Robust Sparse Analysis Regularization
Résumé
This paper studies the properties of L1-analysis regularization for the resolution of linear inverse problems. Most previous works consider sparse synthesis priors where the sparsity is measured as the L1 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 and the fused lasso. We first prove that a solution of analysis regularization is a piecewise affine function of the observations. Similarly, it is a piecewise affine function of the regularization parameter. This allows us to compute the degrees of freedom associated to sparse analysis estimators. Another contribution gives 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. Our last contribution defines a stronger sufficient 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. We illustrate these theoritical results on practical examples to study the robustness of the total variation and the fused lasso regularizations.
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