# Unfolded Algorithms for Deep Phase Retrieval
## 2 System Model and Problem Formulation  
In this section, we provide a mathematical formulation of the phase retrieval problem. Additionally, we present a summary of existing classical model-based phase retrieval algorithms, setting the stage for the proposed hybrid model-based and data-driven methodology. A phase retrieval system can be mathematically formalized by examining the following encoding module (i.e., the data-acquisition system):  
\[\text{Encoding Module:}\quad\mathbf{y}=|\mathbf{Ax}|, \tag{2}\]where \(\mathbf{x}\in\mathbb{C}^{n}\) denotes the underlying signal of interest, \(\mathbf{A}=[\mathbf{a}_{1},\mathbf{a}_{2},\cdots,\mathbf{a}_{m}]\in\mathbb{C}^{n\times m}\) denotes the sensing matrix, and \(\mathbf{y}\in\mathbb{R}^{m}\) represents the captured phase-less measurements of the signal (\(\mathbf{x}\)). In a traditional phase retrieval problem, the objective is to recover the signal of interest (\(\mathbf{x}\)) from the phase-less measurements (\(\mathbf{y}\)), given the sensing matrix (\(\mathbf{A}\)). Keeping the non-convexity of the problem in mind, observe that if \(\mathbf{x}^{\dagger}\) is a feasible point, then \(e^{-j\gamma}\mathbf{x}^{\dagger}\) is also a solution to the problem for an arbitrary phase constant (\(\gamma\)). Hence, it is only possible to recover the underlying signal of interest up to a phase shift factor in a phase retrieval problem. With that in mind, let \(\mathbf{x}^{\dagger}\) be any solution to the phase retrieval problem, and let \(\mathbf{x}^{\diamond}\) represent the true signal of interest. A meaningful quantifying metric of the closeness of the recovered signal (\(\mathbf{x}^{\diamond}\)) to the true signal can then be defined as follows:  
\[\mathrm{D}(\mathbf{x}^{\diamond},\mathbf{x}^{\dagger})=\min_{\gamma\in[0,2\pi)}\|\mathbf{ x}^{\diamond}-e^{-j\gamma}\mathbf{x}^{\dagger}\|_{2}^{2}, \tag{3}\]  
which was first proposed in [15] and can be interpreted as measuring the Euclidean distance of two complex vectors up to a global phase difference. A system design approach for a phase retrieval problem is mainly concerned with finding the best set of parameters of the encoder module that facilitate the recovery of the underlying signal of interest from the phase-less measurements captured through the encoder module (i.e., the data-acquisition system). In particular, a system design perspective for a phase retrieval data-acquisition system can be defined as designing the sensing matrix in a deterministic fashion according to a performance criterion. As an example, Ref. [47] considered the design of the sensing matrix in a deterministic manner, maximizing the mutual information between the signal of interest and the acquired phase-less measurements. However, the development of effective signal reconstruction techniques that can harness this maximal mutual information obtained by a judicious design of the sensing matrix is still an open problem. Hence, a more natural approach to this problem is to consider a joint design of the sensing matrix and the recovery algorithm. We seek to jointly design the sensing matrix (encoder module) and the reconstruction algorithm (decoder module), in contrast to existing methodologies that either consider the development of the reconstruction algorithm or the design of the sensing matrix. Specifically, we propose a hybrid model-based and data-driven methodology for this problem and utilize the notions of auto-encoders and the deep unfolding technique as the main tools to develop our proposed framework. By implementing such a methodology, while the sensing matrix is designed to maximize the signal reconstruction accuracy of the decoder module, the decoder module itself is trained to perform recovery by accounting for the learned sensing matrix, leading to the promised joint design. Furthermore, due to the auto-encoder structure of the proposed method, the recovery algorithm has the crucial role of guiding the optimization of the sensing matrix in the corresponding feasible space.