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

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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).
    Mathematics Subject Classification: Primary: 47J20, 47A52, 47H99, 68U10; Secondary: 90C26,49J52, 41A25, 26B10.

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