# Sensitivity analysis for multidimensional and functional outputs

arXiv: https://arxiv.org/abs/1311.1797 · HTML: https://mlsanity.com/paper/1311.1797/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Nov 2013.

## Built on by (most cited first, 6 of 6)

- [Coupled Input-Output Dimension Reduction: Application to Goal-oriented Bayesian Experimental Design and Global Sensitivity Analysis](https://mlsanity.com/paper/2406.13425/) (2024)
- [Active Subspaces in Infinite Dimension](https://mlsanity.com/paper/2510.11871/) (2025)
- [Global Sensitivity Analysis for Engineering Design Based on Individual Conditional Expectations](https://mlsanity.com/paper/2512.11946/) (2025)
- [Learning to Choose: An Empowerment-Guided Multi-Agent System with semantic communication for Adaptive Method Selection](https://mlsanity.com/paper/2605.30042/) (2026)
- [A Hilbert-Valued Functional Decomposition Framework for Explaining Time-Dependent Outputs](https://mlsanity.com/paper/2609.11295/) (2026)
- [A Hierarchy of Entropy-Shapley Games for Multivariate Predictive Uncertainty](https://mlsanity.com/paper/2609.35217/) (2026)

## Abstract (arXiv)

Let X:=(X1, ldots, Xp) be random objects (the inputs), defined on some probability space (Omega,F, mathbb P) and valued in some measurable space E=E1×ldots × Ep. Further, let Y:=Y = f(X1, ldots, Xp) be the output. Here, f is a measurable function from E to some Hilbert space mathbbH (mathbbH could be either of finite or infinite dimension). In this work, we give a natural generalization of the Sobol indices (that are classically defined when Y∈mathbb R ), when the output belongs to mathbbH. These indices have very nice properties. First, they are invariant. under isometry and scaling. Further they can be, as in dimension 1, easily estimated by using the so-called Pick and Freeze method. We investigate the asymptotic behaviour of such estimation scheme.