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SUMMARY:Geometry of Sparse Representations - Mark Plumbley\, Center for Di
 gital Music\, Queen Mary University of London
DTSTART:20061212T130000Z
DTEND:20061212T140000Z
UID:TALK5887@talks.cam.ac.uk
CONTACT:Taylan Cemgil
DESCRIPTION:Geometrical considerations can often give us insights or new a
 pproaches to tricky problems.  One of these is the "sparse representation"
  problem. Here we would like to represent a given  vector or signal as a\n
 linear combination of basis vectors (or signals)\, using as few "active" v
 ectors as possible in our representation. For example\, we might wish to\n
 represent the  spectrum of a frame of polyphonic music using a small numbe
 r of "active" note spectra.\n\nIn practice\, we often approxmate this diff
 icult problem by trying to approximate our vector y  with y=Ax\, where A i
 s a basis matrix of column vectors\, and x has minimum 1-norm. This proble
 m  is now a linear program (LP)\, and this type of solution is known as th
 e "Basis Pursuit" (BP)  or "Lasso" method. There are also other methods th
 at build up an approximate sparse  representation one basis vector at a ti
 me\, including\nmatching pursuit (MP)\, or orthogonal  matching pursuit (O
 MP)\, although these do not guarantee to find the 1-norm (BP) solution. It
   is possible to find conditions where MP\, OMP and BP methods give the sa
 me solution as the  "minimum number of vectors" (minimum zero-norm) soluti
 on.\n\nIn this talk\, I will give an overview of some of these methods\, a
 nd investigate the operation  of these from a geometrical perspective. In 
 particular\, we will see that the idea of convex  polytopes\, the generali
 zation of polygons to high dimensions\, will be useful. This will lead  us
  to an algorithm to find sparse representations\, called "Polytope Faces P
 ursuit"\, that  builds up a sparse solution like MP\, but will also find t
 he 1-norm solution eventually.\n
LOCATION:LR5\, Engineering\, Department of
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