Hosted by Olga Mula

Abstract

Recovering an unknown function from point samples is an ubiquitous task in various applicative settings: non-parametric regression, machine learning, reduced modeling, response surfaces in computer or physical experiments, data assimilation and inverse problems. In this lecture we discuss the context where the user is allowed to select the measurement points (sometimes refered to as active learning). This allows one to define a notion of optimal sampling point distribution when the approximation is searched in a arbitrary but fixed linear space of finite dimension and computed by weigted-least squares. Here optimal means that the approximation is comparable to the best possible in this space, while the sampling budget only slightly exceeds the dimension. We present simple randomized strategies that provably  generate optimal samples, and discuss several ongoing developments.