Uniform Function Estimators in Reproducing Kernel Hilbert Spaces
This work addresses function estimation for statistical applications, offering a method that avoids the curse of dimensionality, but it appears incremental as it builds on existing kernel-based approaches.
The paper tackles the problem of regression for function reconstruction from noisy observations at random locations in reproducing kernel Hilbert spaces, showing that the estimator converges uniformly and locally to the conditional expectation with a derived convergence rate.
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.