Learning Structured Sparse Matrices for Signal Recovery via Unrolled Optimization

Countless signal processing applications include the reconstruction of an unknown signal from very few indirect linear measurements. Because the measurement operator is commonly constrained by the hardware or the physics of the observation process, finding measurement matrices that enable accurate signal recovery poses a challenging discrete optimization task. Meanwhile, recent advances in the field of machine learning have highlighted the effectiveness of gradient-based optimization methods applied to large computational graphs such as those arising naturally when unrolling iterative algorithms for signal recovery. However, it has remained unclear how to leverage this technique when the set of admissible measurement matrices is both discrete and sparse. In this paper, we tackle this problem and propose an efficient and flexible method for learning structured sparse measurement matrices. Our approach uses unrolled optimization in conjunction with Gumbel reparametrizations. We empirically demonstrate the effectiveness of our method in two prototypical compressed sensing situations.