Countless signal processing applications include the reconstruction of signals from few indirect linear measurements. The design of effective measurement operators is typically constrained by the underlying hardware and physics, posing a challenging and often even discrete optimization task. While the potential of gradient-based learning via the unrolling of iterative recovery algorithms has been demonstrated, it has remained unclear how to leverage this technique when the set of admissible measurement operators is structured and discrete. We tackle this problem by combining unrolled optimization with Gumbel reparametrizations, which enable the computation of low-variance gradient estimates of categorical random variables. Our approach is formalized by GLODISMO (Gradient-based Learning of DIscrete Structured Measurement Operators). This novel method is easy-to-implement, computationally efficient, and extendable due to its compatibility with automatic differentiation. We empirically demonstrate the performance and flexibility of GLODISMO in several prototypical signal recovery applications, verifying that the learned measurement matrices outperform conventional designs based on randomization as well as discrete optimization baselines.
