Optimal Minimum Mean Square Error Research Articles

Optimal monitoring and attack detection of networks modeled by Bayesian attack graphs

Early attack detection is essential to ensure the security of complex networks, especially those in critical infrastructures. This is particularly crucial in networks with multi-stage attacks, where multiple nodes are connected to external sources, through which attacks could enter and quickly spread to other network elements. Bayesian attack graphs (BAGs) are powerful models for security risk assessment and mitigation in complex networks, which provide the probabilistic model of attackers’ behavior and attack progression in the network. Most attack detection techniques developed for BAGs rely on the assumption that network compromises will be detected through routine monitoring, which is unrealistic given the ever-growing complexity of threats. This paper derives the optimal minimum mean square error (MMSE) attack detection and monitoring policy for the most general form of BAGs. By exploiting the structure of BAGs and their partial and imperfect monitoring capacity, the proposed detection policy achieves the MMSE optimality possible only for linear-Gaussian state space models using Kalman filtering. An adaptive resource monitoring policy is also introduced for monitoring nodes if the expected predictive error exceeds a user-defined value. Exact and efficient matrix-form computations of the proposed policies are provided, and their high performance is demonstrated in terms of the accuracy of attack detection and the most efficient use of available resources using synthetic Bayesian attack graphs with different topologies.

Doubly selective channel estimation and equalization based on ICI/ISI mitigation for OQAM-FBMC systems

Over a doubly selective channel, broadband transmission systems face challenges in channel estimation and equalization. High mobility causes inter-carrier interference (ICI), while multipath transmission induces inter-symbol interference (ISI). In this paper, we present a mitigation method of ICI/ISI for the offset quadrature amplitude-modulated filter bank multi-carrier (OQAM-FBMC) system. It features low inherent imaginary interference (IMI) sensitivity and high efficiency. Specifically, a pilot indices optimization algorithm and a sparse adaptive orthogonal subspace pursuit (SAOSP) algorithm are presented based on the 2-D channel modeling scheme. The guard pilots are first added to mitigate the effect of ICI. Then the index optimization and SAOSP algorithms are applied to achieve a high-accuracy estimation of sparse channel coefficients. In addition, a threshold judgment suboptimal minimum mean square error (MMSE) equalization method is presented based on the variability of the interference power. The method uses normalized interference power thresholds to estimate the ISI dimension and reduce the equalization data, thus mitigating the effect of ISI and achieving efficient equalization. To verify the above methods, single-input-single-output (SISO) and multiple-input-multiple-output (MIMO) models are built. Simulation results indicate a 3-5 dB improvement in channel estimation accuracy. The suboptimal MMSE equalization results are close to the optimal MMSE with about four orders of magnitude reduction in complexity.

Deep MMSE Estimation for Data Detection

This letter develops minimum mean-squared error (MMSE) estimators based on deep neural networks for data detection. Since the optimal MMSE is analytically intractable, researchers usually resort to linear MMSE approximations, which often incur performance degradation. To overcome this loss, we develop a near-optimal estimator by exploiting the Donsker-Varadhan representation of mutual information (MI) and the recently discovered derivative relationship between MI and MMSE. The near-optimal MMSE estimator can be computed with a deep neural network (NN). We can train the NN using the mini-batches of input and output samples. We validate this estimator using two examples where the closed-from MMSE is available. We then use this estimator to design an end-to-end communication system. We compare this setup with several conventional techniques and show promising performance.

PCQNet: A Trainable Feedback Scheme of Precoder for the Uplink Multi-User MIMO Systems.

Multi-user multiple-input multiple-output (MU-MIMO) technology can significantly improve the spectral and energy efficiencies of wireless networks. In the uplink MU-MIMO systems, the optimal precoder design at the base station utilizes the Lagrange multipliers method and the centralized iterative algorithm to minimize the mean squared error (MSE) of all users under the power constraint. The precoding matrices need to be fed back to the user equipment to explore the potential benefits of the joint transceiver design. We propose a CNN-based compression network named PCQNet to minimize the feedback overhead. We first illustrate the effect of the trainable compression ratios and feedback bits on the MSE between the original precoding matrices and the recovered ones. We then evaluate the block error rates as the performance measure of the centralized implementation with an optimal minimum mean-squared error (MMSE) transceiver. Numerical results show that the proposed PCQNet achieves near-optimal performance compared with other quantized feedback schemes and significantly reduces the feedback overhead with negligible performance degradation.

Deep Neural Network: An Alternative to Traditional Channel Estimators in Massive MIMO Systems

Fifth-generation (5G) requires a highly accurate estimate of the channel state information (CSI) to exploit the benefits of massive multiple-input-multiple-output (MaMIMO) systems. 5G systems use pilot sequences to estimate channel behaviour using traditional methods like least squares (LS), or minimum mean square error (MMSE) estimation. However, traditional methods do not always obtain reliable estimations: LS exhibits a poor estimation when inadequate channel conditions (i.e., low- signal-to-noise ratio (SNR) region) and MMSE requires prior statistical knowledge of the channel and noise (complex to implement in practice). We present a deep learning framework based on deep neural networks (DNNs) for fifth-generation (5G) MaMIMO channel estimation. After a first preliminary scheme with which we verify the good estimation capacity of our DNN-based approach, we propose two different models, which differ in the information processed by the DNN and benefit from lower computational complexity or greater flexibility for any reference signal pattern, respectively. The results show that, compared to the LS-based channel estimation, the DNN approach decreases the mean square error (MSE) and the system’s spectral efficiency (SE) increases, especially in the low- SNR region. Our approach provides results close to optimal MMSE estimation but benefits from not requiring any prior channel statistics information.

Deep Residual Learning for Channel Estimation in Intelligent Reflecting Surface-Assisted Multi-User Communications

Channel estimation is one of the main tasks in realizing practical intelligent reflecting surface-assisted multi-user communication (IRS-MC) systems. However, different from traditional communication systems, an IRS-MC system generally involves a cascaded channel with a sophisticated statistical distribution. In this case, the optimal minimum mean square error (MMSE) estimator requires the calculation of a multidimensional integration which is intractable to be implemented in practice. To further improve the channel estimation performance, in this paper, we model the channel estimation as a denoising problem and adopt a deep residual learning (DReL) approach to implicitly learn the residual noise for recovering the channel coefficients from the noisy pilot-based observations. To this end, we first develop a versatile DReL-based channel estimation framework where a deep residual network (DRN)-based MMSE estimator is derived in terms of Bayesian philosophy. As a realization of the developed DReL framework, a convolutional neural network (CNN)-based DRN (CDRN) is then proposed for channel estimation in IRS-MC systems, in which a CNN denoising block equipped with an element-wise subtraction structure is specifically designed to exploit both the spatial features of the noisy channel matrices and the additive nature of the noise simultaneously. In particular, an explicit expression of the proposed CDRN is derived and analyzed in terms of Bayesian estimation to characterize its properties theoretically. Finally, simulation results demonstrate that the performance of the proposed method approaches that of the optimal MMSE estimator requiring the availability of the prior probability density function of channel.

Bayesian Learning-Based Linear Decentralized Sparse Parameter Estimation in MIMO Wireless Sensor Networks Relying on Imperfect CSI

Optimal linear minimum mean square error (MMSE) transceiver design techniques are proposed for Bayesian learning (BL)-based sparse parameter vector estimation in a multiple-input multiple-output (MIMO) wireless sensor network (WSN). Our proposed transceiver designs rely on majorization theory and hyperparameter estimates obtained from the BL module for minimizing the mean square error (MSE) of parameter estimation at the fusion center (FC). The linear transceiver design framework is initially proposed for the general scenario with arbitrary SNR sensor observations, followed by a special case with high-SNR sensor observations scenario. Our analysis also incorporates the channel correlation. The MMSE channel estimates are determined for the sensors (SNs), followed by a robust transceiver design procedure that is resilient to the channel state information (CSI) uncertainty arising due to the channel estimation error, an aberration that is unavoidable in practical implementations. Our simulation results demonstrate the improved performance of the proposed BL framework and optimal MMSE transceiver design in sparse parameter estimation relying on realistic imperfect channel estimates over the benchmarks.

Generalized Space–Time Line Code With Receive Combining for MIMO Systems

In this article, a generalized space–time line code (GSTLC) is proposed. The proposed GSTLC concurrently transmits multiple data streams obtained by space–time line code encoding and precoding to provide spatial multiplexing and diversity. When full channel state information (CSI) is available to the transmitter and limited CSI is available to the receiver, an optimal precoding matrix is derived for GSTLC with linear combining that is based on the minimum mean square error (MMSE) criterion. Moreover, the detection signal-to-interference-plus-noise ratio (SINR) is derived corresponding to the optimal MMSE precoder. Using the SINR, we propose a receiver combining scheme that selects the best combining matrix, thereby minimizing the bit error rate (BER) from a codebook that includes candidate combining matrices. Through numerical simulations, we demonstrate that the proposed GSTLC with the optimal combiner achieves the theoretical lower bound of BER. In addition, we show that the GSTLC with combiner selection performs better than conventional precoded spatial multiplexing methods in terms of the BER.

Channel Estimation for Reconfigurable Intelligent Surface Aided MISO Communications: From LMMSE to Deep Learning Solutions

We consider multi-antenna wireless systems aided by reconfigurable intelligent surfaces (RIS). RIS presents a new physical layer technology for improving coverage and energy efficiency by intelligently controlling the propagation environment. In practice however, achieving the anticipated gains of RIS requires accurate channel estimation. Recent attempts to solve this problem have considered the least-squares (LS) approach, which is simple but also sub-optimal. The optimal channel estimator, based on the minimum mean-squared-error (MMSE) criterion, is challenging to obtain and is non-linear due to the non-Gaussianity of the effective channel seen at the receiver. Here we present approaches to approximate the optimal MMSE channel estimator. As a first approach, we analytically develop the best linear estimator, the LMMSE, together with a corresponding majorization-minimization-based algorithm designed to optimize the RIS phase shift matrix during the training phase. This estimator is shown to yield improved accuracy over the LS approach by exploiting second-order statistical properties of the wireless channel and the noise. To further improve performance and better approximate the globally-optimal MMSE channel estimator, we propose data-driven non-linear solutions based on deep learning. Specifically, by posing the MMSE channel estimation problem as an image denoising problem, we propose two convolutional neural network (CNN)-based methods to perform the denoising and approximate the optimal MMSE channel estimation solution. Our numerical results show that these CNN-based estimators give superior performance compared with linear estimation approaches. They also have low computational complexity requirements, thereby motivating their potential use in future RIS-aided wireless communication systems.

Optimal state estimation for finite-field networks with stochastic disturbances

This paper investigates the optimal state estimation of finite-field networks (FFNs) with stochastic disturbances. The stochastic disturbances are regarded as independent noise process. The optimal state estimation problem of FFNs in the sense of minimum mean square error (MMSE) is proposed, and a necessary and sufficient condition is proposed to find the optimal state estimator by minimizing the mean square error (MSE). Moreover, the multi-valued Kalman filter is established, including prediction step and update step, based on which, both optimal state estimator and MMSE can be calculated effectively.

TBM performance prediction with Bayesian optimization and automated machine learning

Accurately predicting the performance of a tunnel boring machine (TBM) is important to safe and efficient tunneling. The application of machine learning algorithms to TBM performance prediction creates several challenges. Such prediction is a nontrivial task involving procedures such as data preprocessing, selection of a machine learning algorithm and optimization of the related hyperparameters. The demand for expert knowledge has restricted the application of machine learning methods to TBM performance prediction, and it is meaningful to study predicting TBM performance automatically. In this paper, we explore three approaches to TBM performance prediction using Bayesian optimization and automated machine learning (AutoML). In the first study, Bayesian optimization is used to determine the optimal hyperparameters of various machine learning algorithms, including support vector regression (SVR), decision tree, bagging tree, random forest and AdaBoost. We attain the minimum mean squared error (MSE) values of 3.135×10-2 and 3.177×10-2 for a decision tree and SVR, respectively. In the second approach called the neural architecture search (NAS), the optimal combination of architecture, hyperparameters and the training procedure of an artificial neural network is found in a single operation. We obtain the optimal results of 3.514×10-2 and 3.237×10-2 if complete and simplified NAS are used, respectively. In the third method, the best combination of a data preprocessing method, a machine learning model and the related hyperparameters is found, and an optimal MSE value of 3.148×10-2 is obtained using AutoML. In all three studies, we obtain state-of-the-art prediction results that are superior to a previous best prediction result of 3.500×10-2. The prediction results prove that Bayesian optimization and AutoML are powerful tools that can not only effectively predict TBM performance but also reduce the demand for expert knowledge of machine learning.

Reconstruction of Clipped Signals in Quantized Uplink Massive MIMO Systems

This paper considers the uplink of a single-cell multiuser massive multiple-input multiple-output system. Each receiver antenna of the base station (BS) is assumed to be equipped with a pair of analog-to-digital converters to quantize the real and imaginary part of the received signal. We propose a novel clipping-aware receiver (CA-MMSE), which performs minimum mean square error (MMSE) reconstruction only on the clipped received samples, while the granular samples are left unchanged after the quantization. On this basis, we present an iterative algorithm to implement the CA-MMSE receiver and derive a sufficient condition for its geometrical convergence to a fixed point. We show that as long as the number of BS antennas or the quantization resolution is sufficiently high, then, the performance of the CA-MMSE is as good as the optimal MMSE receiver which reconstructs all quantized received symbols. Additionally, we propose a novel Bussgang-based weighted zero-forcing (B-WZF) receiver which distinguishes the clipping and granular distortion and it is shown that as long as the received training symbols per antenna are correlated, the CA-MMSE brings significant improvements compared to conventional receivers in the literature while for users that do not experience deep large-scale fading the simpler B-WZF is near to the CA-MMSE for sufficiently high signal-to-noise ratio and quantization resolution.

Two-Stage Algorithm for Estimation of Nonlinear Functions of State Vector in Linear Gaussian Continuous Dynamical Systems

This paper focuses on the optimal minimum mean square error estimation of a nonlinear function of state (NFS) in linear Gaussian continuous-time stochastic systems. The NFS represents a multivariate function of state variables which carries useful information of a target system for control. The main idea of the proposed optimal estimation algorithm includes two stages: the optimal Kalman estimate of a state vector computed at the first stage is nonlinearly transformed at the second stage based on the NFS and the minimum mean square error (MMSE) criterion. Some challenging theoretical aspects of analytic calculation of the optimal MMSE estimate are solved by usage of the multivariate Gaussian integrals for the special NFS such as the Euclidean norm, maximum and absolute value. The polynomial functions are studied in detail. In this case the polynomial MMSE estimator has a simple closed form and it is easy to implement in practice. We derive effective matrix formulas for the true mean square error of the optimal and suboptimal quadratic estimators. The obtained results we demonstrate on theoretical and practical examples with different types of NFS. Comparison analysis of the optimal and suboptimal nonlinear estimators is presented. The subsequent application of the proposed estimators demonstrates their effectiveness.

Boolean Kalman filter and smoother under model uncertainty

Partially-observed Boolean dynamical systems (POBDS) are a general class of nonlinear state-space models that provide a rich framework for modeling many complex dynamical systems. The model consists of a hidden Boolean state process, observed through an arbitrary noisy mapping to a measurement space. The optimal minimum mean-square error (MMSE) POBDS state estimators are the Boolean Kalman Filter and Smoother. However, in many practical problems, the system parameters are not fully known and must be estimated. In this paper, for POBDS under model uncertainty, we derive an optimal Bayesian estimator for state and parameter estimation. The exact algorithms are derived for the case of discrete and finite parameter space, and for general parameter spaces, an approximate Markov-Chain Monte-Carlo (MCMC) implementation is introduced. We demonstrate the performance of the proposed methodology by means of numerical experiments with POBDS models of gene regulatory networks observed through noisy measurements.

Making Cell-Free Massive MIMO Competitive With MMSE Processing and Centralized Implementation

Cell-free Massive MIMO is considered as a promising technology for satisfying the increasing number of users and high rate expectations in beyond-5G networks. The key idea is to let many distributed access points (APs) communicate with all users in the network, possibly by using joint coherent signal processing. The aim of this paper is to provide the first comprehensive analysis of this technology under different degrees of cooperation among the APs. Particularly, the uplink spectral efficiencies of four different cell-free implementations are analyzed, with spatially correlated fading and arbitrary linear processing. It turns out that it is possible to outperform conventional Cellular Massive MIMO and small cell networks by a wide margin, but only using global or local minimum mean-square error (MMSE) combining. This is in sharp contrast to the existing literature, which advocates for maximum-ratio combining. Also, we show that a centralized implementation with optimal MMSE processing not only maximizes the SE but largely reduces the fronthaul signaling compared to the standard distributed approach. This makes it the preferred way to operate Cell-free Massive MIMO networks. Non-linear decoding is also investigated and shown to bring negligible improvements.

A Bayesian Algorithm for Distributed Network Localization Using Distance and Direction Data

A reliable, accurate, and affordable positioning service is highly required in wireless networks. High-resolution estimates of distance and direction data are available in most current and emerging wireless systems. Combining these two sensing modalities can improve the estimation performance and identifiability of the cooperative localization problem, and reduce its sensitivity to the geometry of anchor nodes, i.e., the reference nodes with known locations. However, this is still an open and challenging research problem. In this paper, the novel Message Passing Hybrid Localization (MPHL) algorithm is proposed to solve the problem of cooperative distributed localization using joint distance and direction estimates without any prior information. A statistical model is formulated for the problem, and approximate minimum mean square error (MMSE) estimates of the node locations are computed. The proposed MPHL algorithm is a distributed technique based on belief propagation and Markov chain Monte Carlo sampling. Numerical results are presented showing that the average localization error is significantly reduced in almost every simulation scenario, about 50% in most cases, compared to the state of the art. This improvement in localization performance is due to close approximation of a statistically optimal MMSE estimator.

Adaptive Virtual Waveform Design for Millimeter-Wave Joint Communication–Radar

Joint communication and radar (JCR) waveforms with fully digital baseband generation and processing can now be realized at the millimeter-wave (mmWave) band. Prior work has proposed a mmWave wireless local area network (WLAN)-based JCR that exploits the WLAN preamble for radars. The performance of target velocity estimation, however, was limited. In this paper, we propose a virtual waveform design for an adaptive mmWave JCR. The proposed system transmits a few non-uniformly placed preambles to construct several receive virtual preambles for enhancing velocity estimation accuracy, at the cost of only a small reduction in the communication data rate. We evaluate JCR performance trade-offs using the Cramer-Rao Bound (CRB) metric for radar estimation and a novel distortion minimum mean square error (MMSE) metric for data communication. Additionally, we develop three different MMSE-based optimization problems for the adaptive JCR waveform design. Simulations show that an optimal virtual (non-uniform) waveform achieves a significant performance improvement as compared to a uniform waveform. For a radar CRB constrained optimization, the optimal radar range of operation and the optimal communication distortion MMSE (DMMSE) are improved. For a communication DMMSE constrained optimization with a high DMMSE constraint, the optimal radar CRB is enhanced. For a weighted MMSE average optimization, the advantage of the virtual waveform over the uniform waveform is increased with decreased communication weighting. Comparison of MMSE-based optimization with traditional virtual preamble count-based optimization indicated that the conventional solution converges to the MMSE-based one only for a small number of targets and a high signal-to-noise ratio.

Non-Alternating Globally Optimal MMSE Precoding for Multiuser VLC Downlinks

We study the minimum mean-squared error (MMSE) precoding for multiuser visible light communication downlink systems. Different from radio frequency systems, this problem does not admit a closed-form solution due to the light-emitting diode optical power constraints. To handle the difficult non-convex problem, we transform it into an equivalent convex form via the Charnes–Cooper transformation. In this way, we are able to achieve the globally optimal MMSE precoder by solving only one convex problem, thus requiring much lower complexity than the existing alternating algorithm. The proposed non-alternating method is also extended to robust the MMSE precoder optimization in the presence of statistical channel-state information errors.

Performance Analysis of Distributed Wireless Sensor Networks for Gaussian Source Estimation in the Presence of Impulsive Noise

Abstract—We address the distributed estimation of a scalar Gaussian source in wireless sensor networks (WSNs). The sensor nodes transmit their noisy observations, using the amplifyand- forward relaying strategy through coherent multiple access channel to the fusion center (FC) that reconstructs the source parameter. In this letter, we assume that the received signal at the FC is corrupted by impulsive noise and channel fading, as encountered for instance within power substations. Over Rayleigh fading channel and in presence of Middleton class-A impulsive noise, we derive the minimum mean square error (MMSE) optimal Bayesian estimator along with its mean square error (MSE) performance bounds. From the obtained results, we conclude that the proposed optimal MMSE estimator outperforms the linear MMSE estimator developed for Gaussian noise scenario. Index Terms—Distributed WSN, MMSE optimal Bayesian estimation, Middleton class-A impulsive noise, Rayleigh fading.

Gene regulatory network state estimation from arbitrary correlated measurements

BackgroundAdvancements in gene expression technology allow acquiring cheap and abundant data for analyzing cell behavior. However, these technologies produce noisy, and often correlated, measurements on the transcriptional states of genes. The Boolean network model has been shown to be effective in capturing the complex dynamics of gene regulatory networks (GRNs). It is important in many applications, such as anomaly detection and optimal intervention, to be able to track the evolution of the Boolean states of a gene regulatory network using noisy time-series transcriptional measurements, which may be correlated in time.ResultsWe propose efficient estimators for the Boolean states of GRNs using correlated time-series transcriptional measurements, where the nature of the correlation and of the measurements themselves are entirely arbitrary. More specifically, we propose new algorithms based on a hypothesis tree to compute optimal minimum mean square error (MMSE) filtering and smoothing state estimators for a Partially-Observed Boolean Dynamical System (POBDS) with correlated measurements. The algorithms are exact but may be computationally expensive for large state spaces or long time horizons, in which case a process for pruning the hypothesis tree is employed to obtain an approximation of the optimal MMSE estimators, while keeping computation tractable. Performance is assessed through a comprehensive set of numerical experiments based on the p53-MDM2 negative-feedback loop Boolean regulatory network, where the standard Boolean Kalman Filter (BKF) and Boolean Kalman Smoother (BKS) for uncorrelated measurements are compared to the corresponding new estimators for correlated measurements, called BKF-CORR and BKS-CORR, respectively.