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SUMMARY:Gaussian distributions in symmetric spaces: novel  tools for stati
 stical learning with covariance matrices - Salem Said\, CNRS\, IMS Laborat
 ory\, Bordeaux
DTSTART:20180517T130000Z
DTEND:20180517T140000Z
UID:TALK102004@talks.cam.ac.uk
CONTACT:Alberto Padoan
DESCRIPTION:The concept of Gaussian distribution can be based on several d
 ifferent definitions: maximum entropy\, minimum uncertainty\, the central 
 limit theorem\, or the kinetic theory of gases. When considered in Euclide
 an space\, all of these definitions lead to the same expression of the Gau
 ssian distribution\, but in more general spaces\, different definitions le
 ad to different expressions. This talk will propose an original definition
  of the concept of Gaussian distribution\, which is valid in Riemannian sy
 mmetric spaces of negative curvature. Namely\, the definition is given by 
 the property that maximum likelihood is equivalent to Riemannian barycentr
 e. There are no good or bad definitions\, only more or less useful ones. T
 he proposed definition offers two advantages (1) many spaces of covariance
  matrices (real\, complex\, quaternion\, Toeplitz\, block-Toeplitz) are Ri
 emannian symmetric spaces of negative curvature (2) it provides a statisti
 cal foundation to the use of Riemannian barycentres\, which is a popular t
 echnique in many applications. The talk will compare the proposed definiti
 on to other possible definitions\, develop its theoretical consequences\, 
 and finally explain how it gives rise to new statistical learning algorith
 ms\, specifically adapted to big data and high-dimensional data\, all of t
 his being illustrated by examples. Details may be found in \n\nhttps://arx
 iv.org/abs/1507.01760\n\nhttps://arxiv.org/abs/1607.06929\n\nhttps://arxiv
 .org/abs/1707.07163\n
LOCATION:Cambridge University Engineering Department\, Lecture Room 5
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