Matrix Inversion Algorithm Research Articles

Adaptive digital antenna array signal processing algorithm based on LDL decomposition

Two algorithms for inverting correlation matrices of digital antenna array signals are compared: LDL decomposition and the fringing method. The idea of comparing algorithms was to take into account the limitations on the length of the input data mantissa in the process of space-time processing using FPGAs. It has been found that rounding errors in the signal processing process in FPGAs can lead to the destruction of the directivity pattern. This is due to the fact that when implementing various algorithms for matrix inversion, the distribution of errors in the channels becomes interconnected. This means that when implementing adaptive antenna arrays on FPGAs, it is necessary to carefully choose the length of the mantissa and the conversion algorithm. In this regard, an algorithm for signal processing in an adaptive antenna array is proposed, its difference from the known ones is that when implemented on FPGAs, floating-point arithmetic of limited mantissa bit is used. When comparing two computationally efficient matrix inversion algorithms (the LDL method and the fringing method), the LDL method has higher stability. It was found that with an increase in the rank of the correlation matrix, the effect of rounding errors will be stronger. In this regard, it is necessary to determine in advance at what ratio of power sources and internal noise it is planned to use an adaptive processor.

Design of Low Power Thinned Smart Antenna for 6G Sky Connection

To improve radio access capability, sky connections relying on satellites or unmanned aerial vehicles (UAV), as well as high-altitude platforms (HAP) will be exploited in 6G wireless communication systems, complementing terrestrial networks. For long-distance communication, a large smart antenna will be used that is characterized by high amounts of power consumed by digital beamformers. This paper focuses on reducing power consumption by relying on a thinned smart antenna (TSA). The performance of TSA is investigated in the sub-6 GHz band. The differential evolution (DE) algorithm is used to optimize excitation weights of the individual dipoles in the antenna array and these excitation weights are then used in TSA for beamforming, with signal processing algorithms deployed. The DE technique is used with the least mean square, recursive least square and sample matrix inversion algorithms. The proposed method offers almost the same directivity, simultaneously ensuring lower side lobes (SLL) and reduced power consumption. For a TSA of 20, 31, and 64 dipoles, the power savings are 20%, 19.4%, and 17.2%, respectively. SLL reductions achieved, in turn, vary from 5.2 dB to 8.1 dB.

Conflict-Free Parallel Data Access Technology for Matrix Calculation in Memory System of ASIP of 5G/6G Macro Base Stations

Among the physical layer baseband algorithms in macro base stations, the matrix processing has the dom-inant computing cost, large data access overhead and com-plicated addressing mode. The existing data access methods are not the best solution for memory system of ASIP (Ap-plication Specific Instruction Set Processor) in 5G/6G macro stations. We hence proposed a parallel conflict-free data access method for matrix. Moreover, we proposed a non-redundant access method for positive definite matrix that stores only trigonometric part, and the corresponding parallel addressing method with low storage overhead. Our method solves the problem of data conflict and minimizes the memory space of ASIP, and supports ASIP to approach the performance limit of architecture to the maximum ex-tent under the constraint of architecture and data parallel-ism. We implemented the proposed method as a static memory optimizer, which provides a tool for the design and optimization of memory system of ASIP in 5G/6G macro base stations with low overhead. Experimental results show that for 64W64 positive definite matrix inversion algorithm based on Cholesky decomposition, our addressing method can save 32% of the execution time, and the cost of memory space is reduced by half.

Spatial cell-type enrichment predicts mouse brain connectivity

A fundamental neuroscience topic is the link between the brain's molecular, cellular, and cytoarchitectonic properties and structural connectivity. Recent studies relate inter-regional connectivity to gene expression, but the relationship to regional cell-type distributions remains understudied. Here, we utilize whole-brain mapping of neuronal and non-neuronal subtypes via the matrix inversion and subset selection algorithm to model inter-regional connectivity as a function of regional cell-type composition with machine learning. We deployed random forest algorithms for predicting connectivity from cell-type densities, demonstrating surprisingly strong prediction accuracy of cell types in general, and particular non-neuronal cells such as oligodendrocytes. We found evidence of a strong distance dependency in the cell connectivity relationship, with layer-specific excitatory neurons contributing the most for long-range connectivity, while vascular and astroglia were salient for short-range connections. Our results demonstrate a link between cell types and connectivity, providing a roadmap for examining this relationship in other species, including humans.

Quantum algorithm for the covariance matrix preparation and its application

Performing the eigendecomposition of the covariance matrix of the dataset is of great significance in the field of machine learning. However, classical operations will become time-consuming when involving large data sets. In this paper, in order to address this problem, we design an efficient quantum algorithm to prepare the covariance matrix state by means of quantum amplitude estimation. After that, we research on its application in principal component analysis and Mahalanobis distance calculation. Specifically, we obtain the transformation matrix for quantum principal component analysis based on the singular value estimation algorithm and the amplitude amplification algorithm. Furthermore, we invoke the quantum matrix inversion algorithm to calculate the Mahalanobis distance. The final complexity analysis shows that our proposed algorithms can achieve speedup compared to their classical counterparts under certain conditions.

Antenna Beampattern Matched Optimum Coherent Detection in High-Resolution Mechanically Scanning Maritime Surveillance Radars

Mechanically scanning radars are still widely used in ocean guard and surveillance areas nowadays, owing to simple structure, light weight and low cost. The antenna rotation causes amplitude modulation of target returns along pulses by the antenna azimuth beampattern. The traditional coherent detection that ignores this modulation suffers from some performance loss. In this paper, an antenna beampattern matched optimum coherent detector and a dual-threshold detection scheme are proposed for high-resolution mechanically scanning maritime surveillance radars. Firstly, it is shown that the modulation of target returns is determined by the antenna azimuth beampattern, scan rate, and pulse repetition interval (PRI) and the antenna scanning also affect the temporal correlation of sea clutter. Secondly, the antenna beampattern matched optimum coherent detector is given in compound-Gaussian sea clutter with inverse Gamma distributed texture and the pulse scanning mode is used to deal with unknown target azimuth angle in the matched filter. Thirdly, to mitigate the high computational cost of the pulse scanning mode, a dual-threshold scheme is given, which searches all pulse-range cells by short-time integration and further confirms suspected cells whose test statistics are between the two lower and higher thresholds in the search by long-time integration. Moreover, two fast recursive clutter covariance matrix inversion algorithms along pulses and range cells are presented, which bring a great reduction in computational complexity. Finally, real measured radar data is used to verify the proposed detector and scheme and the experimental results show that it behaves better in detection of small targets than the traditional detectors.

Quantum Regularized Least Squares

Linear regression is a widely used technique to fit linear models and finds widespread applications across different areas such as machine learning and statistics. In most real-world scenarios, however, linear regression problems are often ill-posed or the underlying model suffers from overfitting, leading to erroneous or trivial solutions. This is often dealt with by adding extra constraints, known as regularization. In this paper, we use the frameworks of block-encoding and quantum singular value transformation (QSVT) to design the first quantum algorithms for quantum least squares with general ℓ2-regularization. These include regularized versions of quantum ordinary least squares, quantum weighted least squares, and quantum generalized least squares. Our quantum algorithms substantially improve upon prior results on quantum ridge regression (polynomial improvement in the condition number and an exponential improvement in accuracy), which is a particular case of our result.To this end, we assume approximate block-encodings of the underlying matrices as input and use robust QSVT algorithms for various linear algebra operations. In particular, we develop a variable-time quantum algorithm for matrix inversion using QSVT, where we use quantum eigenvalue discrimination as a subroutine instead of gapped phase estimation. This ensures that substantially fewer ancilla qubits are required for this procedure than prior results. Owing to the generality of the block-encoding framework, our algorithms are applicable to a variety of input models and can also be seen as improved and generalized versions of prior results on standard (non-regularized) quantum least squares algorithms.

Direct Adaption Methods for Linear and Circular Antenna Arrays

Introduction. The mitigation of interferences that degrade the performance of navigation systems constitutes one of the most significant problems of contemporary satellite navigation. This problem is conventionally solved using digital adaptive space filters. Depending on a particular radio technical system, the mathematical description of digital signal processing methods may involve specific calculation structures implemented using specific calculation algorithms. For example, the use of centrosymmetric linear and circular antenna arrays in a radio navigation system allows the description of such systems in terms of Toeplitz and circulant sample covariance matrices, respectively, and the inversion of such matrices by means of special numerical methods in order to design a digital filter.Aim. A comparative analysis of the performance of space signal processing algorithms is carried out along with an estimation of Toeplitz and circulant sample covariance matrices and numerical methods of their inversion. The previously obtained results in this field are clarified.Materials and methods. An analysis of algorithm performance was carried out in the MATLAB environment using experimental recordings of satellite navigation signals and jammers obtained by an actual radio technical system.Results. A new expression was derived for estimating circulant sample covariance matrices. Formulae that describe a modification of the Bareiss numerical Toeplitz matrix inversion algorithm for the case of complex Hermitian matrix were introduced. An analysis of the results of computer simulation allowed the algorithms with the highest performance to be indicated. The amount of time taken by the algorithms based on Toeplitz and circulant matrices did not exceed 2.5 10⋅ −3 s and 0.04 s, respectively. The carrier-to-noise ratio in the processed signal was at least 46 dB. Conclusion. The formulae obtained and the algorithms analyzed can be used when implementing adaptive digital filtering of satellite navigation signals.

Inverse-coefficient black-box quantum state preparation

Black-box quantum state preparation is a fundamental building block for many higher-level quantum algorithms. The basic task of black-box state preparation is to transduce the data encoded as computational basis of quantum state into the amplitude. In the present work, we address the problem of transducing the reciprocal of the data, not the data itself into the amplitude, which is called the inverse-coefficient problem. This algorithm can be used directly as a subroutine in the matrix inversion algorithms. Furthermore, we extend this approach to address the more general nonlinear-coefficient problem in black-box state preparation. Our algorithm is based on the technique of inequality test. It can greatly relieve the need to do quantum arithmetic and the error is only resulted from the truncated error of binary string. The present algorithms enrich the algorithm library of black-box quantum state preparation and will be useful ingredients of quantum algorithm to implement non-linear quantum state transformations.

The Study of the Possibility of Applying Parallel Programming to the Algorithms of Space-Time Adaptive Processing

The article presents the description, assumptions and subsequent steps of the space-time adaptive processing (STAP) algorithms used as a signal processing tool in radars. The possibilities of object detection using the Sample Matrix Inversion (SMI) and Data Domain Least Squares (DDLS) algorithms were compared and showned. The article shows the impact of the use of parallel programming on the computation time of both algorithms. The main aim of this study was to propose an efficient method for the real-time implementation of the STAP algorithm in airborne radar systems. The idea of using parallel programming in STAP, supported only by the preliminary research results presented above, gives a real chance for the casual implementation of the STAP algorithm in a radar operating in close to real time mode.

Calculating the Moore–Penrose Generalized Inverse on Massively Parallel Systems

In this work, we consider the problem of calculating the generalized Moore–Penrose inverse, which is essential in many applications of graph theory. We propose an algorithm for the massively parallel systems based on the recursive algorithm for the generalized Moore–Penrose inverse, the generalized Cholesky factorization, and Strassen’s matrix inversion algorithm. Computational experiments with our new algorithm based on a parallel computing architecture known as the Compute Unified Device Architecture (CUDA) on a graphic processing unit (GPU) show the significant advantages of using GPU for large matrices (with millions of elements) in comparison with the CPU implementation from the OpenCV library (Intel, Santa Clara, CA, USA).

An improved quantum-inspired algorithm for linear regression

We give a classical algorithm for linear regression analogous to the quantum matrix inversion algorithm [Harrow, Hassidim, and Lloyd, Physical Review Letters'09] for low-rank matrices [Wossnig, Zhao, and Prakash, Physical Review Letters'18], when the input matrix A is stored in a data structure applicable for QRAM-based state preparation.Namely, suppose we are given an A∈Cm×n with minimum non-zero singular value σ which supports certain efficient ℓ2-norm importance sampling queries, along with a b∈Cm. Then, for some x∈Cn satisfying ‖x–A+b‖≤ε‖A+b‖, we can output a measurement of |x⟩ in the computational basis and output an entry of x with classical algorithms that run in O~(‖A‖F6‖A‖6σ12ε4) and O~(‖A‖F6‖A‖2σ8ε4) time, respectively. This improves on previous "quantum-inspired" algorithms in this line of research by at least a factor of ‖A‖16σ16ε2 [Chia, Gilyén, Li, Lin, Tang, and Wang, STOC'20]. As a consequence, we show that quantum computers can achieve at most a factor-of-12 speedup for linear regression in this QRAM data structure setting and related settings. Our work applies techniques from sketching algorithms and optimization to the quantum-inspired literature. Unlike earlier works, this is a promising avenue that could lead to feasible implementations of classical regression in a quantum-inspired settings, for comparison against future quantum computers.

Study of digital diagram formation for optimum interference and noise reduction in antenna arrays of different shapes with directional radiators

In the paper the digital beamforming is examined in azimuth and elevation, which allows more accurate formation of nulls and maximum of the radiation pattern. The minimum variance distortionless response beamformer, the sample matrix inversion algorithm using regularization, and the null steering algorithms at the output of cylindrical, hemispherical and planar lattices are considered. The ratio of the useful signal power to the active interference plus noise resulting power and the bit error ratio at the output of digital antenna arrays are estimated depending on the directivity of the antenna elements and the number of averaging samples. It has been established that a hemispherical antenna array can significantly increase the transmission reliability in comparison with the considered ones, which will reduce the computational load without using complicated diagramming algorithms.

Сравнительный анализ методов калибровки каналов сигналов антенн моноимпульсной радиолокации

Today, antenna arrays (ARs) are increasingly used as antennas in modern radio engineering systems. With the help of AR, you can control the shape of the radiation pattern (DN) and the position of its main lobe, called the beam of such an antenna system. Each AA channel contains microwave components for controlling the amplitude and phase of the signal passing through it. These components are characterized by errors in setting the amplitude and phase, which negatively affects the shape of the pattern and its characteristics. Therefore, to compensate for these errors, it is necessary to use the AP calibration at the production stage, and also during its operation. The paper presents various calibration methods; the most suitable and promising methods for AA calibration were determined; the principles of operation of these methods are considered, the advantages and disadvantages of the methods are identified, a comparative analysis of all methods is carried out, as well as the conditions under which it is more expedient to use one or another method. The following calibration methods are considered: Phaseless methods: (Rotating-Element Electric Field Vector method - REV method, Measurement of Two Elements method - MTE method, Sorace algorithm, Levita (Leavitt), Correlation algorithm Adaptive methods: (Music method, Capon method, SMI (Sample Matrix Inversion algorithm), least mean square error (LMS) algorithm, Quasi-Newton algorithm); Autocalibration methods: (Friedlander-Weiss method, Wang-Kedzou method, Estelie-Swindleharst-Otterson method), as well as the proposed two-phase AR calibration method. These methods are the most attractive from a practical point of view, since they can be used not only for factory calibration of the AA, but also during operation.

Quantum support vector machine based on regularized Newton method

An elegant quantum version of least-square support vector machine, which is exponentially faster than the classical counterpart, was given by Rebentrost et al. using the matrix inversion algorithm (HHL). However, the application of the HHL algorithm is restricted when the structure of the input matrix is not well. The iteration algorithms such as the Newton method are widespread in training the classical support vector machine. This paper demonstrates a quantum support vector machine based on the regularized Newton method (RN-QSVM), which achieves an exponential speed-up over classical algorithm. At first, the regularized quantum Newton algorithm is proposed to get rid of the constraint of input matrix. Then we train the RN-QSVM by using the regularized quantum Newton algorithm and classify a query sample by constructing the quantum state. Experiments demonstrate that RN-QSVM respectively provides advantages in terms of accuracy, robustness, and complexity compared to QSLS-SVM, LS-QSVM, and the classical method.

Fine‐grain task‐parallel algorithms for matrix factorizations and inversion on many‐threaded CPUs

AbstractWe extend a two‐level task partitioning previously applied to the inversion of dense matrices via Gauss–Jordan elimination to the more challenging QR factorization as well as the initial orthogonal reduction to band form found in the singular value decomposition. Our new task‐parallel algorithms leverage the tasking mechanism currently available in OpenMP to exploit “nested” task parallelism, with a first outer level that operates on matrix panels and a second inner level that processes the matrix either by ‐panels or by tiles, in order to expose a large number of independent tasks. We present a detailed performance analysis, including execution traces, which shows that the two‐level refinement into fine grain tasks allows for an improved load balancing and delivers high performance on current general‐purpose many‐core processors (CPUs) from Intel and AMD.

Recursive versus nonrecursive Richardson algorithms: systematic overview, unified frameworks and application to electric grid power quality monitoring

Sufficiently accurate, fast and computationally efficient solution of the system of linear equations is required in many estimation problems. Richardson iteration is one of the main solvers for linear equations, which provides optimization possibilities for time critical and accuracy critical applications. Convergence rate improvement and reduction of the computational complexity of the Richardson iteration are the most important problems in the area. The introduction of Newton–Schulz iterations is the efficient way for convergence rate improvement and the paper starts with systematic overview of the high-order Newton–Schulz matrix inversion algorithms. In addition, the unified framework for recursive computationally efficient convergence accelerators and error models for a number of combinations of Richardson and Newton–Schulz iterations is developed. A new nonrecursive parameter estimation concept is introduced and compared in this paper with recursive estimation. Recursive and nonrecursive Richardson algorithms together with the standard LU decomposition method were applied to the electric grid power quality monitoring problem. The algorithms were tested for the detection of the sag and swell signatures in the voltage and current signals on real data in three-phase power system. Nonrecursive Richardson algorithms which save close to half of the computational time compared to LU decomposition method were recommended for power quality monitoring applications.

An improved algorithm for computing hitting probabilities of quantum walks

Quantum walks are the quantum corresponding version of classical random walks, providing polynomial and even exponential acceleration for some classical difficult problems. In recent years, there have been numerous researches on quantum walks, among which the hitting probabilities problem is another crucial problem. In 2021, Guan et al. have proposed an HHL-based algorithm to calculate the hitting probabilities of quantum walks, which can effectively transform the original problem into an inverse matrix problem, then achieve an acceleration effect on the corresponding classical algorithm under some specific conditions. However, we propose a new algorithm consisting of an improved quantum matrix inversion algorithm with complexity Oκ2log1.5(κ/ε)polylog(n), where κ is the condition number of the matrix, n+1 is the number of positions of the one-dimensional quantum walks, and ε is the expected accuracy of the output state. Compared to the HHL-based algorithm, our improved algorithm achieves an exponential improvement in the dependence on accuracy.

Gaussian Belief Propagation Solvers for Nonsymmetric Systems of Linear Equations

In this paper, we argue for the utility of deterministic inference in the classical problem of numerical linear algebra, that of solving a linear system. We show how the Gaussian belief propagation solver, known to work for symmetric matrices can be modified to handle nonsymmetric matrices. Furthermore, we introduce a new algorithm for matrix inversion that corresponds to the generalized belief propagation derived from the cluster variation method (or Kikuchi approximation). We relate these algorithms to LU and block LU decompositions and provide certain guarantees based on theorems from the theory of belief propagation. All proposed algorithms are compared with classical solvers (e.g., Gauss-Seidel, BiCGSTAB) with application to linear elliptic equations. We also show how the Gaussian belief propagation can be used as multigrid smoother, resulting in a substantially more robust solver than the one based on the Gauss-Seidel iterative method.

Response Times Reconstructor Based on Mathematical Expectation Quotient for a High Priority Task over RT-Linux

Every computer task generates response times depending on the computer hardware and software. The response times of tasks executed in real-time operating systems such as RT-Linux can vary as their instances evolve even though they always execute the same algorithm. This variation decreases as the priority of the tasks increases; however, the minimum and maximum response times are still present in the same task, and this complicates its monitoring, decreasing its level of predictability in case of contingency or overload, as well as making resource sizing difficult. Therefore, the need arises to propose a model capable of reconstructing the dynamics of response times for the instances of a task with high priority in order to analyze their offline behavior under specific working conditions. For this purpose, we develop the necessary theory to build the response time reconstruction model. Then, to test the proposed model, we set up a workbench consisting of a single board computer, PREEMPT_RT, and a high priority task generated by the execution of a matrix inversion algorithm. This work demonstrates the application of the theory in an experimental process, presenting a way to model and reconstruct the dynamics of response times by a high-priority task on RT-Linux.