# Unfolded Algorithms for Deep Phase Retrieval
## 1 Introduction  
Phase retrieval focuses on recovering a complex or real-valued signal of interest (\(\mathbf{x}\in\mathbb{R}^{n}/\mathbb{C}^{n}\)) from \(m\) linear phase-less measurements, typically in the form of  
\[y_{i}=|\left\langle\mathbf{a}_{i};\mathbf{x}\right\rangle|,\text{ for }i\in\{1,2, \cdots,m\}, \tag{1}\]  
where the set of sensing vectors \(\{\mathbf{a}_{i}\in\mathbb{R}^{n}/\mathbb{C}^{n}\}_{i=1}^{m}\) is assumed to be known _a priori_. The long-standing phase retrieval problem has fostered the development of countless methodologies, algorithms, and systems in an attempt to solve its many challenges. This is partially due to phase retrieval's applications in numerous fields, such as imaging and optics [1, 2]. Direct applications of phase retrieval have become integral, even in space exploration, where they assist space telescopes like NASA's Hubble Space Telescope (HST) and James Webb Space Telescope (JWST) [3, 4, 5, 6]. Many modern methodologies and applications of phase retrieval stem from the pioneering works of [PERSON], [PERSON], and [PERSON] [7, 8, 9, 10, 11], which are not only still widely referenced in research but are also still implemented in various forms in applications such as the recently launched JWST [4]. Despite its impact, the Gerchberg-Saxton algorithm's struggles with efficiently finding the optimal solution have spawned a myriad of new research avenues. Convex methodologies have faced challenges in this area [12, 13, 14], leading to the development of alternative approaches thattackle the phase retrieval problem through the use of a non-convex lens. This is clearly demonstrated by the Wirtinger flow (WF) ecosystem of algorithms, where WF and its extensions, such as reshaped Wirtinger flow (RWF), truncated Wirtinger flow (TWF), and incremental reshaped Wirtinger flow (ITWF), have shown the potential to address the problem accurately and efficiently [15; 16; 17; 18]. The Wirtinger flow-based algorithms' ability to demonstrate the effectiveness of spectral initialization and updating has led to continuous improvements in initialization and updating for methodologies like incremental reshaped Wirtinger flow (IRWF) [15; 18]. Approaches such as TWF address the challenge of the high number of required measurements in the phase retrieval problem by using block-sparse solvers [17]. This is particularly relevant in scenarios where the signal is either inherently sparse or can be made sparse through design, a characteristic common in many practical applications [19; 20]. Performance has been further enhanced by algorithms like Sparse Truncated Amplitude Flow (SPARTA) [20], which utilizes power iterations for initialization and truncated gradient calculations via thresholds, proving to be highly effective. However, many well-performing phase retrieval algorithms, including improved versions of Wirtinger flow, struggle to optimize internal parameters [18]. This issue is especially evident when determining the optimal step size for signal recovery in cases of small sample sizes, where many approaches tend to fall into local minima. As a result, computational inefficiency remains a significant challenge in applying these phase retrieval algorithms to large-scale or real-time scenarios. An additional categorization of phase retrieval algorithms is rooted in their model-based or data-driven nature. The previously discussed methodologies are model-based, where the goal is to design algorithms through a preliminary modeling of the problem, incorporating the reasoning that emerges from the model. Data-driven approaches, on the other hand, rely on data to validate the functionality of the algorithms and harness the representational capabilities of deep neural networks. These networks are trained in such a way that, given a measurement vector (\(\mathbf{y}=|\mathbf{Ax}|\),) the resulting model functions as an estimator of the true signal. Fienup's method and hybrid input-output (HIO) are among the few phase retrieval algorithms that have incorporated deep learning techniques [21; 22].