## 1 Introduction
However, this type of random selection can lead to information loss and performance degradation as demonstrated in [17; 18]. Information theory is another widely used framework for optimizing the CMM in massive MIMO systems. These principles of information theory provide a mathematical framework for quantifying the amount of information that can be transmitted over a communication channel. In [18; 19], the CMM is optimized by maximizing the mutual information between the compressed measurement and the signal DOAs. This approach is based on considering the availability of a coarse prior distribution of the DOAs. By reducing the dimension of the received signal, the required number of front-end circuits is effectively reduced with minimal performance loss. Reference [20] extends this idea by developing a general compressive measurement scheme that combines the CMM and the sparse array. The framework can consider any arbitrary sparse array as the receive antennas and use the CMM to compress the dimension. As a result, it can effectively reduce both the number of physical antennas and the front-end circuits. They also optimize the CMM by maximizing the mutual information of the compressed measurements and DOA distribution, while considering the availability of the prior distribution of DOAs. In many practical cases, however, the required a priori distribution may not available. To address this issue, an iterative optimization approach is developed in [21]. Starting with no prior information on the DOA distribution, the CMM is optimized based on mutual information maximization and then used to estimate the DOA spectrum. The estimated normalized DOA spectrum is subsequently used as the prior information for the next iteration, thus iteratively improving the accuracy of the estimated DOA spectrum. Optimizing the CMM in a sequential adaptive manner may lead to better performance compared to non-adaptive schemes [22; 23]. However, using optimization techniques, such as projected gradient descent or simplified versions of projected coordinate descent, to obtain the desired CMM can be computationally expensive [24]. On the other hand, codebook-based methods, such as the hierarchical codebook developed in [23] and the hierarchical posterior matching (hiePM) strategy developed in [5], can reduce the computational burden. Nonetheless, the performance of codebook-based methods relies heavily on the quality of the codebooks and may be inferior to codebook-free approaches. Recently, deep learning methods have emerged as a popular approach for effectively addressing complex optimization problems in various wireless communication and signal processing applications, including massive MIMO beamforming [25; 26], intelligent reflecting surface [27; 28], DOA estimation [29; 30], and wireless channel estimation [31; 32; 33]. In a prior study [34], we developed a deep learning method for sequentially updating the CMM. Specifically, we trained a neural network without any prior information to obtain the optimized CMM, which was then used to update the posterior distribution of signal DOAs by leveraging the subsequent measurement. However, this approach faces two challenging issues. First, for each snapshot of the impinging signal, the CMM must be updated, the compressed measurement computed, and the posterior distribution updated. As such, it incurs high-computational costs, especially for updating the posterior distribution at each snapshot. Second, the posterior update relies on the accuracy of the estimated spatial spectrum, and any inaccuracies in this estimation can lead to performance degradation and slow convergence. Conversely, any inaccuracy or change in the posterior estimation will affect the spectrum estimation performance. In [35], LSTM neural networks are used in various communication system problems, including adaptive beamformer design for mmWave initial beam alignment applications. However, this study was limited to single-channel and single-user scenarios.