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SUMMARY:The Geometry of Phase Transitions in Convex Optimization - Lotz\, 
 M (University of Manchester)
DTSTART:20130719T133000Z
DTEND:20130719T140000Z
UID:TALK46300@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Recent empirical research indicates that many convex optimizat
 ion problems with random constraints exhibit a phase transition as the num
 ber of constraints increases. For example\, this phenomenon emerges in the
  l1 minimization method for identifying a sparse vector from random linear
  samples. Indeed\, this approach succeeds with high probability when the n
 umber of samples exceeds a threshold that depends on the sparsity level\; 
 otherwise\, it fails with high probability. This is joint work with D. Ame
 lunxen\, M. McCoy and J. Tropp.\n\nWe present the first rigorous analysis 
 that explains why phase transitions are ubiquitous in random convex optimi
 zation problems. We also describe tools for making reliable predictions ab
 out the quantitative aspects of the transition\, including the location an
 d the width of the transition region. These techniques apply to regularize
 d linear inverse problems with random measurements\, to demixing problems 
 under a random incoherence model\, and also to cone programs with random a
 ffine constraints. \n\nThese applications depend on foundational research 
 in conic geometry. A new summary parameter\, called the statistical dimens
 ion\, canonically extends the dimension of a linear subspace to the class 
 of convex cones. The main result demonstrates that the sequence of conic i
 ntrinsic volumes of a convex cone concentrates sharply near the statistica
 l dimension. This fact leads to an approximate version of the conic kinema
 tic formula that gives bounds on the probability that a randomly oriented 
 cone shares a ray with a fixed cone.\n\n\n
LOCATION:Seminar Room 1\, Newton Institute
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