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Shearlet-based regularization in statistical inverse learning with an application to X-ray tomography

Statistical inverse learning theory, a field that lies at the intersection of inverse problems and statistical learning, has lately gained more and more attention. In an effort to steer this interplay more towards the variational regularization framework, convergence rates have recently been proved for a class of convex, $p$-homogeneous regularizers with $p \in (1,2]$, in the symmetric Bregman distance. Following this path, we take a further step towards the study of sparsity-promoting regularization and extend the aforementioned convergence rates to work with $\ell^p$-norm regularization, with $p \in (1,2)$, for a special class of non-tight Banach frames, called shearlets, and possibly constrained to some convex set. The $p = 1$ case is approached as the limit case $(1,2) \ni p \rightarrow 1$, by complementing numerical evidence with a (partial) theoretical analysis, based on arguments from $Γ$-convergence theory. We numerically demonstrate our theoretical results in the context of X-ray tomography, under random sampling of the imaging angles, using both simulated and measured data.