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CATEGORIES:Signal Processing and Communications Lab Seminars
SUMMARY:Geometry of Sparse Representations - Mark Plumbley
\, Center for Digital Music\, Queen Mary Universit
y of London
DTSTART;TZID=Europe/London:20061212T130000
DTEND;TZID=Europe/London:20061212T140000
UID:TALK5887AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/5887
DESCRIPTION:Geometrical considerations can often give us insig
hts or new approaches to tricky problems. One of
these is the "sparse representation" problem. Here
we would like to represent a given vector or sig
nal as a\nlinear combination of basis vectors (or
signals)\, using as few "active" vectors as possib
le in our representation. For example\, we might w
ish to\nrepresent the spectrum of a frame of poly
phonic music using a small number of "active" note
spectra.\n\nIn practice\, we often approxmate thi
s difficult problem by trying to approximate our v
ector y with y=Ax\, where A is a basis matrix of
column vectors\, and x has minimum 1-norm. This pr
oblem is now a linear program (LP)\, and this typ
e of solution is known as the "Basis Pursuit" (BP)
or "Lasso" method. There are also other methods
that build up an approximate sparse representatio
n one basis vector at a time\, 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 cond
itions where MP\, OMP and BP methods give the same
solution as the "minimum number of vectors" (min
imum zero-norm) solution.\n\nIn this talk\, I will
give an overview of some of these methods\, and i
nvestigate the operation of these from a geometri
cal perspective. In particular\, we will see that
the idea of convex polytopes\, the generalization
of polygons to high dimensions\, will be useful.
This will lead us to an algorithm to find sparse
representations\, called "Polytope Faces Pursuit"\
, that builds up a sparse solution like MP\, but
will also find the 1-norm solution eventually.\n
LOCATION:LR5\, Engineering\, Department of
CONTACT:Taylan Cemgil
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