# Unfolded Algorithms for Deep Phase Retrieval
## 4 Numerical Results and Discussion
### Performance of the Proposed UPR-SPARTA  
In this part, we evaluate the performance of the proposed UPR-SPARTA framework for the case of recovering a \(k\)-sparse real-valued Gaussian signal. For training purposes, we generate a training dataset of size 2048, where each data point \(\mathbf{x}\in\mathbb{R}^{n}\) in the dataset follows a standard Gaussian distribution whose \((n-k)\) entries set to zero are chosen randomly and uniformly. Accordingly, a testing dataset of size 2048 with the same settings is generated to evaluate the performance of the proposed UPR-SPARTA and the baseline SPARTA algorithm. For all experiments in this part, UPR-SPARTA is implemented with \(L=20\) layers (iterations), and we let the baseline SPARTA algorithm run for the same number of iterations. We first provide the exact recovery performance of UPR-SPARTA in terms of the previously define ESR metric based on 100 Monte Carlo trials. For this purpose, we set the signal dimension to \(n=100\) and the sparsity level to \(k=5\), while the ratio \((m/n)\) increases from 0.1 to 3.0. In order to show the effectiveness of the learned parameters, we compare the performance of the proposed approach with that of the standard SPARTA algorithm in the following scenarios:* _Case 1_: The standard SPARTA algorithm with a randomly generated measurement matrix (\(A\sim\mathcal{N}(\mathbf{0},I)\)) and a fixed step size of \(G^{i}=\mu I\), where \(\mu=1\) for the iterations as reported in [20]. * _Case 2_: The UPR-SPARTA algorithm with a learned measurement matrix (\(A\)) and a fixed step size whose value is similar to the previous case. * _Case 3_: The UPR-SPARTA algorithm with the learned pre-conditioning matrices (\(\{G_{i}\}_{i}\)) for a fixed, randomly generated \(A\), as in _Case 1_. * _Case 4_: The UPR-SPARTA algorithm with the learned measurement matrix (\(A\)) and pre-conditioning matrices (\(\{G_{i}\}_{i}\)). It should be noted that Cases 1-4 for both algorithms in this section are gradual steps in that they start from the baseline algorithm and make their way to multiple learned parameters in an unfolded network. An unfolded version of just the algorithm without learning any parameters would constitute a reorganization of the way the algorithm is computed and would not yield meaningful results. In other words, a lack of learning would only involve unfolding iterations of the algorithm into layers of the deep network. Figure 1 demonstrates the ESR versus the number of measurements \((m/n)\) for the above scenario. It can be observed from Figure 1 that the proposed methodology significantly outperforms the standard state-of-the-art SPARTA algorithm in recovering a sparse signal in all considered cases, which is further evidence for the effectiveness of the learned parameters. Moreover, a comparison between the UPR-SPARTA with learned preconditioning matrices (Case 2) and the standard SPARTA algorithm reveals that for applications where the measurement matrix is imposed by the physics of the system, the learning of the preconditioning matrices can significantly increase the performance of the recovery. On the other hand, considering a case of learning over the measurement matrix (\(A\)) only and employing fixed step sizes (Case 3) further indicates the effectiveness of learning task-specific measurement matrices tailored to the application at hand and can significantly increase the performance of the recovery algorithm. Finally, it can be observed from Figure 1 that the proposed UPR-SPARTA, while allowing for learning over all system parameters, outperforms all other cases and achieves a significantly higher ESR metric along all measurement ratios \((m/n)\). We note that such a significant gain achieved by the proposed approach for Cases 2-4 is due to the hybrid model-based and data-driven nature of the proposed methodology. We further note that learning the preconditioning matrices must lead to the acceleration of the convergence of the proposed methodology.