Robust Sparse Analysis Regularization

This paper investigates the theoretical guarantees of
$\ell^{1}$
-analysis regularization when solving linear inverse problems. Most of previous works in the literature have mainly focused on the sparse synthesis prior where the sparsity is measured as the
$\ell^{1}$
norm of the coefficients that synthesize the signal from a given dictionary. In contrast, the more general analysis regularization minimizes the
$\ell^{1}$
norm of the correlations between the signal and the atoms in the dictionary, where these correlations define the analysis support. The corresponding variational problem encompasses several well-known regularizations such as the discrete total variation and the fused Lasso. Our main contributions consist in deriving sufficient conditions that guarantee exact or partial analysis support recovery of the true signal in presence of noise. More precisely, we give a sufficient condition to ensure that a signal is the unique solution of the
$\ell^{1}$
-analysis regularization in the noiseless case. The same condition also guarantees exact analysis support recovery and
$\ell^{2}$
-robustness of the
$\ell^{1}$
-analysis minimizer vis-à-vis an enough small noise in the measurements. This condition turns to be sharp for the robustness of the sign pattern. To show partial support recovery and
$\ell^{2}$
-robustness to an arbitrary bounded noise, we introduce a stronger sufficient condition. When specialized to the
$\ell^{1}$
-synthesis regularization, our results recover some corresponding recovery and robustness guarantees previously known in the literature. From this perspective, our work is a generalization of these results. We finally illustrate these theoretical findings on several examples to study the robustness of the 1-D total variation, shift-invariant Haar dictionary, and fused Lasso regularizations.