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.

翻译：我们研究基于有限次函数求值的函数逼近或恢复问题。这是最优恢复、机器学习以及数值分析领域中一个被广泛研究的问题，但许多基础性洞见直到最近才得以揭示。我们讨论了由有限数据量引起的信息论极限的不同方面，以及尽可能接近该极限的算法与采样策略。我们还从更广义的角度探讨了（最优）采样，允许包含非线性、自适应和随机等其他类型的测量方式，并基于信息基复杂度的思想，呈现了不同设定之间的若干关联。我们希望本文既能提供该主题的基础性介绍，又能对当前研究现状作出当代性总结。