New insights into the feature maps of Sobolev kernels: application in global sensitivity analysis
Abstract
As part of the study of an input-output numerical simulator, performing a sensitivity analysis allows to identify the input parameters having the greatest influence on the output variability. Since the variance-based approach (also known as the ANOVA framework) is too expensive, and the kernel-based approach (leading to HSIC indices) lacks interpretability, the HSIC-ANOVA framework has recently emerged to marry the advantages of both. A major particularity of this new methodology is the need to use the unanchored Sobolev kernels. This paper investigates how sensitivity is measured according to the chosen Sobolev kernel. To achieve this, at least one explicit feature map is extracted from each Sobolev kernel and this helps identify the dependence patterns captured by HSIC-ANOVA indices. For the Sobolev kernel of order r = 1, three different proof techniques are proposed to disclose its Mercer feature map. For higher-order Sobolev kernels (r ≥ 2), it is proved that the Mercer feature map does not have a closed-form expression. In response, a slightly relaxed feature map is obtained after considering a sub-kernel decomposition. This latter feature map allows to justify why the Sobolev kernels of order r ≥ 2 should not be used to estimate HSIC-ANOVA indices.
Related
- Feature tracking in physics-informed neural networks via joint optimization of nonlinear deformation manifolds: application to shocks
- Causal sensitivity analyses for indirect and spillover mediation effects in cluster-randomized trials: Application to a pragmatic trial on chronic pain management
- Bridging Markov Chain Monte Carlo Techniques and Tierney–Kadane Approximations for Progressively Censored Garhy Reliability Models: Simulation Insights and a Medical Application
- Nonparametric hazard rate estimation with associated kernels and minimax bandwidth choice
- Sharp estimates for the Jacobi and trigonometric Dunkl heat kernels
BibTeX
@misc{sarazin2026insights,
title = {{New insights into the feature maps of Sobolev kernels: application in global sensitivity analysis}},
author = {Gabriel Sarazin and Amandine Marrel and Sebastien da Veiga and Vincent Chabridon},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://cea.hal.science/cea-04320711v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - New insights into the feature maps of Sobolev kernels: application in global sensitivity analysis
AU - Gabriel Sarazin
AU - Amandine Marrel
AU - Sebastien da Veiga
AU - Vincent Chabridon
PY - 2026
DA - 2026/10/06
PB - HAL
UR - https://cea.hal.science/cea-04320711v2
AB - As part of the study of an input-output numerical simulator, performing a sensitivity analysis allows to identify the input parameters having the greatest influence on the output variability. Since the variance-based approach (also known as the ANOVA framework) is too expensive, and the kernel-based approach (leading to HSIC indices) lacks interpretability, the HSIC-ANOVA framework has recently emerged to marry the advantages of both. A major particularity of this new methodology is the need to use the unanchored Sobolev kernels. This paper investigates how sensitivity is measured according to the chosen Sobolev kernel. To achieve this, at least one explicit feature map is extracted from each Sobolev kernel and this helps identify the dependence patterns captured by HSIC-ANOVA indices. For the Sobolev kernel of order r = 1, three different proof techniques are proposed to disclose its Mercer feature map. For higher-order Sobolev kernels (r ≥ 2), it is proved that the Mercer feature map does not have a closed-form expression. In response, a slightly relaxed feature map is obtained after considering a sub-kernel decomposition. This latter feature map allows to justify why the Sobolev kernels of order r ≥ 2 should not be used to estimate HSIC-ANOVA indices.
ER -