Convex Regularization and Representer Theorems

12/11/2018

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by   Claire Boyer, et al.

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We establish a result which states that regularizing an inverse problem with the gauge of a convex set C yields solutions which are linear combinations of a few extreme points or elements of the extreme rays of C. These can be understood as the atoms of the regularizer. We then explicit that general principle by using a few popular applications. In particular, we relate it to the common wisdom that total gradient variation minimization favors the reconstruction of piecewise constant images.