This paper addresses the problem of sparse data recovery. This is an important task in many signal and image processing areas, and is often solved via ℓ 1 regularization. Nowadays, many efficient algorithms are available to solve the corresponding optimization problem. While their convergence is based on a well established mathematical theory, their practical perfomance is heavily determined by the selection of appropriate hyperparameters (e.g. the stepsize). This choice is often difficult and time consuming. In this paper we propose a deep network architecture based on the unrolling of a primaldual algorithm. This allows us to learn the hyperparameters involved in the algorithm automatically in a data driven way. The proposed network has an interpretable structure where each layer mimics one iteration of the primal-dual algorithm. The method is computationally efficient thanks to deep learning framework accelerations. Through an example arising from spectroscopy signal restoration, we show that unrolling improves the restoration performance with respect to state-ofthe-art, including the iterative implementation of primal-dual algorithm.