🤖 AI Summary
This study addresses the fundamental challenge of achieving optimal function approximation under limited evaluation data—a central problem in numerical analysis and machine learning. From the perspective of information-based complexity, the work systematically investigates function recovery under generalized sampling by integrating information-theoretic analysis, optimal recovery theory, and nonlinear, adaptive, and randomized sampling mechanisms. It uncovers intrinsic connections among diverse sampling strategies, characterizes the information-theoretic limits of function approximation given finite data, and proposes efficient algorithms and sampling schemes that approach these limits. The results provide foundational insights and a unified framework for optimal sampling theory.
📝 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.