Asymptotic Theory for Linear Functionals of Kernel Ridge Regression

An asymptotic theory is established for linear functionals of the predictive function given by kernel ridge regression, when the reproducing kernel Hilbert space is equivalent to a Sobolev space.The theory covers a wide variety of linear functionals, including point evaluations, evaluation of derivatives, L2 inner products, etc.We establish the upper and lower bounds of the estimates and their asymptotic normality.We show the asymptotic normality of these estimators under mild conditions, which enables uncertainty quantification of a wide range of frequently used plug-in estimators.The theory also implies that the minimax L error of kernel ridge regression can be attained under n -1 log n.