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

Presented by: 
Michael Unser
Friday 8th September 2017 - 09:50 to 10:40
INI Seminar Room 1
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.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons