Diffeomorphism-Based Feature Learning Using Poincaré Inequalities

Abstract

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 that achieves superior accuracy on small datasets.

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-learned features. Specifically, we learn the feature map by minimizing an error bound obtained using Poincaré’s inequality applied either in input space or in feature space.

This results in two distinct strategies, which we compare both theoretically and numerically, and which we evaluate in relation to existing methods in the literature. In particular, we prove that if we define the nonlinear feature map as the first components of a C1-diffeomorphism, then our strategy is theoretically guaranteed. Our strategy for learning the C1-diffeomorphism is based on coupling flows, a specific 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 state-of-the-art competitors in terms of accuracy on small datasets.

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 earned her Ph.D. in applied mathematics from the University of Cergy-Pontoise in 2001. She began her career as an assistant professor at INSA Toulouse. Since 2008, she has held a full professorship at the University of Grenoble Alpes. Her research focuses on uncertainty quantification, sensitivity analysis, model order reduction, nonparametric inference, and risk analysis, primarily with environmental applications. In 2015, she was awarded the Blaise Pascal Prize by the French Academy of Sciences, a prize established by the Société de Mathématiques Appliquées et Industrielles (SMAI). She is actively involved in initiatives to promote scientific careers among young women.