Optimal Linear and Nonlinear Dimensionality Reduction

Abstract

Understanding how to optimally approximate general compact sets using 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 suited to linear approximation methods. Interest in n-width has recently been revived by the approximation of parametrized/stochastic PDEs and the development of reduced basis methods. We briefly review some now-classical results.

We then focus on analogous concepts for nonlinear approximation that remain the subject of current research, driven in particular by the development of neural networks and potential applications to hyperbolic parametrized PDEs for which linear methods are ineffective. We discuss a general framework that encompasses 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 the Laboratoire Jacques-Louis Lions at Sorbonne University in Paris, France. After early work focused on the development of the theory of wavelet bases in relation to algorithms used in signal and image processing, his research has shifted toward various applications, all grounded in the theoretical foundations of nonlinear approximation theory and harmonic analysis. In particular, this 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, and the numerical solution of partial differential equations. His more recent research focuses on problems involving a very large number of variables, where efficient numerical solutions are challenged by the curse of dimensionality, as well as on model reduction strategies for 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.