# Unfolded Algorithms for Deep Phase Retrieval
## 1 Introduction
These same issues arise with the additional body of work performed using convolutional neural networks such as prDeep [23], which ultimately leads to a separate class of architectures not versatile enough to improve existing algorithms due to being built around a large number of extremely specific parameters. In fact, some of [PERSON]'s recent work on space telescopes takes advantage of smart initializations obtained from convolutional neural networks for phase retrieval [24]. Said limitations directly relate to the two major pitfalls of data-driven approaches, i.e., (i) their need for a relatively large amount of data for training purposes and (ii) their inherent lack of interpretability, even after training. In the context of this work, the proposed methodology is a model-driven neural network, meaning that the specific computations that are carried out by the network are in full accordance with the underlying mathematical signal processing algorithm. Still, data-driven approaches such as deep learning techniques have become quite useful in recent years for the handling of challenging signal processing problems. As a result, it is not difficult to see why a hybrid model-based and data-driven model has the potential to improve the data acquisition model and signal recovery algorithms even further by effectively dealing with each approach's weaknesses. For complex systems, such a hybrid model is promising due to its ability to integrate parameterized and non-parameterized mathematical models. In addition, the employment of particular activation functions for model-based deep architectures allows them to be differentiated from conventional deep learning methodologies. Specifically, the activation functions can be designed to mimic non-linearities already present in traditional optimization algorithms. The deep unfolding technique is an effective amalgam of model-based and data-driven approaches. [PERSON] et al. first introduced this notion in [25]; it was then extended thoroughly by [PERSON] et al. in [26]. Deep unfolding facilitates the design of model-aware deep architectures based on well-established iterative signal processing techniques. Deep unfolded networks have repeatedly displayed great promise in the field of signal processing and can make use of the substantial amounts of data, along with the domain knowledge gleaned from the underlying problem at hand [27; 28; 29; 30; 31; 32; 33; 34]. Furthermore, they have the ability to utilize the adaptability and reliability of model-based methods while also taking advantage of the expressive power of deep neural networks. As a result, they are an ideal candidate for problems such as phase retrieval, particularly in non-convex settings, where researchers struggle with bounding the complexity of signal processing algorithms while keeping them effective. This problem particularly arises when applying iterative optimization techniques. In this context, first-order methods are widely used as iterative optimization techniques with low per-iteration complexity. The drawback, on the other hand, stems from their tendency to suffer from a slow speed of convergence, since they usually require many iterations to converge. It should also be noted that predicting the number of iterations required for convergence is generally a difficult task. As a result, they may not be recommended for low-latency and reliable signal processing in real-time applications. Therefore, it is logical to consider fixing the total number of iterations of such algorithms (say, \(L\) iterations) while, at the same time, seeking to optimize the (parameters of the) iterations in order to achieve the best improvement in the underlying objective function at hand. Accordingly, our goal in this paper is to improve existing first-order methods by meta optimizing the SPARTA and IRWF iterations when the total computational budget is fixed.