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UID:7460@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20171106T153000
DTEND;TZID=Europe/Paris:20171106T163000
DTSTAMP:20241120T204336Z
URL:https://www.i2m.univ-amu.fr/evenements/principal-component-analysis-a-
 berry-esseen-bound-for-the-spectral-projectors-of-the-covariance-operator/
SUMMARY: (...): Principal Component Analysis: a Berry-Esseen bound for the 
 spectral projectors of the covariance operator
DESCRIPTION:: Principal Component Analysis is a popular technique to study 
 the covariance structure of a random vector. In a recent series of papers\
 , we proved several new results about Principal Component Analysis in an i
 nfinite dimensional setting. One result of interest is about the asymptoti
 c distribution of the empirical spectral projectors. 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 counte
 rpart of $P_r.$ We obtain a Berry-Esseen type bound that quantifies the ac
 curacy of the normal approximation in term of the effective rank ${\\bf r}
 (\\Sigma)$ and another quantity $B_r(\\Sigma)$ characterizing the order of
  magnitude of $\\mathrm{Var}(\\|\\hat P_r-P_r\\|_2^2)$.This is a joint wor
 k with Vladimir Koltchinskii.Bibliography:Koltchinskii\, V. and K. Lounici
  (2014).Asymptotics and concentration bounds for spectral projectors of sa
 mple covariance.Koltchinskii\, V. and K. Lounici (2014).Concentration ineq
 ualities and moment bounds for sample covariance operators.Koltchinskii\, 
 V. and K. Lounici (2015).Normal approximation and concentration of spectra
 l projectors of sample covariance.-http://math.unice.fr/laboratoire/fiche&
 id=745-http://people.math.gatech.edu/~klounici6/--
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