  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 $\Gamma$-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.
