NORMAL APPROXIMATION AND CONCENTRATION OF SPECTRAL PROJECTORS OF SAMPLE COVARIANCE

• 成果类型: Article
• 署名作者: Koltchinskii, Vladimir; Lounici, Karim
• 署名单位: University System of Georgia; Georgia Institute of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/16-AOS1437
• 发表日期: 2017
• 页码: 121-157
• 关键词: CENTRAL-LIMIT-THEOREM; sparse pca; asymptotics; bounds

摘要

Let X, X-1,...,X-n be i.i.d. Gaussian random variables in a separable Hilbert space H with zero mean and covariance operator Sigma = E(X circle times X), and let (Sigma) over cap := n(-1) Sigma(n)(j=1) (X-i circle times X-j) be the sample (empirical) covariance operator based on (XI,..,Xn). Denote by P-r the spectral projector of Sigma corresponding to its rth eigenvalue mu(r) and by (P-r) over cap the empirical counterpart of P-r. The main goal of the paper is to obtain tight bounds on sup(x is an element of R)vertical bar p{parallel to(P-r) over cap - P-r parallel to(2)(2) - E parallel to(P-r) over cap - P-r parallel to(2)(2)/Var(1/2) (parallel to(P-r) over cap - P-r parallel to(2)(2)) <= x} - Phi(x)vertical bar, where parallel to . parallel to(2) denotes the Hilbert-Schmidt norm and Phi is the standard normal distribution function. Such accuracy of normal approximation of the distribution of squared Hilbert-Schmidt error is characterized in terms of so-called effective rank of Sigma defined as r(Sigma) = tr(Sigma)/parallel to Sigma parallel to(infinity) , where tr(Sigma) is the trace of Sigma and parallel to Sigma parallel to(infinity) is its operator norm, as well as another parameter characterizing the size of Var(parallel to(P-r) over cap - P-r parallel to(2)(2)). Other results include nonasymptotic bounds and asymptotic representations for the mean squared Hilbert-Schmidt norm error E parallel to(P-r) over cap - P-r parallel to(2)(2) and the variance Var(parallel to(P-r) over cap - P-r parallel to(2)(2)), and concentration inequalities for parallel to(P-r) over cap- P-r parallel to(2)(2) around its expectation.