A local approach to parameter space reduction for regression and classification tasks

TL;DR

This paper introduces local active subspaces (LAS), a novel dimension reduction method combining active subspaces with supervised clustering, to improve surrogate modeling in high-dimensional parameter spaces.

Contribution

The paper proposes LAS, integrating active subspaces with clustering algorithms to enhance local dimension reduction for surrogate models in complex parameter spaces.

Findings

LAS effectively identifies subdomains with lower variation in objective functions.

Clustering based on LAS improves surrogate model accuracy.

Method handles vectorial outputs and classifies regions by local active subspace dimension.

Abstract

Parameter space reduction has been proved to be a crucial tool to speed-up the execution of many numerical tasks such as optimization, inverse problems, sensitivity analysis, and surrogate models' design, especially when in presence of high-dimensional parametrized systems. In this work we propose a new method called local active subspaces (LAS), which explores the synergies of active subspaces with supervised clustering techniques in order to carry out a more efficient dimension reduction in the parameter space. The clustering is performed without losing the input-output relations by introducing a distance metric induced by the global active subspace. We present two possible clustering algorithms: K-medoids and a hierarchical top-down approach, which is able to impose a variety of subdivision criteria specifically tailored for parameter space reduction tasks. This method is particularly useful for the community working on surrogate modelling. Frequently, the parameter space presents subdomains where the objective function of interest varies less on average along different directions. So, it could be approximated more accurately if restricted to those subdomains and studied separately. We tested the new method over several numerical experiments of increasing complexity, we show how to deal with vectorial outputs, and how to classify the different regions with respect to the local active subspace dimension. Employing this classification technique as a preprocessing step in the parameter space, or output space in case of vectorial outputs, brings remarkable results for the purpose of surrogate modelling.