A class of optimal estimators for the covariance operator in reproducing kernel Hilbert spaces
type: Item
instance of: scholarly article
title: A class of optimal estimators for the covariance operator in reproducing kernel Hilbert spaces
zbMATH Open document ID: 1410.62143
author: Yang Zhou
author: Di-Rong Chen
author: Wei Huang
published in: Journal of Multivariate Analysis
publication date: 2019-01-04T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 62J10
Mathematics Subject Classification ID: 47B32
Mathematics Subject Classification ID: 62C20
Mathematics Subject Classification ID: 62G20
zbMATH DE Number: 6997751
zbMATH Keywords: covariance operator
zbMATH Keywords: minimax lower bound
zbMATH Keywords: rate of convergence
zbMATH Keywords: reproducing kernel Hilbert space
zbMATH Keywords: shrinkage estimator
reviewed by: A. Kukush
describes a project that uses: KPCA plus LDA
MaRDI profile type: Publication
full work available at URL: j.jmva.2018.09.003
OpenAlex ID: W2892153240
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Wikidata QID: Q129227719
DOI: 10.1016/J.JMVA.2018.09.003
description: scientific article; zbMATH DE number 6997751