  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.
