## 1 Introduction
### Previous Research and Limitations  
The issue of channel estimation has attracted broad interest in the past few years, and the existing algorithms can be classified as least square (LS)/minimum mean square error (MMSE)-based, compressive sensing (CS)-based, message passing (MP)-based, deep learning (DL)-based, or VI-based [10; 11; 12]. Conventional channel estimation algorithms typically include LS, MMSE, or expectation maximization (EM)-based methods. In 2005, [PERSON] et al. [13] researched into iterative LS-based channel estimation algorithms, and showed that LS works well in mobile networks. In 2010, [PERSON] et al. [14] proposed an approximate linear minimum mean square error (ALMMSE) fast fading CE algorithm. In 2012, [PERSON] et al. [15] proposed a soft-output MMSE channel estimation algorithm under correlated Rician fading MIMO channels. In 2013, [PERSON] et al. [16] proposed a distributed angle estimation (or channel estimation) algorithm based on space-alternating generalized expectation maximization. Compared with the LS method, MMSE, including linear minimum mean square error (LMMSE, [17]), considers noise and therefore improves the accuracy. However, with matrix inverse computations, LMMSE has higher computation complexity. Therefore, the conventional channel estimation algorithms (LS, MMSE, etc.) are obviously flawed with their problems of high complexity and inability to scale, which can rapidly lead to failure in massive MIMO scenarios. In a massive MIMO, as the number of antennas in each base station (BS) increases, the downlink pilot training overhead and uplink channel state information (CSI) feedback overhead can quickly become overwhelming. Therefore, the pilots for evaluating CSI are inadequate. A CS-based approach can be used to solve the above issue with conventional channel estimation algorithms. From the simulated study we can see that limited by the number of scatterers in the BS, the user channel matrix tends to be sparse in massive MIMO scenarios. In this case, it is ineffective to use long training symbols to estimate CSI. CS can mitigate this disadvantage and make use ofuser channel matrix sparsity. In 2014, [PERSON] et al. [18] proposed a distributed compressive transmitter side channel state information (CSIT) estimation scheme. In their work, they locally observed the compressed measurements while recovering the CSIT at the base station. In 2016, [PERSON] et al. [19] proposed a new CS-based downlink CSI estimation scheme for frequency division duplexing (FDD) massive MIMO systems (CE for FDD). However, there are still challenges in CS-based channel estimation algorithms, such as the difficulty to exploit joint channel sparsity among multiple users, and to decide how sparsity can affect performance. Although some researchers have tried to analyze this issue, in general, these are specific analyses, and thus cannot be extended. Owing to its convergence guarantee property and distributed nature, the MP algorithm is also used to implement channel estimation. In 2005, [PERSON] et al. [20] proposed a robust distributed sensor network inference architecture and exploited MP to compute the global parameter after forming a spanning tree and a junction tree. In 2019, [PERSON] et al. [21] proposed a generalized approximate message passing (GAMP) algorithm to find the ground truth of the mmWave MIMO channel coefficient matrix. However, MP-based CE algorithms are too specific to generalize in different types of networks. Some researchers use DL to estimate channel due to its strong computing power. In 2017, [PERSON] et al. [22] proposed a DL-based CE algorithm to train a deep neural network (DNN) under different channel conditions, and then applied it online to recover the transmitted data. In 2018, [PERSON] et al. [23] proposed a DL-based channel estimation for beam-space mmWave massive MIMO systems. It is interesting that the authors also use a learned denoising-based approximate message passing network (LDAMP), i.e., they combined the use of DL and MP. In 2020, [PERSON] et al.