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What is hidden behind the Sobolev kernels involved in the HSIC-ANOVA decomposition?

Abstract : In many industrial fields, the numerical simulators that model physical phenomena are very expensive from a computational viewpoint. In this context, only a very limited number of computer experiments can be planned. Statistical learning techniques helps build a surrogate model but this task becomes excessively challenging when the simulation code takes a high number (dozens, if not hundreds) of uncertain parameters as input variables. To escape from this deadlock, global sensitivity analysis (GSA) may be used as a tool for dimension reduction. In particular, the sensitivity measures which are based on the Hilbert-Schmidt independence criterion (HSIC) have been increasingly used over the past few years because they pave the way to several variable selection procedures. However, the so-called HSIC indices still suffer from their lack of interpretability, especially among non-specialists who rather ask for percentage-like indicators. To ease interpretation, HSIC-ANOVA indices have been recently introduced to allow for a strict separation of main effects and interactions. This breakthrough was obtained after assuming mutual independence among input variables and provided that ANOVA kernels, such as Sobolev kernels, are used to estimate HSIC-ANOVA indices. Very little is said in the literature about Sobolev kernels and their properties. In this talk, their feature maps are investigated in three different ways. Firstly, a simulation-based approach named kernel feature analysis (KFA) allows to distinguish different behaviors depending on the integer value given to their smoothness parameter. Secondly, the eigenvalue problem that goes with the Mercer decomposition of Sobolev kernels is transformed into a Cauchy problem that can be solved in one specific case. Thirdly, a series expansion of the translation-invariant term in Sobolev kernels discloses a fully analytical characterization of the underlying reproducing kernel Hilbert spaces (RKHS). All these theoretical results allow to prove that all Sobolev kernels are characteristic, which means HSIC-ANOVA indices are well-adapted association measures to detect independence among input-output pairs.
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https://hal-cea.archives-ouvertes.fr/cea-03701074
Contributor : Amandine Marrel Connect in order to contact the contributor
Submitted on : Thursday, June 23, 2022 - 9:44:49 AM
Last modification on : Friday, June 24, 2022 - 3:56:00 AM

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  • HAL Id : cea-03701074, version 1

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Gabriel Sarazin, Amandine Marrel, Sébastien da Veiga, Vincent Chabridon. What is hidden behind the Sobolev kernels involved in the HSIC-ANOVA decomposition?. 2022 SAMO Conference - 10th International Conference on Sensitivity Analysis of Model Output, Florida State University, Mar 2022, Tallahassee, United States. ⟨cea-03701074⟩

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