Diffeomorphism-Based Feature Learning Using Poincaré Inequalities

Résumé

Joint work with Romain Verdière (Inria Grenoble) and Olivier Zahm (Inria Grenoble).

During this talk, I will present a gradient-enhanced algorithm for high-dimensional function approximation which achieves outperforming accuracy on small data sets.

This algorithm, introduced in [1], proceeds in two steps: first, we reduce the input dimension by learning the relevant input features from gradient evaluations and, second, we regress the function output against the pre-learnt features. Specifically, we learn the feature map by minimizing an error bound obtained using Poincaré inequality applied either in input space or in feature space.

This results in two different strategies which we compare both theoretically and numerically, and which we position in relation to existing methods from the literature. In particular, we prove that if we seek the nonlinear feature map as the first components of a C1-diffeomorphism, then our strategy is theoretically guaranteed. Our strategy to learn the C1-diffeomorphism is based on coupling flows, a particular class of invertible neural networks defined as the composition of block-triangular maps.

Finally I will present several numerical experiments to demonstrate that the algorithm we propose outperforms the state-of-the-art competitors in terms of accuracy with little data sets.

References

[1] R Verdière, C Prieur, O Zahm : Diffeomorphism-based feature learning using Poincaré inequalities on augmented input space Journal of Machine Learning Research 26 (139), 1-31.

Clémentine Prieur

Clémentine Prieur graduated a PhD thesis in applied mathematics at Cergy-Pontoise University in 2001. She started her career as assistant professor at INSA Toulouse. Since 2008 she has a full professor position at Grenoble Alpes University. Her research focuses on uncertainty quantification, sensitivity analysis, model order reduction, nonparametric inference, and risk analysis with mainy environmental applications. In 2015, she was laureate of the Blaise Pascal prize awarded by the French Académie des Sciences and founded by the Société de Mathématiques Appliquées et Industrielles (SMAI). She contributes to actions to promote scientific careers among young women.