Optimal sampling for least-squares approximation

Least-squares approximation is one of the most important methods for recovering an unknown function from data. While in many applications the data is fixed, in many others there is substantial freedom to choose where to sample. In this paper, we review recent progress on near-optimal random sampling strategies for (weighted) least-squares approximation in arbitrary linear spaces. We introduce the Christoffel function as a key quantity in the analysis of (weighted) least-squares approximation from random samples, then show how it can be used to construct a random sampling strategy, termed Christoffel sampling, that possesses near-optimal sample complexity: namely, the number of samples scales log-linearly in the dimension of the approximation space n. We discuss a series of variations, extensions and further topics, and throughout highlight connections to approximation theory, machine learning, information-based complexity and numerical linear algebra. Finally, motivated by various contemporary applications, we consider a generalization of the classical setting where the samples need not be pointwise samples of a scalar-valued function, and the approximation space need not be linear. We show that, even in this significantly more general setting, suitable generalizations of Christoffel function still determine the sample complexity. Consequently, these can be used to design enhanced, Christoffel sampling strategies in a unified way for general recovery problems. This article is largely self-contained, and intended to be accessible to nonspecialists.