Representer theorems for ill-posed inverse problems: Tikhonov vs. generalized total-variation regularization

Add to your list(s) Download to your calendar using vCal

• Michael Unser (EPFL - Ecole Polytechnique Fédérale de Lausanne)
• Friday 08 September 2017, 09:50-10:40
• Seminar Room 1, Newton Institute.

If you have a question about this talk, please contact INI IT.

VMVW01 - Variational methods, new optimisation techniques and new fast numerical algorithms

In practice, ill-posed inverse problems are often dealt with by introducing a suitable regularization functional. The idea is to stabilize the problem 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 present two representer theorems that characterize the 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 some differential operator L. We prove that the extreme points of the corresponding minimization problem 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 observation and explains why the solution is intrinsically sparse. The powerful aspect of this characterization is that it applies to any linear inverse problem.

This talk is part of the Isaac Newton Institute Seminar Series series.

This talk is included in these lists:

• All CMS events
• Featured lists
• INI info aggregator
• Isaac Newton Institute Seminar Series
• School of Physical Sciences
• Seminar Room 1, Newton Institute
• bld31

Note that ex-directory lists are not shown.