Reproducing Kernel Hilbert Spaces for Penalized Regression: A Tutorial

Penalized regression procedures have become very popular ways to estimate complicated functions. The smoothing spline, for example, is the solution of a minimization problem in a functional space. If such a minimization problem is posed on a reproducing kernel Hilbert space (RKHS), the solution is guaranteed to exist, is unique, and has a very simple form. There are excellent books and articles about RKHS and their applications in statistics; however, this existing literature is very dense. This article provides a friendly reference for a reader approaching this subject for the first time. It begins with a simple problem, a system of linear equations, and then gives an intuitive motivation for reproducing kernels. Armed with the intuition gained from our first examples, we take the reader from vector spaces to Banach spaces and to RKHS. Finally, we present some statistical estimation problems that can be solved using the mathematical machinery discussed. After reading this tutorial, the reader will be ready to study more advanced texts and articles about the subject, such as those by Wahba or Gu. Online supplements are available for this article.