A Primer on Reproducing Kernel Hilbert Spaces

TL;DR

This paper provides an accessible introduction to reproducing kernel Hilbert spaces, emphasizing their geometric interpretation and utility in studying continuously varying problems, serving as a bridge to infinite-dimensional linear algebra.

Contribution

It offers a novel geometric perspective on reproducing kernel Hilbert spaces and clarifies their role in infinite-dimensional linear algebra, with improved pedagogic explanations.

Findings

Emphasizes the extrinsic geometric viewpoint of RKHS.

Shows RKHS as a bridge to infinite-dimensional linear algebra.

Highlights the continuous variation of coordinate systems with geometry.

Abstract

Reproducing kernel Hilbert spaces are elucidated without assuming prior familiarity with Hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel Hilbert spaces and explaining when and why these spaces are efficacious. The novel viewpoint is that reproducing kernel Hilbert space theory studies extrinsic geometry, associating with each geometric configuration a canonical overdetermined coordinate system. This coordinate system varies continuously with changing geometric configurations, making it well-suited for studying problems whose solutions also vary continuously with changing geometry. This primer can also serve as an introduction to infinite-dimensional linear algebra because reproducing kernel Hilbert spaces have more properties in common with Euclidean spaces than do more general Hilbert spaces.