Iterative Algorithm Induced Deep-Unfolding Neural NetworksPrecoding Design for MultiuserMIMOSystem 1
1.Abstract
Abstract—Optimization theory assisted algorithms have received great attention for precoding design in multiuser multiple-input multiple output (MU-MIMO) systems. Although the resultant optimization algorithms are able to provide excellent performance, they generally require considerable computational complexity, which gets in the way of their practical application in real-time systems. In this work, in order to address this issue, we first propose a framework for deep-unfolding, where a general form of iterative algorithm induced deep-unfolding neural network (IAIDNN) is developed in matrix form to better solve the problems in communication systems. Then, we implement the proposed deep-unfolding framework to solve the sum-rate maximization problem for precoding design in MU-MIMO systems. An efficient IAIDNN based on the structure of the classic weighted minimum mean-square error (WMMSE) iterative algorithm is developed. Specifically, the iterative WMMSE algorithm is unfolded into a layer-wise structure, where a number of trainable parameters are introduced to replace the high complexity operations in the forward propagation. To train the network, a generalized chain rule of the IAIDNN is proposed to depict the recurrence relation of gradients between two adjacent layers in the back propagation. Moreover, we discuss the computational complexity and generalization ability of the proposed scheme. Simulation results show that the proposed IAIDNN efficiently achieves the performance of the iterative WMMSE algorithm with reduced computational complexity.
- 提出了一个deep-unfolding framework。
- 实现了这个框架来解决和速率最大问题，基于WMMSE迭代算法
- 提出了反向传播的梯度递推关系的广义链法则
2.introduction
Multiuser multiple-input multiple-output (MU-MIMO) systems have received great attention in wireless communications, since they can dramatically increase the spectrum efficiency[1]–[5]. In order to maximize the spectrum efficiency, a number of efficient iterative precoding design algorithms which rely on the optimization theory have been proposed for the downlink of MU-MIMO systems [6]–[9]. An iterative water-filling algorithm (IWFA) has been developed for MIMO interference systems in [6]. The authors of [7] applied semidefinite relaxation (SDR) to design the transmit precoding for MIMO multicasting systems. In [8], a weighted minimum mean square error (WMMSE) iterative algorithm has been proposed for precoding design in MU-MIMO systems, where the sum rate maximization problem is first equivalently transformed into an MMSE problem and then a block coordinate descent (BCD) method is proposed to solve the resultant MMSE problem. The authors of [9] have proposed an iterative hybrid precoding algorithm based on a novel penalty dual decomposition (PDD) optimization framework. Although these iterative precoding algorithms provide approaching theoretical bound performance, they require very high computational complexity due to the large-dimensional matrix inversion and the large number of iterations, especially for the massive MU-MIMO systems in the upcoming 5G communications, which hinders their applications in real-time systems.
描述了一些传统的做法，尽管可以取得很好的效果，却是以付出巨大的计算资源为代价取得的。
Recently, a number of studies have developed machine learning based algorithms to solve computationally intensive and time sensitive signal processing tasks for communications[10]. The main idea of this method is to treat the iterative algorithm as a black-box, and learn the mapping between the input and the output by employing the deep neural network (DNN) and the convolutional neural network (CNN) [11]. Some representative studies can be found in [12]–[18] for different applications, such as resource allocation and channel estimation. The first try came from [12] and [13], where the authors have applied the multi-layer perceptron (MLP) and CNN to approximate the iterative WMMSE algorithm in a multiuser single-input single-output system. The authors of [14] have proposed an efficient power allocation algorithm by employing unsupervised learning to achieve better performance. With the aid of underlying topology of wireless networks, several resource allocation schemes based on the spatial convolution and the graph neural network (GNN) have been proposed in [15] and [16], respectively. Furthermore, the authors of [17] and [18] have applied the DNN and CNN in channel estimation and channel state information (CSI) feedback.
描述了机器学习在通信领域应用取得的一些成绩。
However, the black-box based neural networks (NNs) suffer from poor interpretability and generalization ability, and likely its performance cannot be guaranteed. The data-driven black box based NN requires a lot of training samples, which incurs a long training time. To overcome such drawbacks, a number of studies [19]–[24] have been proposed to unfold the iterations into a layer-wise structure analogous to a NN based on the existing iterative algorithms. This method is referred to as deep unfolding [25] and has a wide range of applications in communications, such as detection and coding [26]–[29], resource allocation and channel estimation [30]–[32]. For MIMO detection, the authors of [26] have designed the deep unfolding NN based on the projected gradient algorithm and a model-driven deep learning NN has been developed in [27]. The authors of [28] have applied a multi-layer network to approximate the iterative soft-threshold algorithm (ISTA) for sparse coding. In [29], a deep-unfolding based hybrid decoder designed for polar code has been proposed. In addition, an approximate message passing (AMP) inspired NN has been developed in [30] for massive MIMO channel estimation. In [31], a primal-dual method that learns the parameters of the primal and dual variables has been proposed to solve the constrained resource allocation problem, and the authors of [32] have extended it to the scenario of distributed optimization.
描述了unfolding的一些应用。
To the best of our knowledge, the deep-unfolding based NNs have not been well investigated for precoding design in MU-MIMO systems. Moreover, the design of existing deep unfolding NNs mainly focus on the optimization of scalar variables. In this work, we first propose a general framework for deep-unfolding, where a general form of an iterative algorithm induced deep-unfolding neural network (IAIDNN) is developed in matrix form to better solve the problems in communication systems. Based on a general iterative algorithm, the structure of IAIDNN is designed in the forward propagation (FP), where a number of trainable parameters are introduced. In the back propagation (BP), the generalized chain rule (GCR) of the IAIDNN in matrix form is proposed, which depicts the recurrence relation of gradients between two adjacent layers. The gradients of the trainable parameters in different layers are calculated based on the GCR. It extends the chain rule in DNN, which is the basis of the famous platform “tensorflow”, and we show that the existing chain rule is a special case of our proposed GCR.
根据迭代算法提出了神经网络的架构，推导出了反向传播的公式，
We implement the proposed deep-unfolding framework to solve the sum-rate maximization problem for precoding design in MU-MIMO systems, where an efficient IAIDNN based on the structure of the classic iterative WMMSE algorithm [8] is developed. Specifically, by integrating the power constraint into the objective function, we obtain an equivalent unconstrained sum-rate maximization problem, the objective function of which is regarded as the loss function in the unsupervised training stage. To design the IAIDNN, the iterative WMMSE algorithm is unfolded into a layer-wise structure with a series of matrix multiplication and non-linear operations. On the one hand, we use much smaller number of iterations, i.e., layers in the IAIDNN, to approximate the iterative WMMSE algorithm, and avoid the matrix inversion to reduce computational complexity. On the other hand, we aim at improving the performance by introducing trainable parameters. In the FP, we apply the element-wise non-linear function and the first-order Taylor expansion structure of the inverse matrix to approximate the matrix inversion operation. In the BP, we employ the proposed GCR to calculate the gradients of the trainable parameters and update them based on the stochastic gradient descent (SGD) method. Moreover, we develop a black-box based CNN as a benchmark, and discuss the computational complexity and generalization ability of the proposed schemes.
（一些details，不过看不懂。。。）
The contributions of this work are summarized as follows.
- We propose a framework for deep-unfolding, where the general form of IAIDNN is developed in matrix form to better solve the problems in communication systems. To train the IAIDNN, the GCR is proposed to calculate the gradients of the trainable parameters.
- We implement the proposed deep-unfolding framework to solve the sum-rate maximization problem for precoding design in MU-MIMO systems. Based on the structure of the iterative WMMSE algorithm, an efficient IAIDNN is developed, where the iterative WMMSE algorithm is unfolded into a layer-wise structure.
- We analyze the computational complexity and generalization ability of the proposed schemes. Simulation results show that the proposed IAIDNN efficiently achieves the performance of the iterative WMMSE algorithm with reduced computational complexity. The contribution becomes more significant in a massive MU-MIMO system