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In this talk I will detail several key properties of L1-analysis regularization for the resolution of linear inverse problems. Most previous theoretical works consider sparse synthesis priors where the sparsity is measured as the norm of the coefficients that synthesize the signal in a given dictionary. In contrast, the more general analysis regularization minimizes the L1 norm of the correlations between the signal and the atoms in the dictionary. The corresponding variational problem includes several well-known regularizations such as the discrete total variation, the fused lasso and sparse correlation with translation invariant wavelets. I will first study the variations of the solution with respect to the observations and the regularization parameter, which enables the computation of the degrees of freedom estimator. I will then give a sufficient condition to ensure that a signal is the unique solution of the analysis regularization when there is no noise in the observations. The same criterion ensures the robustness of the sparse analysis solution to a small noise in the observations. Lastly I will define a stronger condition that ensures robustness to an arbitrary bounded noise. In the special case of synthesis regularization, our contributions recover already known results, that are hence generalized to the analysis setting. I will illustrate these theoretical results on practical examples to study the robustness of the total variation, fused lasso and translation invariant wavelets regularizations. (This is joint work with S. Vaiter, C. Dossal, J. Fadili)
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