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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Representer theorems for ill-posed inverse problem
s: Tikhonov vs. generalized total-variation regula
rization - Michael Unser (EPFL - Ecole Polytechniq
ue Fédérale de Lausanne)
DTSTART;TZID=Europe/London:20170908T095000
DTEND;TZID=Europe/London:20170908T104000
UID:TALK78461AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/78461
DESCRIPTION:In practice\, ill-posed inverse problems are often
dealt with by introducing a suitable regularizati
on functional. The idea is to stabilize the proble
m while promoting "desirable" solutions. Here\, we
are interested in contrasting the effect Tikhonov
vs. total-variation-like regularization. To that
end\, we first consider a discrete setting and pre
sent two representer theorems that characterize th
e solution of general convex minimization problems
subject to $\\ell_2$ vs. $\\ell_1$ regularization
constraints. Next\, we adopt a continuous-domain
formulation where the regularization semi-norm is
a generalized version of total-variation tied to s
ome differential operator L. We prove that the ext
reme points of the corresponding minimization prob
lem are nonuniform L-splines with fewer knots than
the number of measurements. For instance\, when L
is the derivative operator\, then the solution is
piecewise constant\, which confirms a standard ob
servation and explains why the solution is intrins
ically sparse. The powerful aspect of this charact
erization is that it applies to any linear inverse
problem.

LOCATION:Seminar Room 1\, Newton Institute
CONTACT:INI IT
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