Sparse $\ell^q$-regularization of inverse problems with deep learning

We propose a sparse reconstruction framework for solving inverse problems. Opposed to existing sparse reconstruction techniques that are based on linear sparsifying transforms, we train an encoder-decoder network $D \circ E$ with $E$ acting as a nonlinear sparsifying transform. We minimize a Tikhonov functional which used a learned regularization term formed by the $\ell^q$-norm of the encoder coefficients and a penalty for the distance to the data manifold. For this augmented sparse $\ell^q$-approach, we present a full convergence analysis, derive convergence rates and describe a training strategy. As a main ingredient for the analysis we establish the coercivity of the augmented regularization term.