Large Dimensional Random Matrix Theory Research Articles

Large-dimensional random matrix theory and its applications in deep learning and wireless communications

Large-dimensional (LD) random matrix theory, RMT for short, which originates from the research field of quantum physics, has shown tremendous capability in providing deep insights into large-dimensional systems. With the fact that we have entered an unprecedented era full of massive amounts of data and large complex systems, RMT is expected to play more important roles in the analysis and design of modern systems. In this paper, we review the key results of RMT and its applications in two emerging fields: wireless communications and deep learning. In wireless communications, we show that RMT can be exploited to design the spectrum sensing algorithms for cognitive radio systems and to perform the design and asymptotic analysis for large communication systems. In deep learning, RMT can be utilized to analyze the Hessian, input–output Jacobian and data covariance matrix of the deep neural networks, thereby to understand and improve the convergence and the learning speed of the neural networks. Finally, we highlight some challenges and opportunities in applying RMT to the practical large-dimensional systems.

Secrecy rate analysis for reconfigurable intelligent surface-assisted mimo communications with statistical CSI

In this paper, a reconfigurable intelligent surface (RIS)-assisted MIMO wireless secure communication system is considered, in which a base station (BS) equipped with multiple antennas exploits statistical channel state information to communicate with a legitimate multi-antenna user, in the presence of an eavesdropper, also equipped with multiple antennas. We firstly obtain an analytical expression of the ergodic secrecy rate based on the results of large-dimensional random matrix theory. Then, a jointly alternating optimization algorithm with the method of Taylor series expansion and the projected gradient ascent method is proposed to design the transmit covariance matrix at the BS, as well as the diagonal phase- shifting matrix to maximize the ergodic secrecy rate. Simulations are conducted to demonstrate the accuracy of the derived analytical expressions, as well as the superior performance of our proposed algorithm.

Learning block structures in U-statistic-based matrices

SummaryWe introduce a conceptually simple, efficient and easily implemented approach for learning the block structure in a large matrix. Using the properties of U-statistics and large-dimensional random matrix theory, the group structure of many variables can be directly identified based on the eigenvalues and eigenvectors of the scaled sample matrix. We also establish the asymptotic properties of the proposed approach under mild conditions. The finite-sample performance of the approach is examined by extensive simulations and data examples.

Backhaul-Aware Resource Allocation and Optimum Placement for UAV-Assisted Wireless Communication Network

Driven by its agile maneuverability and deployment, the unmanned aerial vehicle (UAV) becomes a potential enabler of the terrestrial networks. In this paper, we consider downlink communications in a UAV-assisted wireless communication network, where a multi-antenna UAV assists the ground base station (GBS) to forward signals to multiple user equipments (UEs). The UAV is associated with the GBS through in-band wireless backhaul, which shares the spectrum resource with the access links between UEs and the UAV. The optimization problem is formulated to maximize the downlink ergodic sum-rate by jointly optimizing UAV placement, spectrum resource allocation and transmit power matrix of the UAV. The deterministic equivalents of UE’s achievable rate and backhaul capacity are first derived by utilizing large-dimensional random matrix theory, in which, only the slowly varying large-scale channel state information is required. An approximation problem of the joint optimization problem is then introduced based on the deterministic equivalents. Finally, an algorithm is proposed to obtain the optimal solution of the approximate problem. Simulation results are provided to validate the accuracy of the deterministic equivalents, and the effectiveness of the proposed method.

Performance of Network-Assisted Full-Duplex for Cell-Free Massive MIMO

Network assisted full-duplex (NAFD) is a spatial-division duplex technique for future wireless networks with cell-free massive multiple-input multiple-output (CF massive MIMO) network, where a large number of remote antenna units (RAUs), either using half-duplex or full-duplex, jointly support truly flexible duplex including time-division duplex, frequency-division duplex and full duplex on demand of uplink and downlink traffic by using network MIMO methods. With NAFD, bi-directional data rates of the wireless network could be increased and end-to-end delay could be reduced. In this paper, the spectral efficiency of NAFD communications in CF massive MIMO network with imperfect channel state information (CSI) is investigated under spatial correlated channels. Based on large dimensional random matrix theory, the deterministic equivalents for the uplink sum-rate with minimum-mean-square-error (MMSE) receiver as well as the downlink sum-rate with zero-forcing (ZF) and regularized zero-forcing (RZF) beamforming are derived. Numerical results show that under various environmental settings, the deterministic equivalents are accurate in both a large-scale system and system with a finite number of antennas. It is also shown that with the downlink-to-uplink interference cancellation, the uplink spectral efficiency of CF massive MIMO with NAFD could be improved. The spectral efficiencies of NAFD with different duplex configurations such as in-band full-duplex, and half-duplex are compared. With the same total numbers of transmit and receive antennas, NAFD with half-duplex RAUs offers a higher spectral efficiency. To alleviate the uplink-to-downlink interference, a novel genetic algorithm based user scheduling strategy (GAS) is proposed. Simulation results show that the achievable downlink sum-rate by using the GAS is greatly improved compared to that by using the random user scheduling.

A Levitation Condition Awareness Architecture for Low-Speed Maglev Train Based on Data-Driven Random Matrix Analysis

Modern low-speed maglev trains typically use multi-node decentralized levitation control modules, which results in a complex levitation control system with coupling interaction. Conducting systematic levitation condition awareness of the levitation control system is still a promising challenge. In this paper, under the hypothesis of levitation residuals following normal distribution, a levitation condition awareness architecture for the levitation control system is proposed based on data-driven random matrix analysis. The proposed architecture consists of an engineering procedure followed by a cascaded mathematical procedure. In the decentralized engineering procedure, the data-driven modeling for individual levitation control modules is achieved by nonlinear autoregressive modeling with an exogenous input neural network, and the unknown parameters are identified by a modified combinatorial genetic algorithm. On this basis, high-dimensional analysis of streaming residual random matrices for the levitation control system is conducted aided by large-dimensional random matrix theory, and the control limits of the constructed indicators are well-designed using the theorical distributions. Based on the comparative analysis of the experimental datasets, the proposed awareness architecture is verified to show the effectiveness of the systematic condition evaluation of the levitation system, and incipient train-guideway interaction vibration abnormalities can be detected in a timely manner.

Large system analysis of downlink C-RAN with phase noise and fronthaul compression

This paper studies the effect of phase noise and fronthaul compression on a downlink cloud radio access network (C-RAN), where several remote radio heads (RRHs) are coordinated to communicate with users by a baseband unit (BBU) on the cloud server. In the system, the baseband signals are precoded at BBU, and then compressed before being transmitted to RRHs through capacity-limited fronthaul links which results in the compressive quantization noise. We assume the regularized zero-forcing precoding is performed with an imperfect channel state information and a compression strategy is applied at BBU. The effect of phase noise arising from nonideal local oscillators both at RRHs and users is considered. We propose an approximate expression for the downlink ergodic sum-rate of considered C-RAN utilizing large dimensional random matrix theory in the large-system regime. From simulation results, the accuracy of the approximate expression is validated, and the effect of phase noise and fronthaul compression can be analyzed theoretically based on the approximate expression.

Active Eavesdropping Detection Based on Large-Dimensional Random Matrix Theory for Massive MIMO-Enabled IoT

The increasing Internet-of-Things (IoT) applications will take a significant share of the services of the fifth generation mobile network (5G). However, IoT devices are vulnerable to security threats due to the limitation of their simple hardware and communication protocol. Massive multiple-input multiple-output (massive MIMO) is recognized as a promising technique to support massive connections of IoT devices, but it faces potential physical layer breaches. An active eavesdropper can compromises the communication security of massive MIMO systems by purposely contaminating the uplink pilots. According to the random matrix theory (RMT), the eigenvalue distribution of a large dimensional matrix composed of data samples converges to the limit spectrum distribution that can be characterized by matrix dimensions. With the assistance of RMT, we propose an active eavesdropping detection method in this paper. The theoretical limit spectrum distribution is exploited to determine the distribution range of the eigenvalues of a legitimate user signal. In addition the noise components are removed using the Marčenko–Pastur law of RMT. Hypothesis testing is then carried out to determine whether the spread range of eigenvalues is “normal” or not. Simulation results show that, compared with the classical Minimum Description Length (MDL)-based detection algorithm, the proposed method significantly improves active eavesdropping detection performance.

Energy Efficiency of Downlink C-RAN With Edge Caching and Fronthaul Compression

In this letter, we study the impact of edge caching on the energy efficiency (EE) of a downlink cloud radio access network. The fronthaul compression strategy is introduced to tackle the problem of the limited fronthaul capacity. To reduce computational complexity, we resort to the large dimensional random matrix theory to derive the deterministic equivalent of the EE based on which the optimal cache capacity that maximizes the EE is derived. We compare the performance of the fronthual compression scheme with that of the data-sharing scheme and reveal the tradeoff between the EE and the transmit power. Simulation results show that the approximation is accurate and validate the dependency of the EE on the cache capacity.

Coordinated Multicast Precoding for Multi-Cell Massive MIMO Transmission Exploiting Statistical Channel State Information

This paper considers coordinated multi-cell multicast precoding for massive multiple-input-multiple-output transmission where only statistical channel state information of all user terminals (UTs) in the coordinated network is known at the base stations (BSs). We adopt the sum of the achievable ergodic multicast rate as the design objective. We first show the optimal closed-form multicast signalling directions of each BS, which simplifies the coordinated multicast precoding problem into a coordinated beam domain power allocation problem. Via invoking the minorization-maximization framework, we then propose an iterative power allocation algorithm with guaranteed convergence to a stationary point. In addition, we derive the deterministic equivalent of the design objective to further reduce the optimization complexity via invoking the large-dimensional random matrix theory. Numerical results demonstrate the performance gain of the proposed coordinated approach over the conventional uncoordinated approach, especially for cell-edge UTs.

Energy-Efficient Multicast Precoding for Massive MIMO Transmission with Statistical CSI

In this paper, we investigate energy-efficient multicast precoding for massive multiple-input multiple-output (MIMO) transmission. In contrast with most previous work, where instantaneous channel state information (CSI) is exploited to facilitate energy-efficient wireless transmission design, we assume that the base station can only exploit statistical CSI of the user terminals for downlink multicast precoding. First, in terms of maximizing the system energy efficiency, the eigenvectors of the optimal energy-efficient multicast transmit covariance matrix are identified in closed form, which indicates that optimal energy-efficient multicast precoding should be performed in the beam domain in massive MIMO. Then, the large-dimensional matrix-valued precoding design is simplified into an energy-efficient power allocation problem in the beam domain with significantly reduced optimization variables. Using Dinkelbach’s transform, we further propose a sequential beam domain power allocation algorithm which is guaranteed to converge to the global optimum. In addition, we use the large-dimensional random matrix theory to derive the deterministic equivalent of the objective to reduce the computational complexity involved in sample averaging. We present numerical results to illustrate the near-optimal performance of our proposed energy-efficient multicast precoding for massive MIMO.

Joint Optimization of Fronthaul Compression and Bandwidth Allocation in Uplink H-CRAN With Large System Analysis

In this paper, we consider an uplink heterogeneous cloud radio access network (H-CRAN), where a macro base station (BS) coexists with many remote radio heads (RRHs). For cost-savings, only the BS is connected to the baseband unit (BBU) pool via fiber links. The RRHs, however, are associated with the BBU pool through wireless fronthaul links, which share the spectrum resource with radio access networks. Due to the limited capacity of fronthaul, the compress-and-forward scheme is employed, such as point-to-point compression or Wyner-Ziv coding. Different decoding strategies are also considered. This work aims to maximize the uplink ergodic sum-rate (SR) by jointly optimizing quantization noise matrix and bandwidth allocation between radio access networks and fronthaul links, which is a mixed time-scale issue. To reduce computational complexity and communication overhead, we introduce an approximation problem of the joint optimization problem based on large-dimensional random matrix theory, which is a slow time-scale issue because it only depends on statistical channel information. Finally, an algorithm based on Dinkelbach's algorithm is proposed to find the optimal solution to the approximate problem. In summary, this work provides an economic solution to the challenge of constrained fronthaul capacity, and also provides a framework with less computational complexity to study how bandwidth allocation and fronthaul compression can affect the SR maximization problem.

On a spiked model for large volatility matrix estimation from noisy high-frequency data

Recently, inference about high-dimensional integrated covariance matrices (ICVs) based on noisy high-frequency data has emerged as a challenging problem. In the literature, a pre-averaging estimator (PA-RCov) is proposed to deal with the microstructure noise. Using the large-dimensional random matrix theory, it has been established that the eigenvalue distribution of the PA-RCov matrix is intimately linked to that of the ICV through the Marčenko–Pasturequation. Consequently, the spectrum of the ICV can be inferred from that of the PA-RCov. However, extensive data analyses demonstrate that the spectrum of the PA-RCov is spiked, that is, a few large eigenvalues (spikes) stay away from the others which form a rather continuous distribution with a density function (bulk). Therefore, any inference on the ICVs must take into account this spiked structure. As a methodological contribution, a spiked model is proposed for the ICVs where spikes can be inferred from those of the available PA-RCov matrices. The consistency of the inference procedure is established. In addition, the methodology is applied to the real data from the US and Hong Kong markets. It is found that the model clearly outperforms the existing one in predicting the existence of spikes and in mimicking the empirical PA-RCov matrices.

Testing identity of high-dimensional covariance matrix

ABSTRACTTwo new statistics are proposed for testing the identity of high-dimensional covariance matrix. Applying the large dimensional random matrix theory, we study the asymptotic distributions of our proposed statistics under the situation that the dimension p and the sample size n tend to infinity proportionally. The proposed tests can accommodate the situation that the data dimension is much larger than the sample size, and the situation that the population distribution is non-Gaussian. The numerical studies demonstrate that the proposed tests have good performance on the empirical powers for a wide range of dimensions and sample sizes.

Enhanced Group Sparse Beamforming for Green Cloud-RAN: A Random Matrix Approach

Group sparse beamforming is a general framework to minimize the network power consumption for cloud radio access networks, which, however, suffers high computational complexity. In particular, a complex optimization problem needs to be solved to obtain the remote radio head (RRH) ordering criterion in each transmission block, which will help to determine the active RRHs and the associated fronthaul links. In this paper, we propose innovative approaches to reduce the complexity of this key step in group sparse beamforming. Specifically, we first develop a smoothed $\ell _{p}$ -minimization approach with the iterative reweighted- $\ell _{2}$ algorithm to return a Karush–Kuhn–Tucker (KKT) point solution, as well as enhance the capability of inducing group sparsity in the beamforming vectors. By leveraging the Lagrangian duality theory, we obtain closed-form solutions at each iteration to reduce the computational complexity. The well-structured solutions provide opportunities to apply the large-dimensional random matrix theory to derive deterministic approximations for the RRH ordering criterion. Such an approach helps to guide the RRH selection only based on the statistical channel state information, which does not require frequent update, thereby significantly reducing the computation overhead. Simulation results shall demonstrate the performance gains of the proposed $\ell _{p}$ -minimization approach, as well as the effectiveness of the large system analysis-based framework for computing the RRH ordering criterion.

Robust MMSE precoding for massive MIMO transmission with hardware mismatch

Due to hardware mismatch, the channel reciprocity of time-division duplex massive multiple-input multiple-output system is impaired.Under this condition, there exist several different approaches for base station (BS) to obtain downlink (DL) channel information based on the minimum mean-square-error (MMSE) estimation method.In this paper, we show that with the hardware mismatch parameters BS can obtain the same DL channel information via these different approaches.As the obtained DL channel information is usually imperfect,we propose a precoding technique based on the criterion that minimizes the mean-square-error (MSE) of signal detection at the user terminals (UTs).The proposed precoding is robust to the channel estimation error and significantly improves the system performance compared to the conventional regularized zero-forcing precoding.Furthermore, we derive an asymptotic approximation of the ergodic sum rate for the proposed precoding using the large dimensional random matrix theory,which is tight as the number of antennas both at the BS and UT approach infinity with a fixed non-zero and finite ratio.This approximation can provide a reliable sum rate prediction at a much lower computation cost than Monte Carlo simulations.Simulation results show that the approximation is accurate even for a realistic system dimension.

Large System Analysis of Resource Allocation in Heterogeneous Networks With Wireless Backhaul

Small-cell networks and massive multiple-input multiple-output (MIMO) systems are regarded as important candidate techniques for 5G communication systems. This paper considers a heterogeneous network composed of a macrocell tier overlaid with an extremely dense tier of small-cells. In the network, the macrocell base station (BS), which applies massive MIMO, does not only serve macro user equipment units but also provides wireless backhaul for small-cell access points (APs). The wireless backhaul shares the same spectrum resource with radio access networks without creating extra spectrum resources. However, due to the densification of small-cells, the inter- and intra-tier interferences become severe. To mitigate the interferences, we use the regularized zero-forcing precoding combined with a projection technique is used at the BS in downlink (DL) to avoid interference to the APs in uplink (UL). Meanwhile, the joint linear minimum mean square error detection is applied in UL to mitigate the inter-tier interference. We derive deterministic expressions for ergodic UL and DL sum rates (SRs) by leveraging the large-dimensional random matrix theory. The expressions only depend on statistical channel information and can be used to optimize the bandwidth division between radio access links and wireless backhaul, as well as the time allocation between DL and UL operation intervals. Numerical results show that the deterministic SR equivalents are accurate and that the proposed resource allocation method can significantly improve system performance.

Performance Analysis of Correlated Massive MIMO Systems With Spatially Distributed Users

In this paper, we analyze the performance of an uplink large-scale multiple-input-multiple-output system with a single base station (BS) serving spatially distributed multiantenna user devices (UDs) within a fixed coverage area. Stochastic geometry is used to characterize the spatially distributed users while large dimensional random matrix theory is used to achieve deterministic approximations of the sum rate of the system. In particular, the users in the vicinity of the BS are considered to follow a Poisson point process within the fixed coverage area. The sum rate of this system is analyzed with respect to different number of antennas at the BS as well as the intensity of the users within the coverage area of the cell. Closed-form approximations for the deterministic rate at low and high signal-to-noise ratio regimes are derived that have very low computational complexity. The deterministic rate for a general $k$ th ordered user is also derived. It is shown that the deterministic approximations offer a reliable estimate of the ergodic sum rate obtained by Monte Carlo simulations. We also briefly touch on the growing issue of power consumption in wireless systems by analyzing the energy efficiency of the system using a power consumption model, taking into consideration the circuit power consumption, which is a function of the number of antennas of the BS and UDs.

High Dimensional Correlation Matrices: The Central Limit Theorem and Its Applications

Summary Statistical inferences for sample correlation matrices are important in high dimensional data analysis. Motivated by this, the paper establishes a new central limit theorem for a linear spectral statistic of high dimensional sample correlation matrices for the case where the dimension p and the sample size n are comparable. This result is of independent interest in large dimensional random-matrix theory. We also further investigate the sample correlation matrices of a high dimensional vector whose elements have a special correlated structure and the corresponding central limit theorem is developed. Meanwhile, we apply the linear spectral statistic to an independence test for p random variables, and then an equivalence test for p factor loadings and n factors in a factor model. The finite sample performance of the test proposed shows its applicability and effectiveness in practice. An empirical application to test the independence of household incomes from various cities in China is also conducted.

Large System Secrecy Rate Analysis for SWIPT MIMO Wiretap Channels

© 2015 IEEE. In this paper, we study the multiple-input multiple-output wiretap channel for simultaneous wireless information and power transfer, in which there is a base station (BS), an information-decoding (ID) user, and an energy-harvesting (EH) user. The messages intended to the ID user is required to be kept confidential to the EH user. Our objective is to design the optimal transmit covariance matrix at the BS for maximizing the ergodic secrecy rate subject to the harvested energy requirement for the EH user exploiting only statistical channel state information at the BS. To this end, we begin by deriving an approximation for the ergodic secrecy rate using large-dimensional random matrix theory and the method of Taylor series expansion. This approximation enables us to derive the asymptotic-optimal transmit covariance matrix that achieves the tradeoff for ergodic secrecy rate and harvested energy. The simulation results are provided to verify the accuracy of the approximation and show that a bigger rate-energy region can be achieved when the Rician factor increases or the path loss exponent decreases. We also show that when the transmit correlation increases or the distance between the eavesdropper and the BS decreases, the harvested energy will be increased, while the achieved ergodic secrecy rate decreases.