Reproducing kernels based schemes for nonparametric regression

TL;DR

This paper introduces an empirical projection operator scheme using reproducing kernel Hilbert spaces for nonparametric regression, analyzing its error, convergence, and stability with numerical demonstrations.

Contribution

It develops a novel RKHS-based projection scheme for nonparametric regression and provides theoretical error bounds, convergence rates, and stability analysis.

Findings

Error and convergence analysis for the proposed scheme.

Numerical stability and convergence rate of the regularized least squares method.

Numerical simulations demonstrating the effectiveness of the methods.

Abstract

In this work, we develop and study an empirical projection operator scheme for solving nonparametric regression problems. This scheme is based on an approximate projection of the regression function over a suitable reproducing kernel Hilbert space (RKHS). The RKHS considered in this paper are generated by the Mercer kernels given by the Legendre Christoffel-Darboux and convolution Sinc kernels. We provide error and convergence analysis of the proposed scheme under the assumption that the regression function belongs to some suitable functional spaces. We also consider the popular RKHS regularized least square minimization for nonparametric regression. In particular, we check the numerical stability of this second scheme and we provide its convergence rate in the special case of the Sinc kernel. Finally, we illustrate the proposed methods by various numerical simulation.