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.

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.

DOI: 10.5802/afst.1343
Ginsbourger, David 1; Bay, Xavier 2; Roustant, Olivier 2; Carraro, Laurent 3

1 University of Bern, Institute of Mathematical Statistics and Actuarial Science, Alpeneggstrasse 22, CH-3012 Bern, Switzerland
2 École Nationale Supérieure des Mines, Fayol-EMSE, LSTI, 158 cours Fauriel, F-42023 Saint-Etienne, France
3 Télécom Saint-Etienne, 25 rue du Docteur Rémy Annino, F-42000 Saint-Etienne, France
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Ginsbourger, David; Bay, Xavier; Roustant, Olivier; Carraro, Laurent. 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. http://www.numdam.org/articles/10.5802/afst.1343/

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