Statistical performance of optimal scoring in reproducing kernel Hilbert spaces

In this paper we consider a version of optimal scoring in reproducing kernel Hilbert spaces. The estimators are constructed by minimizing regularized (penalized) empirical variances, as previously in penalized optimal scoring. With the cross covariance operators in reproducing kernel Hilbert space, the errors of the estimated functions are bounded by the so called excess error in statistical learning. Then by approaches and tools developed in statistical learning theory, we establish a learning rate Op(nθ−1∕2) for the proposed algorithm under some mild conditions, where n is the sample size and θ>0 is an arbitrarily fixed number. The learning rate has been not obtained in optimal scoring problem so far.