Model-Driven Deep Learning for Massive Multiuser MIMO Constant Envelope Precoding

RMO has been proposed to solve the optimization problem (3) in [5], which transforms the constrained domain into a Riemannian manifold and solves the optimization problem directly on this specific manifold. We briefly review the RMO-based CG algorithm and recommend [14] for technical details about the RMO method.

Considering the constant amplitude constraint for each element of ${\mathbf{x}}$ in problem (3) and assuming ${P_{\rm t}=1}$, the constraint domain of CE precoding problem can be transformed into a Riemannian manifold given by

Given a point ${{\mathbf{x}_{k}}\in\mathcal{M}}$ of the $k$th iteration, the tangent space at point ${\mathbf{x}_{k}}$ is defined as ${\mathcal{T}_{{{\mathbf{x}}_{k}}}}\mathcal{M}=\{\mathbf{z}\in{{\mathbb{C}}^{{{N}_{\rm t}}}}:\mathfrak{Re}\{\mathbf{z}\circ\mathbf{x}_{k}^{*}\}={{\mathbf{0}}_{{{N}_{\rm t}}}}\}$. The search step size and direction of the $k$th iteration are assumed as ${{\alpha}_{k}}$ and ${\mathbf{d}_{k}\in\mathcal{T}_{{\mathbf{x}}_{k}}\mathcal{M}}$, respectively. The point ${\mathbf{x}_{k+1}}$ is obtained by projecting the point ${\mathbf{x}_{k}+{{\alpha}_{k}\mathbf{d}_{k}}}$ back to the manifold as follows:

Next, we introduce how to determine the search direction and step size. Specifically, the CG algorithm is used to determine the search direction. The gradient direction in Euclidean space is denoted by ${{\nabla f({{\mathbf{x}}_{k}})=-2{{\mathbf{H}}^{\rm H}}(\mathbf{s}-\mathbf{H}{{\mathbf{x}}_{k}})}}$, which should be projected onto the tangent space at point ${\mathbf{x}_{k}}$ as

Similarly, the search direction ${{\mathbf{d}_{k-1}}\in\mathcal{T}_{{\mathbf{x}}_{k-1}}\mathcal{M}}$ also needs to be projected onto the tangent plane at point ${\mathbf{x}_{k}}$ as

Then, the search direction ${{\mathbf{d}_{k}}}$ is given by

where ${{\beta}_{k}}$ is the weight calculated by Polak-Ribière formula as

In addition, the search step size ${{\alpha}_{k}}$ restricted to the direction ${{\mathbf{d}_{k}}}$ can be determined according to Armijo backtracking line search rule, that is,

where ${{{c}_{1}}\in(0,~{}1)}$. A large step ${{\alpha}_{k}}$ is usually initialized and attenuated by a factor of ${\tau}$ until (10) is satisfied. The RMO-based CE precoding method is summarized in Algorithm 1. We elaborate the RMO-based CEPNet design in the following subsection.

Two primary factors can increase the computational overhead of Algorithm 1 significantly. First, the step that determines the search step size ${{\alpha}_{k}}$ takes up almost half the time of one iteration, indicating an excessive and unbearable latency overhead when the number of iterations is large. Second, the projection and retraction operations in Algorithm 1 affect the convergence speed of the CG algorithm, thereby increasing computational complexity. Therefore, reducing the backtracking line search overhead when determining the step size as much as possible and adjusting the search direction appropriately to accelerate the convergence speed of the CG algorithm are crucial. To this end, we introduce trainable variables to the algorithm and employ DL tools.

Two improvements are applied to the CG algorithm for search step size and search direction:

1. 1.

   Given the ${k}$th iteration, a trainable scalar ${{w}_{k}^{\alpha}\in{{\mathbb{R}}}}$ is defined as the search step size of the iteration ${k}$. All trainable scalars for the search step size constitute the set ${{{\Theta}^{\alpha}}=\{{w}_{k-1}^{\alpha}:k=1,~{}2,~{}\ldots,~{}K\}}$, where ${K}$ denotes the number of units that represent ${K}$ iterations. Each element of the set ${{\Theta}^{\alpha}}$ is randomly and uniformly initialized between ${{3}\times{10}^{-3}}$ and ${{2}\times{10}^{-2}}$ and trained using the stochastic GD (SGD) algorithm¹¹1The initialization interval is obtained by statistical analysis of the search step size determined by the traditional Armijo backtracking line search rule.. All trainable variables are fixed after training so that they can be used directly during testing without researching on the basis of the Arimijo backtracking line search rule.

2. 2.

   We focus on the weight ${{\beta}_{k}}$ calculated by Polak-Ribière formula in (9) to adjust the search direction of the iteration ${k+1}$. In particular, we calculate ${{\beta}_{k}}$ according to (9) to obtain a reasonable initial weight value. In addition, a trainable scalar ${{w}_{k}^{\beta}\in{{\mathbb{R}}}}$ is defined and multiplied by ${{\beta}_{k}}$ to determine a new weight factor. All trainable scalars for adjusting the search direction constitute the set ${{{\Theta}^{\beta}}=\{{w}_{k}^{\beta}:k=1,~{}2,~{}\ldots,~{}K\}}$. Each element of the set ${{\Theta}^{\beta}}$ is initialized to 1 and trained with SGD. Similarly, the weights are fixed after training.

Finally, we can rewrite (5) as

where ${\mathbf{d}_{k}^{\star}}$ is calculated as

As such, we obtain a CE precoding network named CEPNet, which combines the traditional RMO-based CG algorithm with DL. The overview of the proposed DL architecture is shown in Fig. 1, in which the network inputs are $\mathbf{H}$ and ${\mathbf{s}}$, and the output is ${\mathbf{x}}$. Each unit can be regarded as an iteration of the traditional RMO method. Each unit contains two trainable variables, and the number of total trainable variables in the proposed CEPNet is ${2K}$. Alternatively, each unit contains two active neural layers, and the total number of neural layers in the CEPNet shown in Fig. 1 is ${2K}$. On this basis, we do not recommend that the CEPNet contain too many units because excessive neural layers cause the network to be too “deep,” which may lead to tricky gradient vanishing or exploding. The proposed network is easy to train because only few trainable variables need to be optimized.

To train CEPNet, supervised learning design is inflexible and inadequate because the optimal label is unknown. Obtaining labels through existing algorithms is one approach. However, it only makes the network learn the existing algorithms. Therefore, in our design, an unsupervised learning algorithm is used to train CEPNet effectively. The set of trainable variables is denoted as ${\Theta=\{{\Theta}^{\alpha},{\Theta}^{\beta}\}}$. The inputs of CEPNet are $\mathbf{H}$ and ${\mathbf{s}}$, and the transmitted vector is denoted by $\mathbf{x}=g(\mathbf{s};\mathbf{H};\Theta)$. To improve the robustness of the CEPNet to signal-to-noise ratio (SNR), we use MUI as the loss function directly rather than the average achievable rate or the bit-error rate (BER) that is related to the SNR. Specifically, the loss function is calculated as follows:

where ${{N}_{\rm s}}$ denotes the total number of samples in the training set. ${\mathbf{s}_{i}}$ and ${\mathbf{H}_{i}}$ represent the information symbol and channel vectors associated with the ith symbol, respectively.

The CEPNet is constructed on top of the TensorFlow framework, and an NVIDIA GeForce GTX 1080 Ti GPU is used for accelerated training. The training, validation, and testing sets contain $40,000$, $20,000$, and $60,000$ samples, respectively. We perform simulation experiments with the multipath channel, and the channel vector of the $\mu$th UT is determined by

where ${L^{\mu}}$ is the number of propagation paths of user $\mu$, and $g_{l}^{\mu}$ denotes the complex gain of the $l$th propagation path in the $\mu$th UT’s channel, and

denotes the steering vector of the $\mu$th user, where $d$, $\lambda$, and $\theta_{l}^{\mu}$ denote the distance between two horizontally or vertically adjacent antenna elements, the carrier wavelength, and the angle of departure of the $l$th propagation path in the $\mu$th UT’s channel, respectively. The channel vector $\mathbf{H}={[{{\mathbf{h}}^{1}},~{}{{\mathbf{h}}^{2}},~{}\dots,{{\mathbf{h}}^{N_{\rm u}}}]}^{\rm T}$. We set $L^{\mu}=8$. $g_{l}^{\mu}$ is drawn from ${{\mathcal{N}_{\mathbb{C}}}(0,~{}1)}$, and $\theta_{l}^{\mu}$ is uniformly generated between $0$ and $\pi$. The data sets are formed in ${(\mathbf{s}_{i},\mathbf{H}_{i})}$ pair. The channel vector ${{\mathbf{H}}_{i}}$ is set as block fading, and one transmitted vector ${{\mathbf{s}}_{i}}$ corresponds to one channel vector ${{\mathbf{H}}_{i}}$, where each ${{\mathbf{H}}_{i}}$ is generated independently and each element of the transmitted vector ${{\mathbf{s}}_{i}}$ is drawn from the 16-QAM constellation. We set ${K=20}$ for the trade-off between MUI performance and computational complexity. The set of trainable variables is updated by the ADAM optimizer [15]. The training epochs, learning rate, and batch size are set as $500$, $0.0002$, and $4,000$, respectively.