Optimal Sampling in Weighted Least-Squares Methods - Application to High-Dimensional Approximation

Explore optimal sampling techniques in weighted least-squares methods and their application to high-dimensional approximation in this 48-minute lecture from the Alan Turing Institute. Delve into the challenges of reconstructing complex processes with numerous parameters, addressing the curse of dimensionality in function approximation. Learn about modern approaches that overcome limitations by leveraging structural assumptions like low intrinsic dimensionality, partial separability, and sparse representations. Examine the mathematical foundations of high-dimensional approximation, covering topics such as multivariate approximation theory, high-dimensional integration, and non-parametric regression. Gain insights into key concepts including least-square methods, deterministic counterparts, noise levels, concentration inequalities, stability regimes, and adaptive methods. Discover how these techniques contribute to solving common problems in science and engineering involving complex, parameter-dependent processes.

Introduction
Highdimensional approximation
Common problems
Leastsquare method
General question
Deterministic counterpart
Goal
Key ingredient
Noise level
Concentration inequality
Stability regime
Highdimensional PDS
Curse of dimensionality
Adaptive method
References