  We propose a sparse reconstruction framework (aNETT) for solving inverse
problems. Opposed to existing sparse reconstruction techniques that are based
on linear sparsifying transforms, we train an autoencoder network $D \circ E$
with $E$ acting as a nonlinear sparsifying transform and minimize a Tikhonov
functional with learned regularizer formed by the $\ell^q$-norm of the encoder
coefficients and a penalty for the distance to the data manifold. We propose a
strategy for training an autoencoder based on a sample set of the underlying
image class such that the autoencoder is independent of the forward operator
and is subsequently adapted to the specific forward model. Numerical results
are presented for sparse view CT, which clearly demonstrate the feasibility,
robustness and the improved generalization capability and stability of aNETT
over post-processing networks.
