Linear and nonlinear Gradient-Based dimension reduction

Keywords :

• uncertainty quantification
• function approximation
• neural networks
• autoencoders
• mollifiers

High-dimensional models are everywhere in science and engineering, yet they are often too expensive to evaluate directly. This thesis develops new gradient-based dimension-reduction methods that make surrogate modelling more efficient, especially when only a small number of simulations are available. The core idea is to use a majorize-then-minimize strategy, where the reconstruction error is controlled through Poincaré-type inequalities.

The first part of the presentation introduces a nonlinear dimension-reduction framework in which the feature map is defined as the leading components of a C1-diffeomorphism parameterized by invertible neural networks (affine coupling flows). A dimension-augmentation trick increases expressivity while remaining within the theoretical framework. The methodology extends naturally to vector-valued outputs and autoencoder architectures. Numerical experiments on benchmark models demonstrate that this approach outperforms existing state-of-the-art gradient-based methods.

The second part of the presentation addresses the difficulties of linear dimension reduction on highly oscillatory models by proposing Mollified Active Subspace (MAS). By smoothing gradients before analysis, MAS provides a computable error bound and leads to more reliable feature selection in this context. 

Overall, this work advances both linear and nonlinear gradient-based dimension reduction, providing new theoretical insights and practical algorithms for high-dimensional uncertainty quantification.

Directors:

• Professeur Clémentine Prieur (Université Grenoble Alpes )
• Docteur Olivier Zahm (Université Grenoble Alpes )

Reporters:

• Professeur Yohann De Castro (Centrale Lyon )
• Professeur Alain Celisse (Université Paris 1 Panthéon-Sorbonne )

Examinators:

• Professeur Rémi Gribonval (ENS Lyon )
• Professeur Olivier Roustant (INSA Toulouse )
• Professeur Houman Owhadi (Caltech university )
• Docteur Simon Barthelmé (CNRS, GIPSA-lab )