New asymptotic results in principal component analysis

Jeudi 29 juin 2017 14:00-15:00 - Karim Lounici - Unice

Résumé : Let $X$ be a mean zero Gaussian random vector in a separable Hilbert space $\mathbb H$ with covariance operator $\Sigma :=\mathbb E(X\otimes X).$ Let $\Sigma=\sum_r\geq 1\mu_r P_r$ be the spectral decomposition of $\Sigma$ with distinct eigenvalues $\mu_1>\mu_2> \dots$ and the corresponding spectral projectors $P_1, P_2, \dots.$ Given a sample $X_1,\dots, X_n$ of size $n$ of i.i.d. copies of $X,$ the sample
covariance operator is defined as $\hat \Sigma_n := n^-1\sum_j=1^n X_j\otimes X_j.$ The main goal of principal component analysis is to estimate spectral projectors $P_1, P_2, \dots$ by their empirical counterparts $\hat P_1, \hat P_2, \dots$ properly defined in terms of spectral decomposition of the sample covariance operator $\hat \Sigma_n.$ The aim of this paper is to study asymptotic distributions of important statistics related to this problem, in particular, of statistic $|\hat P_r-P_r|_2^2,$ where $|\cdot|_2^2$
is the squared Hilbert—Schmidt norm. This is done in a ``high-complexity" asymptotic framework in which the so called effective rank $\bf r(\Sigma) :=\frac\rm tr(\Sigma)|\Sigma|_\infty$ ($\rm tr(\cdot)$ being the trace and $|\cdot|_\infty$ being the operator norm) of the true covariance $\Sigma$ is becoming large simultaneously with the sample size $n,$ but $\bf r(\Sigma)=o(n)$ as $n\to\infty.$

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