Normal approximation and concentration of spectral projectors of sample covariance

TL;DR

This paper establishes normal approximation bounds and concentration inequalities for the spectral projectors of sample covariance operators in Hilbert spaces, using effective rank and variance parameters.

Contribution

It provides the first tight bounds on the normal approximation of the spectral projector error distribution in high-dimensional settings.

Findings

Normal approximation accuracy characterized by effective rank.

Non-asymptotic bounds for mean squared error.

Concentration inequalities for spectral projector errors.

Abstract

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 $$ \sup_{x\in {\mathbb R}} \left|{\mathbb P}\left\{\frac{\|\hat P_r-P_r\|_2^2-{\mathbb E}\|\hat P_r-P_r\|_2^2}{{\rm Var}^{1/2}(\|\hat P_r-P_r\|_2^2)}\leq x\right\}-\Phi(x)\right|, $$ where $\|\cdot\|_2$ 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 ${\bf r}(\Sigma)=\frac{{\rm tr}(\Sigma)}{\|\Sigma\|_{\infty}},$ where ${\rm tr}(\Sigma)$ is the trace of $\Sigma$ and $\|\Sigma\|_{\infty}$ is its operator norm, as well as another parameter characterizing the size of ${\rm Var}(\|\hat P_r-P_r\|_2^2).$ Other results include non-asymptotic bounds and asymptotic representations for the mean squared Hilbert--Schmidt norm error ${\mathbb E}\|\hat P_r-P_r\|_2^2$ and the variance ${\rm Var}(\|\hat P_r-P_r\|_2^2),$ and concentration inequalities for $\|\hat P_r-P_r\|_2^2$ around its expectation.