evaluations to integrate $d$-variate functions reproduced by Gaussian kernels and prove the exponential or super-algebraic decay of their worst-case errors. In contrast to earlier work, no constraints are placed on the length-scale parameter of the Gaussian kernel. The first class of algorithms is obtained via an appropriate scaling of the classical Gauss–Hermite rules. For these algorithms we derive lower and upper bounds on the worst-case error of the forms $\exp (-c_1 N^{1/d}) N^{1/(4d)}$ and $\exp (-c_2 N^{1/d}) N^{-1/(4d)}$, respectively, for positive constants $c_1 > c_2$. The second class of algorithms we construct is more flexible and uses worst-case optimal weights for points that may be taken as a nested sequence. For these algorithms we derive upper bounds of the form $\exp (-c_3 N^{1/(2d)})$ for a positive constant $c_3$.


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Additional Information

Toni Karvonen
Affiliation: The Alan Turing Institute, London NW1 2DB, United Kingdom
MR Author ID: 1256975
ORCID: 0000-0002-5984-7295

Chris J. Oates
Affiliation: The Alan Turing Institute, London NW1 2DB, United Kingdom; and School of Mathematics, Statistics & Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom
MR Author ID: 998570

Mark Girolami
Affiliation: The Alan Turing Institute, London NW1 2DB, United Kingdom; and Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom
MR Author ID: 665653
ORCID: 0000-0003-3008-253X

Received by editor(s): July 1, 2020
Received by editor(s) in revised form: December 16, 2020
Published electronically: June 18, 2021
Additional Notes: The authors were supported by the Lloyd’s Register Foundation programme on data-centric engineering at the Alan Turing Institute, United Kingdom
Article copyright: © Copyright 2021 American Mathematical Society