Analytical bounds on the minimizers of (nonconvex) regularized least-squares

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February 2008, 2(1): 133-149. doi: 10.3934/ipi.2008.2.133

Analytical bounds on the minimizers of (nonconvex) regularized least-squares

1.	CMLA, ENS Cachan, CNRS, PRES UniverSud, 61 Av. President Wilson, F-94230 Cachan

Received April 2007 Published January 2008

Abstract
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This is a theoretical study on the minimizers of cost-functions composed of an l ₂ data-fidelity term and a possibly nonsmooth or nonconvex regularization term acting on the differences or the discrete gradients of the image or the signal to restore. More precisely, we derive general nonasymptotic analytical bounds characterizing the local and the global minimizers of these cost-functions. We first derive bounds that compare the restored data with the noisy data. For edge-preserving regularization, we exhibit a tight data-independent bound on the l _∞ norm of the residual (the estimate of the noise), even if its l ₂ norm is being minimized. Then we focus on the smoothing incurred by the (local) minimizers in terms of the differences or the discrete gradient of the restored image (or signal).

Keywords: Regularization, Non-convex analysis, Inverse problems, Variational methods, Edge restoration, Image restoration, Signal restoration, Non-smooth analysis..

Mathematics Subject Classification: Primary: 47J20, 47A52, 47H99, 68U10; Secondary: 90C26,49J52, 41A25, 26B1.

Citation: Mila Nikolova. Analytical bounds on the minimizers of (nonconvex) regularized least-squares. Inverse Problems & Imaging, 2008, 2 (1) : 133-149. doi: 10.3934/ipi.2008.2.133

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