The Annals of Statistics

Normal approximation and concentration of spectral projectors of sample covariance

Vladimir Koltchinskii and Karim Lounici

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Let $X,X_{1},\dots,X_{n}$ be i.i.d. Gaussian random variables in a separable Hilbert space $\mathbb{H}$ with zero mean and covariance operator $\Sigma=\mathbb{E}(X\otimes X)$, and let $\hat{\Sigma}:=n^{-1}\sum_{j=1}^{n}(X_{j}\otimes X_{j})$ be the sample (empirical) covariance operator based on $(X_{1},\dots,X_{n})$. Denote by $P_{r}$ the spectral projector of $\Sigma$ corresponding to its $r$th eigenvalue $\mu_{r}$ and by $\hat{P}_{r}$ the empirical counterpart of $P_{r}$. The main goal of the paper is to obtain tight bounds on

\[\sup_{x\in\mathbb{R}}\vert\mathbb{P} \{\frac{\Vert \hat{P}_{r}-P_{r}\Vert_{2}^{2}-\mathbb{E}\Vert \hat{P}_{r}-P_{r}\Vert_{2}^{2}}{\operatorname{Var}^{1/2}(\Vert \hat{P}_{r}-P_{r}\Vert_{2}^{2})}\leq x\}-\Phi (x)\vert ,\] where $\Vert \cdot \Vert_{2}$ 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 ${\mathbf{r}}(\Sigma)=\frac{\operatorname{tr}(\Sigma)}{\Vert \Sigma \Vert_{\infty}}$, where $\operatorname{tr}(\Sigma)$ is the trace of $\Sigma$ and $\Vert \Sigma \Vert_{\infty}$ is its operator norm, as well as another parameter characterizing the size of $\operatorname{Var}(\Vert \hat{P}_{r}-P_{r}\Vert_{2}^{2})$. Other results include nonasymptotic bounds and asymptotic representations for the mean squared Hilbert–Schmidt norm error $\mathbb{E}\Vert \hat{P}_{r}-P_{r}\Vert_{2}^{2}$ and the variance $\operatorname{Var}(\Vert \hat{P}_{r}-P_{r}\Vert_{2}^{2})$, and concentration inequalities for $\Vert \hat{P}_{r}-P_{r}\Vert_{2}^{2}$ around its expectation.

Article information

Ann. Statist., Volume 45, Number 1 (2017), 121-157.

Received: September 2015
Revised: January 2016
First available in Project Euclid: 21 February 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation

Sample covariance spectral projectors effective rank principal component analysis concentration inequalities normal approximation perturbation theory


Koltchinskii, Vladimir; Lounici, Karim. Normal approximation and concentration of spectral projectors of sample covariance. Ann. Statist. 45 (2017), no. 1, 121--157. doi:10.1214/16-AOS1437.

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