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SUMMARY:High-Dimensional Covariance Structure Estimation - Zhao Ren\, Yale
  University
DTSTART:20140228T140000Z
DTEND:20140228T150000Z
UID:TALK51223@talks.cam.ac.uk
CONTACT:20082
DESCRIPTION:Covariance matrices play a central role in multivariate statis
 tical analysis. A wide range of statistical methodologies\, including clus
 tering analysis\, principal component analysis\, linear and quadratic disc
 riminant analysis\, Gaussian graphical models require the knowledge of the
  covariance or precision structure.\n\nThe first half of the talk is prese
 nted in a chalk talk style. We talk about some motivations and challenges 
 of estimating covariance matrices in the high-dimensional setting. Then we
  briefly review some of the developments in estimation of covariance and p
 recision matrices in the past decade. Several classes of covariance and pr
 ecision matrices are discussed. We pay special attention to optimality the
 ory.\n\nThe second half of the talk focuses on one specific class: Toeplit
 z covariance structure\, which is used in the analysis of stationary time 
 series and a wide range of applications including radar imaging\, target d
 etection and speech recognition. We consider optimal estimation of large T
 oeplitz covariance matrices under the spectral norm. Minimax rate of conve
 rgence is established for two commonly used parameter spaces. The minimax 
 upper bound is obtained by studying the properties of tapering and banding
  estimators. The minimax lower bound is obtained by first constructing a m
 ore informative model for which independent random variables are observed\
 , and then deriving a lower bound for the more informative model by carefu
 lly constructing a collection of least favorable spectral densities and ap
 plying Fano's Lemma.
LOCATION:MR12\,  Centre for Mathematical Sciences\, Wilberforce Road\, Cam
 bridge
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