# Unfolded Algorithms for Deep Phase Retrieval
## 4 Numerical Results and Discussion
### Performance of the Proposed UPR-IRWF  
In this part, we investigate the performance of the proposed UPR-IRWF specifically tailored for conventional phase retrieval applications where no prior knowledge is assumed on the underlying signal of interest (e.g., sparsity). Similar to the previous subsection, we focus on the recovery of a real-valued Gaussian signal (\(\mathbf{x}\in\mathds{R}^{n}\)) from the phase-less measurements. In particular, we employ the UPR-IRWF framework with \(L=50\) layers (iterations), and we let the baseline IRWF algorithm run for the same number of iterations for a fair comparison. For the first experiment, we evaluate the performance of the proposed UPR-IRWF in terms of its ESR versus measurement-to-signal length ratio \((m/n)\) and compare its performance with that of the standard state-of-the-art IRWF algorithm in the following scenarios:  
* _Case 1_: The standard IRWF algorithm with a randomly generated measurement matrix (\(\mathbf{A}\sim\mathcal{N}(\mathbf{0},\mathbf{I})\)) and a fixed step size for the iterations as reported in [18]. * _Case 2_: The UPR-IRWF algorithm with a learned measurement matrix (\(\mathbf{A}\)) and a fixed step size whose value is similar to the previous case. * _Case 3_: The UPR-IRWF algorithm with the learned preconditioning matrices (\(\{\mathbf{G}_{i}\}_{i}\)) for a fixed, randomly generated (\(\mathbf{A}\)), as in _Case 1_. * _Case 4_: The UPR-IRWF algorithm with the learned measurement matrix (\(\mathbf{A}\)) and preconditioning matrices (\(\{\mathbf{G}_{i}\}_{i}\)). Figure 4 exhibits the ESR versus measurement-to signal length ratio \((m/n)\) when the signal length is fixed at \(n=100\). Figure 4 illustrates that the proposed methodology significantly outperforms the baseline IRWF algorithm in terms of ESR in all scenarios. As for Case 3, in which we consider a scenario where the measurement matrix is fixed (i.e., imposed by the physics of the system) and known a priori, the learning of the preconditioning matrices significantly improves the performance of the underlying signal recovery algorithm. This supports the proposition that a judicious design of the preconditioning matrices, when the number of layers (iterations) is fixed, can, indeed, result in accelerated convergence, immensely improving the performance. On the other hand, Case 2, in which we only learn the measurement matrix while employing a fixed scalar step size, further  
Figure 3: Empirical success rate versus the sparsity level (\(k\)) for \(\mathbf{x}\in\mathds{R}^{n}\), where \(n=m=150\). reveals the effectiveness of learning a proper task-specific, data-driven measurement matrix akin to the problem of interest. Finally, it is evident that the proposed UPR-IRWF with both learned \(\mathbf{A}\) and \(\{\mathbf{G}_{i}\}_{i}\) outperforms all other cases. Now, we numerically investigate the convergence properties of the proposed UPR-IRWF as compared to the standard IRWF algorithm via per-layer relative MSE analysis. We, again, stress that such an evaluation is possible due to the model-based nature of the proposed methodology--something that cannot be achieved when using conventional, purely data-driven black-box techniques. For this experiment, we set the signal dimension and number of measurements as \((n,m)=(100,600)\). Figure 5 demonstrates the relative MSE versus iteration (layer) number. It can be clearly observed from Figure 5 that the proposed methodology benefits from accelerated convergence properties and is able to converge to an optimal point of the objective function with as few as \(L=30\) layers. This, in turn, reveals that the proposed methodology can be further truncated to have many fewer iterations, reducing the computational cost of the overall algorithm. Table 1 displays the number of layers/iterations it takes to achieve consistent successful recovery trials for the baseline IRWF and SPARTA algorithms, along with their UPR counterparts.