Argumentwise invariant kernels for the approximation of invariant functions

David Ginsbourger[1]; Xavier Bay[2]; Olivier Roustant[2]; Laurent Carraro[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

Annales de la faculté des sciences de Toulouse Mathématiques (2012)

  • Volume: 21, Issue: 3, page 501-527
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Ginsbourger, David, et al. "Argumentwise invariant kernels for the approximation of invariant functions." Annales de la faculté des sciences de Toulouse Mathématiques 21.3 (2012): 501-527. <http://eudml.org/doc/250993>.

@article{Ginsbourger2012,
abstract = {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.},
affiliation = {University of Bern, Institute of Mathematical Statistics and Actuarial Science, Alpeneggstrasse 22, CH-3012 Bern, Switzerland; École Nationale Supérieure des Mines, Fayol-EMSE, LSTI, 158 cours Fauriel, F-42023 Saint-Etienne, France; École Nationale Supérieure des Mines, Fayol-EMSE, LSTI, 158 cours Fauriel, F-42023 Saint-Etienne, France; Télécom Saint-Etienne, 25 rue du Docteur Rémy Annino, F-42000 Saint-Etienne, France},
author = {Ginsbourger, David, Bay, Xavier, Roustant, Olivier, Carraro, Laurent},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {reproducing kernel Hilbert spaces; invariant random fields},
language = {eng},
month = {4},
number = {3},
pages = {501-527},
publisher = {Université Paul Sabatier, Toulouse},
title = {Argumentwise invariant kernels for the approximation of invariant functions},
url = {http://eudml.org/doc/250993},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Ginsbourger, David
AU - Bay, Xavier
AU - Roustant, Olivier
AU - Carraro, Laurent
TI - Argumentwise invariant kernels for the approximation of invariant functions
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2012/4//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - 3
SP - 501
EP - 527
AB - 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.
LA - eng
KW - reproducing kernel Hilbert spaces; invariant random fields
UR - http://eudml.org/doc/250993
ER -

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