Machine Learning Methods for
Spectral Efficiency Prediction in Massive MIMO Systems

We denote a concatenation of user channel matrices as $H=[H_{1},\dots,H_{K}]\in\mathbb{C}^{R{\times}T}$. Then we make a singular-value decomposition of each matrix as $H_{k}=U_{k}^{\mathrm{H}}S_{k}V_{k}$, where $U_{k}\in\mathbb{C}^{R_{k}{\times}R_{k}}$ is a unitary matrix, $S_{k}\in\mathbb{R}^{R_{k}{\times}R_{k}}$ is a diagonal matrix and $V_{k}\in\mathbb{C}^{R_{k}{\times}T}$ is a semi-unitary matrix. In such way we obtain the decomposition: $H=[U_{1}^{\mathrm{H}}S_{1}V_{1},\dots,U_{K}^{\mathrm{H}}S_{K}V_{K}]\in\mathbb{C}^{R{\times}T}$. For each $V_{k}$ we denote a sub-matrix $\widetilde{V}_{k}\in\mathbb{C}^{L_{k}{\times}T}$ with rows corresponding to the $L_{k}$ largest singular values from $S_{k}$. We denote a concatenation of all $\widetilde{V}_{k}$ as $\widetilde{V}=[\widetilde{V}_{1},\dots,\widetilde{V}_{K}]\in\mathbb{C}^{L{\times}T}$ .

There are some well-known heuristic precoding algorithms [21, 26, 42, 39, 25]:

The diagonal matrix $P\in\mathbb{C}^{L{\times}L}$ is a column-wise power normalization of precoding matrix, $\mu$ is a scalar power normalization constant for meeting per-antenna power constraints (8). Normalization by constant $\mu$ is done as the last step, after normalization by $P$. The value $\sigma^{2}$ is a variance of Gaussian noise $n_{k}$ during transmission (1). We also consider LBFGS precoding optimization scheme [5].

The base station chooses an optimal precoding matrix $W$ based on measured channel $H$, which maximizes the so-called Spectral Efficiency (SE) [37, 10, 2]. Function of SE is closely related to Signal-to-Interference-and-Noise Ratio and is expressed as :

The set of symbol indexes targeted to $k$-th user is denoted as $\mathcal{L}_{k}$:

where $w_{l}\in\mathbb{C}^{T}$ is a precoding for the $l$-th symbol, $g_{l}\in\mathbb{C}^{R_{k}}$ is a detection vector of the $l$-th symbol, $\sigma^{2}$ is a variance of Gaussian noise (1). The total power of transmitting antennas without loss of generality is assumed to be equal to one.

Additionally, we consider the function of Single-User SINR (SUSINR) :

The formula (7) describes channel quality for the specified user without taking into account the others. The matrix $\widetilde{S}\in\mathbb{C}^{L\times L}$ contains all singular values of $\widetilde{S}_{k}$ on the main diagonal: $\widetilde{S}=\text{diag}(\widetilde{S}_{1}\dots\widetilde{S}_{K})\in\mathbb{C}^{L\times L}$, and $\widetilde{S}_{k}\in\mathbb{R}^{L_{k}\times L_{k}}$ is a diagonal sub-matrix of $S_{k}$ consisting of $L_{k}$ largest singular values of the $k$-th user, and $s_{l}$ are the corresponding elements of $\widetilde{S}$ [3].

It is important to note that each transmitting antenna has a restriction on power of the transmitter. Assuming all the symbols being transmitted are properly normalized, this results in the following constraints on the precoding matrix $W$:

The total power is assumed to be equal to one.

A detection matrix $G_{k}$ is specific to each user. Below we consider the main detection algorithms: MMSE and MMSE-IRC. In the case of MMSE, the detection matrix $G_{k}$ is calculated as [18]:

where matrix $H_{k}W_{k}$ is estimated using pilot signals on the user side.

In this subsection, we give a background on machine learning (ML) tasks and methods. In ML, we are given a training dataset $\mathcal{D}^{tr}=\{(x^{tr}_{i},y^{tr}_{i})\}_{i=1}^{N^{tr}}$ of $N^{tr}$ pairs (input object $x$, target $y$). The problem is to learn an algorithm $a(x)$ (also called prediction function) that predicts target $y$ for any new object $x$. Usually $x\in\mathbb{R}^{d}$, however, complex inputs could also be used. Conventionally used targets include $y\in\mathbb{R}$ (regression problem) or $y\in\{1,\dots,C\}$ (classification problem, $C$ is the number of classes). We are also given a set of $N^{te}$ testing objects, $\mathcal{D}^{te}=\{(x^{te}_{i},y^{te}_{i})\}_{i=1}^{N^{te}}$, and a metric $Q(a)=\sum_{i=1}^{N^{te}}q(y_{i},a(x_{i}))$ that measures the quality of predicting target for new objects. This metric should satisfy the problem-specific requirements and could be non-differentiable. Summarizing, in order to use machine learning methods we need to specify objects, targets, metrics, and collect the dataset of (object, target) pairs.

Firstly, we consider the problem of predicting the SE: given $T$ antennas at the base station, $K$ users with $R_{k}$ antennas and $L_{k}$ symbols each, and set of channel matrices $\{H_{k}\}_{k=1}^{K}$ (input object), the problem is to predict the spectral efficiency SE (4) that could be achieved with some precoding algorithm (precoding is not modeled in this problem). We assume that $T=64$, $R_{k}=4$, $L_{k}=2$ for all $k=1\dots K$ are fixed, while the number of users $K$ could be variable. Since $R_{k}$ is fixed, each object $\{H_{k}\in\mathbb{C}^{R_{k}\times T}\}_{k=1}^{K}$, which serves as an input for our ML algorithms, can be represented by three dimensional tensor $\mathcal{H}\in\mathbb{C}^{K\times 4\times 64}$.

Since spectral efficiency prediction is a regression problem, we can use metrics known from this class of tasks. Namely, we use Mean Average Percentage Error (MAPE):

where $\mathrm{SE}_{i}$ is the spectral efficiency computed using specific precoding, and $a(\mathcal{H}_{i})$ is spectral efficiency predicted by the algorithm $a$ for the $i$-th object $\mathcal{H}_{i}$.

Given the dataset $\{\mathcal{H}_{i},\mathrm{SE}_{i}\}_{i=1}^{N^{tr}}$, our goal is to train an algorithm $a(\mathcal{H}_{i})$ that predicts spectral efficiency SE based on the object $\mathcal{H}$. The algorithm may also take additional information as input, e. g. the level of noise $\sigma^{2}$ or SUSINR value. We also consider an alternative problem statement with user-wise spectral efficiency prediction. Our research questions are as follows:

• •

  Can we predict the spectral efficiency (average or user-wise) with acceptable quality, e. g.  with error on the test dataset, which is an order of magnitude less than the value we predict, i. e.  $\mathrm{MAPE}<10\%$ (10)?

• •

  Which machine learning algorithm out of the considered classes (linear models, gradient boosting, neural networks) is the most suitable for working with channel data?

To obtain a dataset for this problem, we generate input channel matrices $H$, find precoding $W$ for each case using a certain precoding scheme, fixed for this particular dataset, and compute the target spectral efficiency SE using (4). To generate channel coefficients, we use Quadriga [17], an open-source software for generating realistic radio channel matrices. We consider two Quadriga scenarios, namely Urban and Rural. For each scenario, we generate random sets of user positions and compute channel matrices for the obtained user configurations. We describe the generation process in detail in our work [3].

In the majority of the experiments, we consider three precoding algorithms: Maximum Ratio Transmission (MRT) [21], Zero Forcing (ZF) [42] and LBFGS Optimization [5]. The first two classic algorithms are quite simple, while the last one achieves higher spectral efficiency, however being slower. We provide proof-of-concept results for Interference Rejection Combiner (IRC) SE (11) [28].

We consider well-known machine learning algorithms for regression task, namely linear models, gradient boosting, and fully connected neural networks. We rely on the following pipeline:

1. 1.

   Take an object $\mathcal{H}$ as input (and possibly other inputs, e. g. SUSINR).

2. 2.

   Extract a feature vector representation $x=f(\mathcal{H})$ based on the raw input $\mathcal{H}$.

3. 3.

   Apply the algorithm $a$ to feature vector $x$ to obtain $a(x)=\mathrm{SE}$.

Further, we discuss how to obtain feature extraction procedure $f$ and prediction function $a$. Motivated by the exact formulas for the case of Maximum Ratio and Zero-Forcing precoding, we select the following default features from the Singular Value Decomposition (SVD) of the $H=U^{\mathrm{H}}SV$ matrix, where $K$ is the number of users, and $L_{i}$ is the $i$-th user number of layers:

• •

  $\{s^{2}_{ij}\}_{i,j=1}^{K,L_{i}}$ – singular values;

• •

  $\{c_{ijkm}\}_{\underset{i\neq j}{i,j,k,m=1}}^{K,K,L_{i},L_{j}}$ - pairwise layer correlations, $c_{ijkm}=corr(V_{ik},V_{jm})=|V_{ik}V_{jm}|^{2}$.

It can be shown that spectral efficiency (4), which is the value being predicted, depends only on the squares of the first $L_{i}$ singular values of $H_{i}$ of all users $i=1\dots K$ and the correlations between the first $L_{i}$ layers of all possible user pairs, which justifies this choice of default features.

There are several issues with these features: 1) the number of features is variable, as the number of users $K$ varies among different objects $\mathcal{H}\in\mathbb{C}^{K\times 4\times 64}$ in the dataset, while the aforementioned ML models take a feature vector of fixed size as input 2) the target objective, SE, is invariant to permutation of users, hence there should be a symmetry in feature representations. To address these issues, we propose two modifications of the default features.

Firstly, we introduce symmetry with respect to user permutations by sorting the feature values along with user indices. For singular values, we sort users by the largest singular value. For pairwise correlations, we sort pairs of users by the largest correlation value of their layers.

Secondly, we alleviate the issue with a variable number of users by introducing polynomial features. For a sequence $x=(x_{1},x_{2},\ldots,x_{n})$ of features, we can apply symmetric polynomial transformation, obtaining fixed number of features $poly_{k}(x)=(e_{1}(x),e_{2}(x),\ldots,e_{k}(x))$:

Therefore, in the following experiments, we consider 3 types of features: default, sorted, poly_k. In some experiments, we also use a SUSINR as an additional feature. Models which are trained on default and sorted features require a dataset with a fixed number of users $K$=const, while poly_k features allow us to train models on a dataset with variable number of users $K$ between objects $\mathcal{H}_{i}$, which is a significant advantage. After extracting features, we use three ML algorithms: linear models, gradient boosting, and fully-connected neural networks. These algorithms are described in Section 4.3.

In this work, we rely on the three most commonly used machine learning algorithms: linear models, gradient boosting, and neural networks. We discuss the positive and negative sides of these algorithms along with the description of how we apply them to 5G data. We compare all the results obtained by these models in Sec. 5.

In Tab. 1, we report the inference time of our best models, namely boosting algorithms with sorted and polynomial features, and of several reference algorithms, e. g. the computational time of true SE values for the Zero-Forcing precoding method. We observe that the time complexity of gradient boosting (including preprocessing) is an order smaller than the time of computing true SE values for the Zero-Forcing algorithm.

In the previous experiments for SE prediction, we predicted SE for the whole pairing of users, i.e. averaged SE. Now, we wish to verify whether it is also possible to obtain reasonable prediction quality for each user separately since it can be useful in practice for selecting modulation coding scheme for each user [4]. We define the target SE_u for each user as SE before averaging by the users: $\mathrm{SE}_{k}=\log_{2}(1+\mathrm{SINR}^{eff}_{k})$.

The features for user-wise SE prediction are the same as for the case of average SE prediction. For each user, we once again use the corresponding singular values, correlations between the chosen user’s layers and the layers of other users, and additional features, such as noise power or equivalently SUSINR. These features characterize the relation of one particular user to all other users.

The results for the Urban scenario, 2/4/8 users (separate models) are shown in Fig. 7. Note that MAPE values in these experiments are not comparable to the values from previous sections, because of different target spaces. The results show that the gradient boosting approach outperforms the linear regression approach, for all considered user counts and precoding algorithms.

We also provide the results for SE-IRC [28] prediction based on MMSE-IRC detection (11), to verify that our method is robust to the change of the detection. For the case of MMSE-IRC, the detection matrix is calculated as:

where the matrix $R_{uu}^{k}$ is related to unit symbol variance and is calculated as follows:

The results for the Urban scenario, 2/4/8 users (separate models) are shown in Fig. 7. From this results we conclude that the LBFGS-IRC composition of precoding and detection allows us to obtain better prediction quality with gradient boosting in comparison with other precoding schemes except MRT for all considered user counts.

To test whether neural networks could improve feature-based spectral efficiency prediction, we train a fully-connected neural network on the default features, for the Urban scenario, 4 users, Zero Forcing precoding method. We consider one and three hidden layers configurations with 200 neurons at each layer. Our results show that while linear models provide MAPE of 0.0679, and gradient boosting – of 0.0383, the one-/three-layer neural networks achieve MAPE of 0.0371 / 0.0372 (see Fig. 8).

Thus, we conclude that using fully-connected neural networks is comparable to using gradient boosting in terms of prediction quality, while the tuning of network hyperparameters and the training procedure itself require significantly more time and memory than those of the other methods. However, these experiments showed that neural networks are in general applicable to channel data.