Gradient-Based Learning of Discrete Structured Measurement Operators for Signal Recovery

TL;DR

This paper introduces GLODISMO, a gradient-based method combining unrolled optimization and Gumbel reparametrizations to learn discrete structured measurement operators, improving signal recovery performance.

Contribution

It presents a novel, efficient approach for learning discrete structured measurement operators using gradient-based methods, extending unrolled optimization with Gumbel reparametrizations.

Findings

Learned measurement matrices outperform random and baseline designs

Method is easy to implement and compatible with automatic differentiation

Demonstrated effectiveness across multiple signal recovery applications

Abstract

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.