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

arXiv: https://arxiv.org/abs/2107.10867 · HTML: https://mlsanity.com/paper/2107.10867/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 64 references that are in the map.
- First posted on arXiv 22 Jul 2021.

## Builds on (most cited first, 8 of 8)

- [Faster k-Medoids Clustering: Improving the PAM, CLARA, and CLARANS Algorithms](https://mlsanity.com/paper/1810.05691/) (2018)
- [Certified dimension reduction in nonlinear Bayesian inverse problems](https://mlsanity.com/paper/1807.03712/) (2018)
- [Data-Driven Aerospace Engineering: Reframing the Industry with Machine Learning](https://mlsanity.com/paper/2008.10740/) (2020)
- [Active Manifolds: A non-linear analogue to Active Subspaces](https://mlsanity.com/paper/1904.13386/) (2019)
- [A supervised learning approach involving active subspaces for an efficient genetic algorithm in high-dimensional optimization problems](https://mlsanity.com/paper/2006.07282/) (2020)
- [Clustered active-subspace based local Gaussian Process emulator for high-dimensional and complex computer models](https://mlsanity.com/paper/2101.00057/) (2020)
- [Model Reduction Framework with a New Take on Active Subspaces for Optimization Problems with Linearized Fluid-Structure Interaction Constraints](https://mlsanity.com/paper/2002.07602/) (2020)
- [Multi-fidelity data fusion for the approximation of scalar functions with low intrinsic dimensionality through active subspaces](https://mlsanity.com/paper/2010.08349/) (2020)

## Abstract (arXiv)

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.