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CATEGORIES:Statistics
SUMMARY:Strong Oracle Optimality of Folded Concave Penaliz
ed Estimation - Jianqing Fan\, Princeton Universit
y
DTSTART;TZID=Europe/London:20130215T160000
DTEND;TZID=Europe/London:20130215T170000
UID:TALK42445AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/42445
DESCRIPTION:Folded concave penalization methods such as SCAD h
ave been shown to enjoy the strong oracle property
for high-dimensional sparse estimation.\n\nHoweve
r\, a folded concave penalization problem usually
has multiple local solutions and the oracle proper
ty is established only for one of the unknown loca
l solutions. A challenging fundamental issue still
remains that it is not clear whether the local op
timal solution computed by a given optimization al
gorithm possesses those nice theoretical propertie
s. To close this important theoretical gap in over
a decade\, we provide a unified theory to show ex
plicitly how to obtain the oracle solution using t
he local linear approximation algorithm. For a fo
lded concave penalized estimation problem\, we sho
w that as long as the problem is localizable and t
he oracle estimator is well behaved\, we can obtai
n the oracle estimator by using the one-step local
linear approximation. In addition\, once the orac
le estimator is obtained\, the local linear approx
imation algorithm converges\, namely produces the
same estimator in the next iteration. We show that
the LASSO is a good initial estimator\, which pro
duces the oracle estimator using the one-step LLA
algorithm for folded concave penalization methods.
This is demonstrated by using three classical sp
arse estimation problems\, namely\, the sparse lin
ear regression\, the sparse logistic regression an
d the sparse precision matrix estimation\, and ill
ustrates the power of combining the LASSO and SCAD
to solve sparse inartistic estimation problem.\n\
n(Joint work with Lingzhou Xue and Hui Zou.)
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
B
CONTACT:Richard Samworth
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