Stability of the Minimizers of Least Squares with a Non-Convex Regularization. Part II: Global Behavior |
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Authors: | S. Durand M. Nikolova |
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Affiliation: | (1) LAMFA UMR 6140, Universite de Picardie, 33 rue Saint-Leu, 90039 Amien Cedex, France;(2) CMLA UMR 8536, ENS de Cachan, 61 av. du President Wilson, 94235 Cachan Cedex, France |
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Abstract: | We address estimation problems where the sought-after solution is defined as the minimizer of an objective function composed of a quadratic data-fidelity term and a regularization term. We especially focus on non-convex and possibly non-smooth regularization terms because of their ability to yield good estimates. This work is dedicated to the stability of the minimizers of such piecewise Cm, with m ≥ 2, non-convex objective functions. It is composed of two parts. In the previous part of this work we considered general local minimizers. In this part we derive results on global minimizers. We show that the data domain contains an open, dense subset such that for every data point therein, the objective function has a finite number of local minimizers, and a unique global minimizer. It gives rise to a global minimizer function which is Cm-1 everywhere on an open and dense subset of the data domain. |
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Keywords: | Stability analysis Regularized least squares Non-smooth analysis Non-convex analysis Signal and image processing |
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