Active subspace methods and derivative-based Shapley effects for functions with non-independent variables
Matieyendou Lamboni, Sergei Kucherenko
TL;DR
The paper addresses the challenge of performing dimension reduction and uncertainty quantification for functions with non-independent inputs by extending derivative-based active subspaces (Db-AS) and Shapley effects to account for input dependencies via a generalized gradient grad(f) that incorporates a dependency-informed metric G. It introduces two complementary approaches: (i) Db-active subspaces based on the gradient of f with non-independent inputs, yielding a C' matrix whose eigenvectors define active directions and a dimension-reduction via a conditional expectation; and (ii) sensitivity-based active subspaces (SF-AS) built from total sensitivity functionals, using Σ^{tot} or its dependent analogue to identify active variables that also reduce output variance. Additionally, the paper defines derivative-based Shapley effects under dependence by combining gradient-based terms with gradient cross-interactions, and provides estimators with dimension-free bias and convergence guarantees. Through analytical results and simulations, it demonstrates that Db-AS and SF-AS can offer varying performance depending on the function class and dependency structure, underscoring their complementary roles for variance reduction and efficient dimension reduction in dependent settings.
Abstract
Lower-dimensional subspaces that impact estimates of uncertainty are often described by Linear combinations of input variables, leading to active variables. This paper extends the derivative-based active subspace methods and derivative-based Shapley effects to cope with functions with non-independent variables, and it introduces sensitivity-based active subspaces. While derivative-based subspace methods focus on directions along which the function exhibits significant variation, sensitivity-based subspace methods seek a reduced set of active variables that enables a reduction in the function's variance. We propose both theoretical results using the recent development of gradients of functions with non-independent variables and practical settings by making use of optimal computations of gradients, which admit dimension-free upper-bounds of the biases and the parametric rate of convergence. Simulations show that the relative performance of derivative-based and sensitivity-based active subspaces methods varies across different functions.