Sparse synthesis regularization with deep neural networks

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.