  We propose a sparse reconstruction framework for solving inverse problems.
Opposed to existing sparse regularization techniques that are based on frame
representations, we train an encoder-decoder network by including an
$\ell^1$-penalty. We demonstrate that the trained decoder network allows sparse
signal reconstruction using thresholded encoded coefficients without losing
much quality of the original image. Using the sparse synthesis prior, we
propose minimizing the $\ell^1$-Tikhonov functional, which is the sum of a data
fitting term and the $\ell^1$-norm of the synthesis coefficients, and show that
it provides a regularization method.
