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CATEGORIES:CUED Control Group Seminars
SUMMARY:Gaussian distributions in symmetric spaces: novel
tools for statistical learning with covariance ma
trices - Salem Said\, CNRS\, IMS Laboratory\, Bord
eaux
DTSTART;TZID=Europe/London:20180517T140000
DTEND;TZID=Europe/London:20180517T150000
UID:TALK102004AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/102004
DESCRIPTION:The concept of Gaussian distribution can be based
on several different definitions: maximum entropy\
, minimum uncertainty\, the central limit theorem\
, or the kinetic theory of gases. When considered
in Euclidean space\, all of these definitions lead
to the same expression of the Gaussian distributi
on\, but in more general spaces\, different defini
tions lead to different expressions. This talk wil
l propose an original definition of the concept of
Gaussian distribution\, which is valid in Riemann
ian symmetric spaces of negative curvature. Namely
\, the definition is given by the property that ma
ximum likelihood is equivalent to Riemannian baryc
entre. There are no good or bad definitions\, only
more or less useful ones. The proposed definition
offers two advantages (1) many spaces of covarian
ce matrices (real\, complex\, quaternion\, Toeplit
z\, block-Toeplitz) are Riemannian symmetric space
s of negative curvature (2) it provides a statisti
cal foundation to the use of Riemannian barycentre
s\, which is a popular technique in many applicati
ons. The talk will compare the proposed definition
to other possible definitions\, develop its theor
etical consequences\, and finally explain how it g
ives rise to new statistical learning algorithms\,
specifically adapted to big data and high-dimensi
onal data\, all of this being illustrated by examp
les. Details may be found in \n\nhttps://arxiv.org
/abs/1507.01760\n\nhttps://arxiv.org/abs/1607.0692
9\n\nhttps://arxiv.org/abs/1707.07163\n
LOCATION:Cambridge University Engineering Department\, Lect
ure Room 5
CONTACT:Alberto Padoan
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