Computing active subspaces with Monte Carlo
For practitioners using active subspaces to reduce high-dimensional parameter studies, this work provides rigorous guidance on sample size and accuracy of subspace estimates.
The paper studies a Monte Carlo method for approximating eigenpairs of the active subspace matrix, providing theoretical guarantees and a bootstrap-based practical approach. It offers guidance on required gradient samples and subspace accuracy, validated on quadratic functions and a 100-variable PDE.
Active subspaces can effectively reduce the dimension of high-dimensional parameter studies enabling otherwise infeasible experiments with expensive simulations. The key components of active subspace methods are the eigenvectors of a symmetric, positive semidefinite matrix whose elements are the average products of partial derivatives of the simulation's input/output map. We study a Monte Carlo method for approximating the eigenpairs of this matrix. We offer both theoretical results based on recent non-asymptotic random matrix theory and a practical approach based on the bootstrap. We extend the analysis to the case when the gradients are approximated, for example, with finite differences. Our goal is to provide guidance for two questions that arise in active subspaces: (i) How many gradient samples does one need to accurately approximate the eigenvalues and subspaces? (ii) What can be said about the accuracy of the estimated subspace, both theoretically and practically? We test the approach on both simple quadratic functions where the active subspace is known and a parameterized PDE with 100 variables characterizing the coefficients of the differential operator.