A class of optimal estimators for the covariance operator in reproducing kernel Hilbert spaces
A reproducing kernel Hilbert space is considered with a continuous reproducing kernel defined over a separable topological space \(\mathcal{X}\). The covariance operator \(\Sigma_P\) of a probability distribution \(P\) over \(\mathcal{X}\) is estimated by i.i.d. observations in \(\mathcal{X}\). It is known that the empirical covariance operator has low bias but its high variance leads to a high mean square error. In the paper under review, a data-driven shrinkage estimator of \(\Sigma_P\) is proposed. In finite samples, the estimator outperforms the empirical covariance operator, especially when the data dimension is much larger than the sample size. It is shown that the estimator is \(\sqrt{n}\)-consistent in Hilbert-Schmidt norm. The minimax optimal rate of convergence is established over suitable class of probability distributions and it is shown that the considered shrinkage operators are minimax rate-optimal.
scientific article; zbMATH DE number 6997751
scientific article