Loss Function Design for Deep Unfolded Sparse Signal Recovery: Supervised and Unsupervised Learning

This paper investigates the impact of loss function design in deep unfolding techniques for sparse signal recovery algorithms. We focus on deep unfolded versions of the fundamental iterative shrinkage thresholding algorithm (ISTA) and the iterative hard thresholding algorithm (IHT). To obtain a guideline for the loss function design, we examine the effect of supervised learning using mean squared error and unsupervised learning using the objective function of the original optimization problem. Our simulation results reveal that the effect of loss function design significantly depends on the convexity of the optimization problem. For convex $\ell_1$-regularized problems, supervised-ISTA achieves better final recovery accuracy but fails to minimize the original objective function, whereas we empirically observe that unsupervised-ISTA converges to a nearly identical solution as conventional ISTA but with accelerated convergence. Conversely, for nonconvex $\ell_0$-regularized problems, both supervised-IHT and unsupervised-IHT converge to better local minima than the original IHT, showing similar performance under the training conditions regardless of the loss function employed. However, when the test conditions differ from the training conditions, our results suggest that unsupervised learning offers better robustness to distribution mismatch. These findings provide valuable insights into the design of effective deep unfolded networks for sparse signal recovery applications.