# Unfolded Algorithms for Deep Phase Retrieval
## 1 Introduction
It would then be reasonable to deduce that such a deep neural network can eventually be trained using a small dataset and that the resulting network can be used as an enhanced first-order method to solve the underlying problem at hand [27; 28; 29; 30; 31; 32; 33; 34; 35]. A particular concern with regards to phase retrieval is the usual assumption that the sensing matrix is known a priori. In the existing literature, the design of phase retrieval algorithms has received the primary attention. The design of the measurement matrix, on the other hand, is considerably unique, considering this is the first work to address the joint design problem. A deterministic design of the sensing matrix presents a set of advantages that can counteract the pitfalls that stem from the random selection of matrices. Deterministic designs allow for judicious matrix constructions that can mitigate the efficiency and storage issues that are associated with random matrix designs. In fact, one may consider designing a specially structured sensing matrix that favors low-computational-complexity operations, as well as on-the-fly computation of the entries of the matrix (see [36] and the references therein for a comprehensive overview of the advantages of deterministic matrix design in similar problems). In addition, they can account for the underlying distribution of the signal and system parameters, which is the main shortcoming of a typical random matrix approach. As such, an architecture that can design sensing vectors/matrices in a deterministic setup would be valuable in practical scenarios [37]. Whether astronomy, X-ray imaging, or any other application, designing a sensing matrix for each signal is a difficult task [38; 39]. Existing research on the design of sensing matrices primarily falls within the domain of compressive sensing, with several approaches utilizing learning-based methodologies [40; 41; 42; 43; 44; 45; 46]. In particular, the use of learning techniques like convolutional neural networks and stacked denoising auto-encoders [43; 44; 46] has gained significant attention. For instance, in works such as [47], the measurement matrix is designed using mutual information, but these ideas have not been fully explored for recovery tasks and have not been applied to phase retrieval. A deep architecture capable of simultaneously designing the sensing matrix and performing the recovery task could significantly enhance both the accuracy and efficiency of phase retrieval, surpassing the capabilities of previous methods. Consequently, we propose a deterministic task-specific and data-specific design for the sensing matrix, which can be cascaded to the recovery algorithm to achieve substantial improvements in recovery performance. **Contributions:** In this paper, we propose model-aware deep architectures, referred to as Unfolded Phase Retrieval (UPR)--as a new approach to the problem of phase retrieval. We specifically focus on the conventional phase retrieval problem using IRWF and the sparse phase retrieval problem using SPARTA, both of which have demonstrated significant potential. Furthermore, we propose a general framework for designing task-specific sensing matrices to improve the performance of the underlying recovery algorithm and to outperform multiple state-of-the-art algorithms as a supplementary feature. More explicitly, we consider a joint design of the sensing matrix and the signal processing algorithm and utilize the deep unfolding technique in the process. As a result, our hybrid methodology adopts the advantages of both methodologies. Specifically, the hybrid model-based and data-driven nature of the proposed framework allows for a more efficient utilization of the data for better accuracy and performance. In addition, the resulting model is interpretable (due to its model-based nature) and requires far fewer data and trainable parameters as compared to its black-box, data-driven counterparts (e.g., black-box deep neural networks).