Optimal Linear and Non-Linear Dimensionality Reduction

Résumé

Understanding how to optimally approximate general compact sets by finite dimensional spaces is of central interest for designing efficient numerical methods in forward simulation or inverse problems. The concept of n-width, introduced in 1936 by Kolmogorov, is well tailored to linear approximation methods. The interest for n-width has recently been revived by the approximation of parametrized/stochastic PDEs, and the development of reduced basis methods. We briefly survey some now classical results.

We then focus on analogous concepts for nonlinear approximation which are still the object of current research, motivated in particular by the development of neural networks, and possible applications to hyperbolic parametrized PDEs for which linear methods are not effective. We discuss a general framework that allows to embrace various concepts of linear and nonlinear widths, and present some recent results and relevant open problems within this framework.

Albert Cohen

Albert Cohen is a professor at Laboratoire Jacques-Louis Lions Sorbonne Université, Paris, France. After early works concerned with the development of the theory of wavelet bases in relation with algorithms used in signal and image processing, his research has been oriented towards various applicative directions, with as a common denominator its theoretical foundations in nonlinear approximation theory and harmonic analysis. In particular, it has led to the development and analysis of adaptive and sparsity-based numerical methods in various application contexts such as data compression, statistical estimation and learning theory or the numerical treatment of partial differential equations. His more recent interest lies in problems that involve a very large number of variables, and whose efficient numerical treatment is therefore challenged by the curse of dimensionality, as well as in model reduction strategies for the purpose of forward simulation and inverse problems. Albert Cohen’s research has been supported by the Advanced ERC grant BREAD (Breaking the Curse of Dimensionality in Analysis and Simulation) awarded in 2014. He has been a junior and senior member of the Institut Universitaire de France, and he is a member of the European Academy of Sciences.