## 1 Introduction  
In recent years, sampling theory has made significant progress [1], which has a great impact on signal processing and communication. More and more scholars ignore [PERSON]'s sampling law and use a series of filtering operations to accurately recover the signal from its linear sample. As we know, Fourier analysis and wavelet analysis are essential for these signal processing techniques, and the mathematical tools involving Fourier and wavelet analysis for sampling and reconstruction are very important [2; 3]. Reference [2] used Riesz bases and frames to study stable reconstructions and generalize the concept of oblique double frames to the infinite-dimensional framework. The authors of Reference [3] developed a redundant sampling procedure that can be used to reduce quantization errors when quantizing measurements before reconstruction. Concerning reconstruction techniques taking advantage of sparsity in general, the basic work of processing noise measurements and non-ideal sampling is an important part of recovering unknown signals from random measurements [4; 5]. The problem of linear reconstruction based on Fourieranalysis has also been recently considered [6; 7; 8; 9]. For signal reconstruction techniques that are not based on sparsity assumptions, the random interpolation of linear interpolation can solve this problem. The first attempt to stochastically characterize the sampling via point process (PP) was done in Reference [10], where sample positions are the outcome of a Poisson PP (independently scattered points). Extensions of such a result to the multidimensional domain can be found in References [11; 12; 13] for homogeneous and inhomogeneous sampling with uncertainties, while PPs taking stochastic repulsion/attraction into account (i.e., the Ginibre PP and the Gauss-Poisson PP) have been recently considered [14; 15]. In addition, the machine learning approach based on Gaussian random processes can also be used to solve signal reconstruction problems [16; 17; 18; 19]. In this paper, the method we use to reconstruct the signal is phase retrieval. Phase retrieval is a typical signal reconstruction problem which seeks to recover a signal or image from the magnitude of linear measurements [20]. This problem occurs in many application areas, such as crystallography [21], optical imaging [22], waveform optimization [23], and other areas [24; 25; 26]. In general, because of the loss of Fourier phase information, the problem of phase retrieval is ill-posed and non-convex. The existing approach to address this problem involves exploiting prior knowledge of the signal domain [27; 28; 29; 30]. Here we focus on the recovery of one-dimensional sparse signals from their Fourier measurements, which has been emphasized in recent works. The most popular method for phase retrieval is based on the use of alternating projections between the different constraints in time and the Fourier magnitude constraints [31; 32; 33], pioneered by [PERSON] and [PERSON], and later extended by [PERSON]. The Gerchberg and Saxton (GS) algorithm in [32] uses image amplitude and Fourier transform spectral amplitude as a priori information to reconstruct the image by alternately applying amplitude constraints in the object and frequency domains. However, the GS algorithm converges slowly and is prone to stagnation. [PERSON] et al. have improved the GS algorithm and proposed the addition of error reduction (ER) and hybrid input-output (HIO) [31]. ER and HIO algorithms are used to determine the image pixel estimation value obtained in the object domain by iteratively alternating projection in the object domain and the frequency domain, setting the pixel value that does not satisfy the constraint to zero, and replacing the image Fourier amplitude with the measured value in the frequency domain. Another state-of-the-art method uses the semi-definite programming (SDP) technique and low-rank matrix recovery framework [34; 35; 36]. [PERSON] et al.