We consider the problem of designing adapted kernels for approximating functions invariant under a known finite group action. We introduce the class of argumentwise invariant kernels, and show that they characterize centered square-integrable random fields with invariant paths, as well as Reproducing Kernel Hilbert Spaces of invariant functions. Two subclasses of argumentwise kernels are considered, involving a fundamental domain or a double sum over orbits. We then derive invariance properties for Kriging and conditional simulation based on argumentwise invariant kernels. The applicability and advantages of argumentwise invariant kernels are demonstrated on several examples, including a symmetric function from the reliability literature.
Nous considérons le problème d’approximation par méthodes à noyaux de fonctions invariantes sous l’action d’un groupe fini. Nous introduisons les noyaux doublement invariants, et montrons qu’ils caractérisent les champs aléatoires centrés de carré intégrable à trajectoires invariantes, ainsi que les espaces de Hilbert à noyau reproduisant de fonctions invariantes. Deux classes particulières de noyaux doublement invariants sont considérées, basées respectivement sur un domaine fondamental ou sur une double somme sur les orbites. Nous établissons ensuite des propriétés d’invariance pour les modèles de Krigeage et les simulations consitionnelles associés. L’applicabilité et les avantages de tels noyaux sont illustrés sur plusieurs exemples, incluant une fonction symétrique issue d’un problème de fiabilité des structures.
David Ginsbourger 1; Xavier Bay 2; Olivier Roustant 2; Laurent Carraro 3
@article{AFST_2012_6_21_3_501_0,
author = {David Ginsbourger and Xavier Bay and Olivier Roustant and Laurent Carraro},
title = {Argumentwise invariant kernels for the approximation of invariant functions},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {501--527},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 21},
number = {3},
year = {2012},
doi = {10.5802/afst.1343},
mrnumber = {3076410},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1343/}
}
TY - JOUR AU - David Ginsbourger AU - Xavier Bay AU - Olivier Roustant AU - Laurent Carraro TI - Argumentwise invariant kernels for the approximation of invariant functions JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2012 SP - 501 EP - 527 VL - 21 IS - 3 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1343/ DO - 10.5802/afst.1343 LA - en ID - AFST_2012_6_21_3_501_0 ER -
%0 Journal Article %A David Ginsbourger %A Xavier Bay %A Olivier Roustant %A Laurent Carraro %T Argumentwise invariant kernels for the approximation of invariant functions %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2012 %P 501-527 %V 21 %N 3 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1343/ %R 10.5802/afst.1343 %G en %F AFST_2012_6_21_3_501_0
David Ginsbourger; Xavier Bay; Olivier Roustant; Laurent Carraro. Argumentwise invariant kernels for the approximation of invariant functions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. 3, pp. 501-527. doi : 10.5802/afst.1343. https://afst.centre-mersenne.org/articles/10.5802/afst.1343/
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