Calculation Of Inverse Matrix Research Articles

Reduced-order identification methods: Hierarchical algorithm or variable elimination algorithm

Reduced-order identification algorithms are usually used in machine learning and big data technologies, where the large-scale systems widely exist. For large-scale system identification, traditional least squares algorithm involves high-order matrix inverse calculation, while traditional gradient descent algorithm has slow convergence rates. The reduced-order algorithm proposed in this paper has some advantages over the previous work: (1) via sequential partitioning of the parameter vector, the calculation of the inverse of a high-order matrix can be reduced to low-order matrix inverse calculations; (2) has a better conditioned information matrix than that of the gradient descent algorithm, thus has faster convergence rates; (3) its convergence rates can be increased by using the Aitken acceleration method, therefore the reduced-order based Aitken algorithm is at least quadratic convergent and has no limitation on the step-size. The properties of the reduced-order algorithm are also given. Simulation results demonstrate the effectiveness of the proposed algorithm.

Study on Large-Scale Urban Water Distribution Network Computation Method Based on a GPU Framework

Large-scale urban water distribution network simulation plays a critical role in the construction, monitoring, and maintenance of urban water distribution systems. However, during the simulation process, matrix inversion calculations generate a large amount of computational data and consume significant amounts of time, posing challenges for practical applications. To address this issue, this paper proposes a parallel gradient calculation algorithm based on GPU hardware and the CUDA Toolkit library and compares it with the EPANET model and a model based on CPU hardware and the Armadillo library. The results show that the GPU-based model not only achieves a precision level very close to the EPANET model, reaching 99% accuracy, but also significantly outperforms the CPU-based model. Furthermore, during the simulation, the GPU architecture is able to efficiently handle large-scale data and achieve faster convergence, significantly reducing the overall simulation time. Particularly in handling larger-scale water distribution networks, the GPU architecture can improve computational efficiency by up to 13 times. Further analysis reveals that different GPU models exhibit significant differences in computational efficiency, with memory capacity being a key factor affecting performance. GPU devices with larger memory capacity demonstrate higher computational efficiency when processing large-scale water distribution networks. This study demonstrates the advantages of GPU acceleration technology in the simulation of large-scale urban water distribution networks and provides important theoretical and technical support for practical applications in this field. By carefully selecting and configuring GPU devices, the computational efficiency of large-scale water distribution networks can be significantly improved, providing more efficient solutions for future urban water resource management and planning.

Gradient neural network model for the system of two linear matrix equations and applications

In this paper, the new type of Gradient Neural Network (GNN) model is proposed for the following linear system of matrix equations: AX=C,XB=D. The convergence analysis of given models is shown. The model is applied for the computation of the regular matrix inverse, as well as Moore-Penrose and Drazin generalized inverses. Some illustrative examples and simulations are given to verify theoretical results.

Radial basis point interpolation for strain field calculation in digital image correlation

In order to extract smooth and accurate strain fields from the noisy displacement fields obtained by digital image correlation (DIC), a point interpolation meshless (PIM) method with a radial basis function (RBF) is introduced for full-field strain calculation, which overcomes the problems of slow calculation speed and unstable matrix inverse calculation of the element-free Galerkin method (EFG). The radial basis point interpolation method (RPIM) with three different radial basis functions and the moving least squares (MLS) and pointwise least squares (PLS) methods are compared by analyzing and validating the strain fields with high-strain gradients in simulation experiments. The results indicate that the RPIM is nearly 80% more computationally efficient than the MLS method when a larger support domain is used, and the efficiency of the RPIM is nearly 26% higher than that of the MLS method when a smaller support domain is used; the strain calculation accuracy is slightly lower than that of the MLS method by 0.3–0.5%, but the stability of the calculation is significantly improved. In contrast with the PLS method, which is easily affected by the noise and the size of the strain calculation window, the RPIM is insensitive to the displacement noise and the size of the support domain and can obtain a similar calculation accuracy. The RPIM with multiquadric (MQ) radial basis functions performs well in balancing the computational accuracy and efficiency and is insensitive to shape parameters. The application cases show that the method can effectively compute the strain field at the crack tip, validating its applicability to the study of the plastic region at the crack tip. In conclusion, the proposed RPIM-based method provides an accurate, practical, and robust approach for full-field strain measurements.

Chip-scale all-optical complex-valued matrix inverter

Matrix inversion is a fundamental and widely utilized linear algebraic operation but computationally expensive in digital-clock-based platforms. Optical computing is a new computing paradigm with high speed and energy efficiency, and the computation can be realized through light propagation. However, there is a scarcity of experimentally implemented matrix inverters that exhibit both high integration density and the capability to perform complex-valued operations in existing optical systems. For the first time, we experimentally demonstrated an iterative all-optical chip-scale processor to perform the computation of complex-valued matrix inversion using the Richardson method. Our chip-scale processor achieves an iteration speed of 10 GHz, which can facilitate ultra-fast matrix inversion with the assistance of high-speed Mach–Zehnder interferometer modulators. The convergence can be attained within 20 iterations, yielding an accuracy of 90%. The proposed chip-scale all-optical complex-valued matrix inverter represents a distinctive innovation in the field of all-optical recursive systems, offering significant potential for solving computationally intensive mathematical problems.

HYPERSPECTRAL ANOMALY DETECTION WITH AN IMPROVED APPROACH: INTEGRATION OF GO DECOMPOSITION ALGORITHM AND LAPLACIAN MATRIX MODIFIER

In this study, a hyperspectral anomaly detection method based on Laplacian matrix (HADLAP) is proposed. This paper addresses the problem of determining covariance matrix inversion in high-dimensional data and proposes a new approach for identifying anomalies in hyperspectral images (HSIs). The study’s goals are to find anomalous locations in HSIs and to deal with the problem of calculating the inversion of the covariance matrix of high dimensional data. The method is centered on two main concepts. One of them is decomposition process. The other one is detection process. First, HSI data is decomposed as a low rank and sparse matrices. Second, the sparse component of the data is used to build Mahalanobis Distance (MD). In this study, go decomposition (GoDec) algorithm is employed to decompose the data. Then, the distance is calculated by obtained matrix with aim of detection of anomalous pixels in the HSIs. The method differs from previous studies that covariance matrix in the distance is computed with Laplacian matrix and MD. Experiments conducted on three hyperspectral datasets present the superiority and effectiveness of the proposed framework in terms of detection performance with respect to state-of-the-art methods.

Quantum algorithms for matrix operations and linear systems of equations

Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations. Using the ‘sender-receiver’ model, we propose quantum algorithms for matrix operations such as matrix-vector product, matrix-matrix product, the sum of two matrices, and the calculation of determinant and inverse matrix. We encode the matrix entries into the probability amplitudes of the pure initial states of senders. After applying proper unitary transformation to the complete quantum system, the desired result can be found in certain blocks of the receiver’s density matrix. These quantum protocols can be used as subroutines in other quantum schemes. Furthermore, we present an alternative quantum algorithm for solving linear systems of equations.

Energy demand forecasting of remote areas using linear regression and inverse matrix analysis

Efficient energy demand forecasting is pivotal for addressing energy challenges in remote areas of Bangladesh, where reliable access to energy resources remains a concern. This study proposes an innovative approach that combines linear regression analysis (LRA) and inverse matrix calculation (IMC) to forecast energy demand accurately in these underserved regions. By leveraging historical energy consumption data and pertinent predictors, such as meteorological conditions, population dynamics, economic indicators, and seasonal patterns, the model provides reliable forecasts. The application of the proposed methodology is demonstrated through a case study focused on remote regions of Bangladesh. The results showcase the approach's effectiveness in capturing the intricate dynamics of energy demand and its potential to inform sustainable energy management strategies in these remote areas. This research contributes to the advancement of energy planning and resource allocation in regions facing energy scarcity, fostering a path towards improved energy efficiency and development. These techniques can be applied to estimate short-term electricity demand for any rural or isolated region worldwide.

Efficient and Low-Complex Signal Detection with Iterative Feedback in Wireless MIMO-OFDM Systems.

To solve error propagation and exorbitant computational complexity of signal detection in wireless multiple-input multiple-output-orthogonal frequency division multiplexing (MIMO-OFDM) systems, a low-complex and efficient signal detection with iterative feedback is proposed via a constellation point feedback optimization of minimum mean square error-ordered successive interference cancellation (MMSE-OSIC) to approach the optimal detection. The candidate vectors are formed by selecting the candidate constellation points. Additionally, the vector most approaching received signals is chosen by the maximum likelihood (ML) criterion in formed candidate vectors to reduce the error propagation by previous erroneous decision, thus improving the detection performance. Under a large number of matrix inversion operations in the above iterative MMSE process, effective and fast signal detection is hard to be achieved. Then, a symmetric successive relaxation iterative algorithm is proposed to avoid the complex matrix inversion calculation process. The relaxation factor and initial iteration value are reasonably configured with low computational complexity to achieve good detection close to that of the MMSE with fewer iterations. Simultaneously, the error diffusion and complexity accumulation caused by the successive detection of the subsequent OSIC scheme are also improved. In addition, a method via a parallel coarse and fine detection deals with several layers to both reduce iterations and improve performance. Therefore, the proposed scheme significantly promotes the MIMO-OFDM performance and thus plays an irreplaceable role in the future sixth generation (6G) mobile communications and wireless sensor networks, and so on.

Efficient and Effective Nonconvex Low-Rank Subspace Clustering via SVT-Free Operators

With the growing interest in convex and nonconvex low-rank matrix learning problems, the widely used singular value thresholding (SVT) operators associated with rank relaxation functions often face higher computational complexity, particularly for large-scale data matrices. To improve the efficacy of low-rank subspace clustering and overcome the issue of high computational complexity, this work proposes an efficient and effective method that avoids the need for singular value decomposition (SVD) computations in the iteration scheme. This can be achieved through the use of a computationally efficient and compact formulation, as well as automatic removal of the optimal mean, which reduces time consumption and enhances evaluation performance. A unified clustering framework based on Schatten- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> norm regularized by ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,<i>q</i></sub> -norm can be formulated using this processing way, where inner element suppression can be achieved by choosing appropriate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p, q</i> ∈ (0,1). Additionally, calculating the optimal mean enhances the robustness of the proposed method in the presence of outliers. Unlike the general iteration scheme of the alternating direction method of multiplier (ADMM) algorithms that introduce auxiliary splitting variables, the proposed alternating re-weighted least square (ARwLS) algorithm uses matrix inverse and multiplication computations to obtain analytic solutions, resulting in faster processing speeds for each sub-problem. To further investigate, we provide the computational complexity of each iteration and the theoretical analysis of the convergence property, where the derived solution is a stationary point. Experimental results on synthetic data and several benchmark datasets demonstrate the promising efficiency and efficacy of the proposed clustering method compared to classical and competing algorithms.

Efficient Iterative Detection Based on Conjugate Gradient and Successive Over-Relaxation Methods for Uplink Massive MIMO Systems

Being a crucial aspect of fifth-generation (5G) mobile communications systems, massively multiple-input multiple-output (mMIMO) architectures are expected to help achieve the highest key performance indicators. However, the huge numbers of antennas used in such systems make it difficult to determine the inversion of the signal channel matrix relied upon by several detection methods, hence posing a problem with accurate estimation of the symbols sent. In this paper, conjugate gradient (CG) and successive over-relaxation (SOR) methods are selected to construct a new iterative approach that avoids the matrix inversion computation issue. This suggested approach for uplink mMIMO detection is based on a joint cascade structure of both iterative methods. The CG method is first applied and adjusted for the initial solution, followed by the SOR method in the final iterations for terminal computations, resulting in an algorithm with robust performance and low computational complexity. Furthermore, the new hybrid scheme operates based on the relaxation parameter, whose value has a great impact on error performance and, whose optimal determination is necessary. Numerical simulations reveal that the proposed scheme is capable of significantly improving signal detection accuracy with minimum complexity. The simulation results indicated that the proposed detector outperforms CG and SOR detectors, achieves close to optimal performance, requires fewer iterations, and reduces complexity.

Improved SSOR-based signal detector by using based on gauss-seidel method for large-scale MIMO systems

As the core technology in large-scale MIMO system, the reliability of signal detection has an important impact on the whole system. Traditional linear detectors, such as the zero-forcing (ZF), often involve very complex matrix inverse calculations when the number of antennas in large-scale MIMO systems is too large. In this paper, we first introduce the traditional signal detector based on SSOR method, the iterative operation is used to replace the complex matrix inverse calculation, thus reducing the computational complexity. Then, the Gauss-Seidel method is proposed to calculate the initial iteration value, and the simulation results show that the bit error rate (BER) performance of the improved SSOR method is greatly improved.

Brain Age Prediction With Improved Least Squares Twin SVR.

Alzheimer's disease (AD) is the prevalent form of dementia and shares many aspects with the aging pattern of the abnormal brain. Machine learning models like support vector regression (SVR) based models have been successfully employed in the estimation of brain age. However, SVR is computationally inefficient than twin support vector machine based models. Hence, different twin support vector machine based models like twin SVR (TSVR), ε-TSVR, and Lagrangian TSVR (LTSVR) models have been used for the regression problems. ε-TSVR and LTSVR models seek a pair of ε-insensitive proximal planes for generation of end regressor. However, SVR and TSVR based models have several drawbacks- i) SVR model is computationally inefficient compared to the TSVR based models. ii) Twin SVM based models involve the computation of matrix inverse which is intractable in real world scenario's. iii) Both TSVR and LTSVR models are based on empirical risk minimization principle and hence may be prone to overfitting. iv) TSVR and LTSVR assume that the matrices appearing in their formulation are positive definite which may not be satisfied in real world scenario's. To overcome these issues, we formulate improved least squares twin support vector regression (ILSTSVR). The proposed ILSTSVR modifies the TSVR by replacing the inequality constraints with the equality constraints and minimizes the slack variables using squares of L2 norm instead of L1. Also, we introduce a different Lagrangian function to avoid the computation of matrix inverses. We evaluated the proposed ILSTSVR model on the subjects including cognitively healthy, mild cognitive impairment and Alzheimer's disease for brain-age estimation. Experimental evaluation and statistical tests demonstrate the efficiency of the proposed ILSTSVR model for brain-age prediction.

Joint design of ordered QR precoding and SIC detection for MIMO VLC systems.

Ordered successive interference cancellation (OSIC) detection has been investigated to mitigate the high spatial correlation for multiple-input multiple-output (MIMO) visible light communication (VLC) systems. However, existing OSIC schemes have to perform reordering and matrix inversion for each detected symbol, which may lead to high complexity when a large number of symbols are transmitted. In this work, a joint design of ordered QR decomposition precoding and SIC detection (OQR-SIC) is proposed for MIMO VLC systems. This work jointly investigates OQR precoding and SIC detection to reduce the detection complexity while alleviating spatial correlation issue for MIMO VLC systems. In OQR-SIC, with the upper triangular matrix obtained by QR decomposition, the SIC detector can detect the symbol sequentially without reordering and matrix inversion calculations. To improve the system data rate, we further optimize the ordering of the columns of the channel matrix before QR decomposition and the power allocation of transmitted signals under the constraints of dimming control, total electrical power and reliable SIC detection. Simulation results demonstrate that the proposed OQR-SIC achieves 2 dB and 9.8 dB signal-to-noise ratio gains compared to conventional QR-SIC in 4 × 4 and 9 × 9 MIMO VLC systems, respectively, when the bit error rate is 10-3.

Iterative photonic processor for fast complex-valued matrix inversion

AnN×Niterative photonic processor is proposed for the first time, we believe, for fast computation of complex-valued matrix inversion, a fundamental but computationally expensive linear algebra operation. Compared to traditional digital electronic processing, optical signal processing has a few unparalleled features that could enable higher representational efficiency and faster computing speed. The proposed processor is based on photonic integration platforms–the inclusion of III-V gain blocks offers net neutral loss in the phase-sensitive loops. This is essential for the Richardson iteration method that is adopted in this paper for complex linear systems. Wavelength multiplexing can be used to significantly improve the processing efficiency, allowing the computation of multiple columns of the inverse matrix using a single processor core. Performances of the key building blocks are modeled and simulated, followed by a system-level analysis, which serves as a guideline for designing anN×NRichardson iteration processor. An inversion accuracy of&gt;98%can be predicted for a64×64photonic processor with a&gt;80times faster inversion rate than electronic processors. Including the power consumed by both active components and electronic circuits, the power efficiency of the proposed processor is estimated to be over an order of magnitude more energy-efficient than electronic processors. The proposed iterative photonic integrated processor provides a promising solution for future optical signal processing systems.

Inverse free reduced universum twin support vector machine for imbalanced data classification

Imbalanced datasets are prominent in real-world problems. In such problems, the data samples in one class are significantly higher than in the other classes, even though the other classes might be more important. The standard classification algorithms may classify all the data into the majority class, and this is a significant drawback of most standard learning algorithms, so imbalanced datasets need to be handled carefully. One of the traditional algorithms, twin support vector machines (TSVM), performed well on balanced data classification but poorly on imbalanced datasets classification. In order to improve the TSVM algorithm’s classification ability for imbalanced datasets, recently, driven by the universum twin support vector machine (UTSVM), a reduced universum twin support vector machine for class imbalance learning (RUTSVM) was proposed. The dual problem and finding classifiers involve matrix inverse computation, which is one of RUTSVM’s key drawbacks. In this paper, we improve the RUTSVM and propose an improved reduced universum twin support vector machine for class imbalance learning (IRUTSVM). We offer alternative Lagrangian functions to tackle the primal problems of RUTSVM in the suggested IRUTSVM approach by inserting one of the terms in the objective function into the constraints. As a result, we obtain new dual formulation for each optimization problem so that we need not compute inverse matrices neither in the training process nor in finding the classifiers. Moreover, the smaller size of the rectangular kernel matrices is used to reduce the computational time. Extensive testing is carried out on a variety of synthetic and real-world imbalanced datasets, and the findings show that the IRUTSVM algorithm outperforms the TSVM, UTSVM, and RUTSVM algorithms in terms of generalization performance.

Distributed reconstruction of time-varying graph signals via a modified Newton’s method

This paper develops a distributed reconstruction algorithm, that can be implemented efficiently, for time-varying graph signals. The reconstruction problem is formulated as an unconstrained optimization problem that minimizes the weighted sum of the data fidelity term and the regularization term. The regularizer used is the nonsmoothness measure of the temporal difference signal. The classical Newton’s method can be used to solve the optimization problem. However, computation of the Hessian matrix inverse is required, and this does not scale well with the graph size. Furthermore, a distributed implementation is not possible. An approximation to the inverse Hessian, that exploits the graph topology, is developed here. The resulting iterative algorithm can be implemented in a distributed manner, and scales well with the graph size. Convergence analysis of the algorithm is presented, which shows convergence to the global optimum. Numerical results, using both synthetic and real world datasets, will demonstrate the superiority of the proposed reconstruction algorithm over existing methods.

A Polak-Ribière-Polyak Conjugate Gradient Algorithm Optimized Broad Learning System for Lithium-ion Battery State of Health Estimation

Accurate state of health (SOH) estimation plays a significant role in the battery management system. This paper investigates a Polak-Ribière-Polyak conjugate gradient (PRPCG) algorithm optimized broad learning system (BLS) for lithium-ion battery SOH estimation. Firstly, effective health indicators (HIs) are extracted from the voltage curve in the constant current charge process. Secondly, a hybrid four layers BLS structure with mapped feature nodes and enhancement nodes connecting to the output is established to build both the linear and nonlinear relationships between the HIs and SOH, in which only the output weights require to be trained. Again, the PRPCG algorithm is adopted for searching optimal output weights without matrix inverse calculation during the training process. Furthermore, certain Gaussian noises are added to enhance the training data for solving the locally low accuracy problem. Finally, under the Oxford battery degradation data set, experiments validate the investigated algorithm has high accuracy in SOH estimation with the mean absolute error below 1%. The enhanced data can efficiently improve the model generalization ability.

Deep-Unfolding Beamforming for Intelligent Reflecting Surface Assisted Full-Duplex Systems

In this paper, we investigate an intelligent reflecting surface (IRS) assisted multi-user multiple-input multiple-output (MIMO) full-duplex (FD) system. We jointly optimize the active beamforming matrices at the access point (AP) and uplink users, and the passive beamforming matrix at the IRS to maximize the weighted sum-rate of the system. Since it is practically difficult to acquire the channel state information (CSI) for IRS-related links due to its passive operation and large number of elements, we conceive a mixed-timescale beamforming scheme. Specifically, the high-dimensional passive beamforming matrix at the IRS is updated based on the channel statistics while the active beamforming matrices are optimized relied on the low-dimensional real-time effective CSI at each time slot. We propose an efficient stochastic successive convex approximation (SSCA)-based algorithm for jointly designing the active and passive beamforming matrices. Moreover, due to the high computational complexity caused by the matrix inversion computation in the SSCA-based optimization algorithm, we further develop a deep-unfolding neural network (NN) to address this issue. The proposed deep-unfolding NN maintains the structure of the SSCA-based algorithm but introduces a novel non-linear activation function and some learnable parameters induced by the first-order Taylor expansion to approximate the matrix inversion. In addition, we develop a black-box NN as a benchmark. Simulation results show that the proposed mixed-timescale algorithm outperforms the existing single-timescale algorithm and the proposed deep-unfolding NN approaches the performance of the SSCA-based algorithm with much reduced computational complexity when deployed online.

Invited Tutorial: Analog Matrix Computing With Crosspoint Resistive Memory Arrays

Matrix computation is ubiquitous in modern scientific and engineering fields. Due to the high computational complexity in conventional digital computers, matrix computation represents a heavy workload in many data-intensive applications, e.g., machine learning, scientific computing, and wireless communications. For fast, efficient matrix computations, analog computing with resistive memory arrays has been proven to be a promising solution. In this Tutorial, we present analog matrix computing (AMC) circuits based on crosspoint resistive memory arrays. AMC circuits are able to carry out basic matrix computations, including matrix multiplication, matrix inversion, pseudoinverse and eigenvector computation, all with one single operation. We describe the main design principles of the AMC circuits, such as local/global or negative/positive feedback configurations, with/without external inputs. Mapping strategies for matrices containing negative values will be presented. The underlying requirements for circuit stability will be described via the transfer function analysis, which also defines time complexity of the circuits towards steady-state results. Lastly, typical applications, challenges, and future trends of AMC circuits will be discussed.