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CATEGORIES:Statistics
SUMMARY:Statistically optimal robust estimation of the pre
cision matrix by convex programming - Arnak Dalaya
n (ENSAE)
DTSTART;TZID=Europe/London:20160226T160000
DTEND;TZID=Europe/London:20160226T170000
UID:TALK63644AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/63644
DESCRIPTION:Multivariate Gaussian distribution is often used a
s a first approximation to the distribution of hig
h-dimensional data. Determining the parameters of
this distribution under various constraints is a w
idely studied problem in statistics\, and is often
considered as a prototype for testing new algorit
hms or theoretical frameworks. In this paper\, we
develop a nonasymptotic approach to the problem of
estimating the parameters of a multivariate Gauss
ian distribution when data are corrupted by outlie
rs. We propose an estimator-efficiently computable
by solving a convex program-that robustly estimat
es the population mean and the population covarian
ce matrix even when the sample contains a signific
ant proportion of outliers. In the case where the
dimension $p$ of the data points is of smaller ord
er than the sample size\, our estimator of the cor
ruption matrix is provably rate optimal simultaneo
usly for the entry-wise $l_1$-norm\, the Frobenius
norm and the mixed $l_2/l_1$ norm. Furthermore\,
this optimality is achieved by a penalized square-
root-of-least-squares method with a universal tuni
ng parameter (calibrating the strength of the pena
lization). These results are partly extended to th
e case where $p$ is potentially larger than $n$\,
under the additional condition that the inverse co
variance matrix is sparse.\n \nBased on a joint wo
rk with S. Balmand
LOCATION:MR12\, Centre for Mathematical Sciences\, Wilberfo
rce Road\, Cambridge.
CONTACT:Quentin Berthet
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