Approximation of Functions: Optimal Sampling and Complexity

TL;DR

This paper reviews recent advances in function approximation, focusing on optimal sampling strategies, information-theoretic limits, and various measurement types, providing a comprehensive overview of current research.

Contribution

It offers a broad overview of optimal sampling methods, theoretical limits, and different measurement types in function approximation, integrating recent insights and research developments.

Findings

Analysis of information-theoretic limits in function approximation

Comparison of sampling strategies including nonlinear, adaptive, and random measurements

Summary of current state-of-the-art algorithms and their near-optimality

Abstract

We consider approximation or recovery of functions based on a finite number of function evaluations. This is a well-studied problem in optimal recovery, machine learning, and numerical analysis in general, but many fundamental insights were obtained only recently. We discuss different aspects of the information-theoretic limit that appears because of the limited amount of data available, as well as algorithms and sampling strategies that come as close to it as possible. We also discuss (optimal) sampling in a broader sense, allowing other types of measurements that may be nonlinear, adaptive and random, and present several relations between the different settings in the spirit of information-based complexity. We hope that this article provides both, a basic introduction to the subject and a contemporary summary of the current state of research.