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SUMMARY:Statistically optimal robust estimation of the precision matrix by
  convex programming - Arnak Dalayan (ENSAE)
DTSTART:20160226T160000Z
DTEND:20160226T170000Z
UID:TALK63644@talks.cam.ac.uk
CONTACT:Quentin Berthet
DESCRIPTION:Multivariate Gaussian distribution is often used as a first ap
 proximation to the distribution of high-dimensional data. Determining the 
 parameters of this distribution under various constraints is a widely stud
 ied problem in statistics\, and is often considered as a prototype for tes
 ting new algorithms or theoretical frameworks. In this paper\, we develop 
 a nonasymptotic approach to the problem of estimating the parameters of a 
 multivariate Gaussian distribution when data are corrupted by outliers. We
  propose an estimator-efficiently computable by solving a convex program-t
 hat robustly estimates the population mean and the population covariance m
 atrix even when the sample contains a significant proportion of outliers. 
 In the case where the dimension $p$ of the data points is of smaller order
  than the sample size\, our estimator of the corruption matrix is provably
  rate optimal simultaneously for the entry-wise $l_1$-norm\, the Frobenius
  norm and the mixed $l_2/l_1$ norm. Furthermore\, this optimality is achie
 ved by a penalized square-root-of-least-squares method with a universal tu
 ning parameter (calibrating the strength of the penalization). These resul
 ts are partly extended to the case where $p$ is potentially larger than $n
 $\, under the additional condition that the inverse covariance matrix is s
 parse.\n \nBased on a joint work with S. Balmand
LOCATION:MR12\, Centre for Mathematical Sciences\, Wilberforce Road\, Camb
 ridge.
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