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SUMMARY:A statistical perspective on sparse regularization and geometric m
 odelling - Aykroyd\, R (University of Leeds)
DTSTART:20140207T134500Z
DTEND:20140207T143000Z
UID:TALK50710@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Consider a typical inverse problem where we wish to reconstruc
 t an unknown function from a set of measurements. When the function is dis
 cretized it is usual for the number of data points to be insufficient to u
 niquely determine the unknowns  the problem is ill-posed. One approach is 
 to reduce the size of the set of eligible solutions until it contains only
  a single solutionthe problem is regularized. There are\, however\, infini
 tely many possible restrictions each leading to a unique solution. Hence t
 he choice of regularization is crucial\, but the best choice\, even amongs
 t those commonly used\, is still difficult. Such regularized reconstructio
 n can be placed into a statistical setting where data fidelity becomes a l
 ikelihood function and regularization becomes a prior distribution. Recons
 truction then becomes a statistical inference task solved\, perhaps\, usin
 g the posterior mode. The common regularization approaches then correspond
  to different choices of prior di stribution. In this talk the ideas of re
 gularized estimation\, including ridge\, lasso\, bridge and elastic-net re
 gression methods\, will be defined. Application of sparse regularization t
 o basis function expansions\, and other dictionary methods\, such as wavel
 ets will be discussed. Their link to smooth and sparse regularization\, an
 d to Bayesian estimation\, will be considered. As an alternative to locall
 y constrained reconstruction methods\, geometric models impose a global st
 ructure. Such models are usually problem specific\, compared to more gener
 ic locally constrained methods\, but when the parametric assumptions are r
 easonable they will make better use of the data\, provide simpler models a
 nd can include parameters which may be used directly\, for example in moni
 toring or control\, without the need for extra post-processing. Finally\, 
 the matching of modelling and estimation styles with numerical procedures\
 , to produce efficient algorithms\, will be discussed.\n
LOCATION:Seminar Room 1\, Newton Institute
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