BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:Representer theorems for ill-posed inverse problems: Tikhonov vs. generalized total-variation regularization - Michael Unser (EPFL - Ecole P olytechnique Fédérale de Lausanne) DTSTART:20170908T085000Z DTEND:20170908T094000Z UID:TALK78461@talks.cam.ac.uk CONTACT:INI IT DESCRIPTION:In practice\, ill-posed inverse problems are often dealt with by introducing a suitable regularization functional. The idea is to stabil ize the problem while promoting "desirable" solutions. Here\, we are inter ested in contrasting the effect Tikhonov vs. total-variation-like regulari zation. To that end\, we first consider a discrete setting and present two representer theorems that characterize the solution of general convex min imization problems subject to $\\ell_2$ vs. $\\ell_1$ regularization const raints. Next\, we adopt a continuous-domain formulation where the regulari zation semi-norm is a generalized version of total-variation tied to some differential operator L. We prove that the extreme points of the correspon ding minimization problem are nonuniform L-splines with fewer knots than t he number of measurements. For instance\, when L is the derivative operato r\, then the solution is piecewise constant\, which confirms a standard ob servation and explains why the solution is intrinsically sparse. The power ful aspect of this characterization is that it applies to any linear inver se problem.
LOCATION:Seminar Room 1\, Newton Institute END:VEVENT END:VCALENDAR