Uniform Function Estimators in Reproducing Kernel Hilbert Spaces

TL;DR

This paper studies a kernel-based function estimator in reproducing kernel Hilbert spaces, showing it converges to the true function with favorable properties and is not affected by the curse of dimensionality.

Contribution

It demonstrates the convergence of kernel estimators in RKHS to the conditional expectation and analyzes their statistical properties, including convergence rates.

Findings

Estimator converges in mean norm to the conditional expectation

Local and uniform convergence of the estimator is established

Preselecting the kernel avoids the curse of dimensionality

Abstract

This paper addresses the problem of regression to reconstruct functions, which are observed with superimposed errors at random locations. We address the problem in reproducing kernel Hilbert spaces. It is demonstrated that the estimator, which is often derived by employing Gaussian random fields, converges in the mean norm of the reproducing kernel Hilbert space to the conditional expectation and this implies local and uniform convergence of this function estimator. By preselecting the kernel, the problem does not suffer from the curse of dimensionality. The paper analyzes the statistical properties of the estimator. We derive convergence properties and provide a conservative rate of convergence for increasing sample sizes.