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This paper introduces the VDR decomposition of Chebyshev-Vandermonde matrices, where V represents an ordinary Vandermonde matrix, D is diagonal, and R is upper triangular. Our motivation for this work stems from the study by Brubeck et al. [Vandermonde with Arnoldi. SIAM Rev. 2021;63(2):405–415]. We explore the VDR decomposition and combine it with a QR decomposition of V for Chebyshev-Vandermonde matrices. Furthermore, we demonstrate that the upper triangular factor in the QR decomposition can be recursively obtained from the upper Hessenberg matrix in the Arnoldi process. To compare our approach with the results obtained using the default QR code in Matlab, we present experimental results. Finally, we provide an algorithm for solving linear systems with Chebyshev-Vandermonde matrices as coefficient matrices, and experimental results for an upper bound on the relative error by applying an a posteriori error analysis for the computed solutions of the systems. [ABSTRACT FROM AUTHOR]

The perturbation of the QR decompostion is analyzed from the probalistic point of view. The perturbation error is approximated by a first-order perturbation expansion with high probability where the perturbation is assumed to be random. Different from the previous normwise perturbation bounds using the Frobenius norm, our techniques are used to develop the spectral norm, as well as the entry-wise perturbation bounds for the stochastic perturbation of the QR decomposition. The statistics tends to be tighter (in the sense of the expectation) and more realistic than the classical worst-case perturbation bounds. The novel perturbation bounds are applicable to a wide range of problems in statistics and communications. In this paper, we consider the perturbation bound of the leverage scores under the Gaussian perturbation, the probability guarantees and the error bounds of the low rank matrix recovery, and the upper bound of the errors of the tensor CUR-type decomposition. We also apply our perturbation bounds to improve the robust design of the Tomlinson-Harashima precoding in the Multiple-Input Multiple-Output (MIMO) system. [ABSTRACT FROM AUTHOR]

Solving the Perspective-n-Point (PnP) problem is difficult in low-power systems due to the high computing workload. To handle this challenge, we present an originally designed FPGA implementation of a PnP solver based on Vivado HLS. A matrix operation library and a matrix decomposition library based on QR decomposition have been developed, upon which the EPnP algorithm has been implemented. To enhance the operational speed of the system, we employed pipeline optimization techniques and adjusted the computational process to shorten the calculation time. The experimental results show that when the number of input data points is 300, the proposed system achieves a processing speed of 45.2 fps with a power consumption of 1.7 W and reaches a peak-signal-to-noise ratio of over 70 dB. Our system consumes only 3.9% of the power consumption per calculation compared to desktop-level processors. The proposed system significantly reduces the power consumption required for the PnP solution and is suitable for application in low-power systems. [ABSTRACT FROM AUTHOR]

Ground vibration test is an important part of aircraft model development and strength verification. In aircraft ground vibration test，an appropriate excitation scheme can significantly shorten the test's period and improve the accuracy of mode identification. In this paper，QR decomposition based on the finite element model is used for global planning of exciter placement，and mode participation is introduced to evaluate the plan's exciting efficiency. The multivariate mode indicator function is used to optimize the exciting force vector for closely spaced modes. Finally，these methods are verified in a small general aircraft's ground vibration test，which significantly improves the test's efficiency and reduces the difficulty of closely spaced modes identification. The excitation scheme optimization method adopted in this article has high engineering application value.

To improve the speeds of the traditional nuclear norm minimization methods, a fast tri-factorization method (FTF) was recently proposed for matrix completion, and it received widespread attention in the fields of machine learning, image processing and signal processing. However, its low convergence accuracy became increasingly obvious, limiting its further application. To enhance the accuracy of FTF, a generalized tri-factorization method (GTF) is proposed in this paper. In GTF, the nuclear norm minimization model of FTF is improved to a novel L 1 , p (0 < p < 2) norm minimization model that can be optimized very efficiently by using QR decomposition. Since the L 1 , p norm is a tighter relaxation of the rank function than the nuclear norm, the GTF method is much more accurate than the traditional methods. The experimental results demonstrate that GTF is more accurate and faster than the state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Abstract To solve the problems of the low efficiency of parameter allocation and time-consuming computation of output weights in the hidden layers of stochastic configuration networks (SCNs), an optimization method is proposed to improve the SCNs construction efficiency. Firstly, with the increase in the number of hidden layer nodes, the key parameters that determine the strictness of the supervision mechanism are reconstructed to speed up the configuration efficiency of hidden layer input weights and biases. Then, the incremental mechanism of the SCNs are combined with the QR decomposition method, and the output weights are calculated by iteratively updating the transformation matrix, thus reducing the computational complexity of training the SCNs. Finally, the proposed method is validated on four standard datasets and historical data of a municipal solid waste incineration process. The experimental results show that the proposed method improves the efficiency of SCN construction while guaranteeing the prediction accuracy of SCNs model.

In this article, we determine probabilistic approximations of the entries of random perturbation matrices implementing the Markoff inequality. These approximations are used to derive with prescribed probability asymptotic componentwise perturbation bounds of some orthogonal and unitary matrix decompositions. We show that the probabilistic asymptotic bounds are significantly less conservative than the corresponding deterministic perturbation bounds. As case studies, we consider the determining of probabilistic perturbation bounds of the QR decomposition, the singular value decomposition and the Schur decomposition of a matrix using an unified method for asymptotic componentwise perturbation analysis of these decompositions. It is demonstrated that the probability bounds of the orthogonal transformations, singular values and eigenvalues are much tighter than the corresponding deterministic asymptotic bounds. The probabilistic bounds derived are appropriate for perturbation analysis of large‐order matrix problems. [ABSTRACT FROM AUTHOR]

This article introduces FiGaRo, an algorithm for computing the upper-triangular matrix in the QR decomposition of the matrix defined by the natural join over relational data. FiGaRo 's main novelty is that it pushes the QR decomposition past the join. This leads to several desirable properties. For acyclic joins, it takes time linear in the database size and independent of the join size. Its execution is equivalent to the application of a sequence of Givens rotations proportional to the join size. Its number of rounding errors relative to the classical QR decomposition algorithms is on par with the database size relative to the join output size. The QR decomposition lies at the core of many linear algebra computations including the singular value decomposition (SVD) and the principal component analysis (PCA). We show how FiGaRo can be used to compute the orthogonal matrix in the QR decomposition, the SVD and the PCA of the join output without the need to materialize the join output. A suite of experiments validate that FiGaRo can outperform both in runtime performance and numerical accuracy the LAPACK library Intel MKL by a factor proportional to the gap between the sizes of the join output and input. [ABSTRACT FROM AUTHOR]

In magnetic resonance (MR) image reconstruction, the orthogonal matching pursuit (OMP) is widely recognized for its simplicity and competitive performance. However, OMP designs a termination condition based on some prior information such as sparsity and noise intensity. In practice, the unknown prior information of MR images cannot guarantee accurate reconstruction. To make OMP suitable for magnetic resonance imaging (MRI), we propose an automatic threshold OMP algorithm based on QR decomposition (ATOMP-QR). The termination condition of ATOMP-QR, which utilizes the mutual incoherence property of the sensing matrix, is related to whether the residual vector includes the orthogonal projection component of measurements. Then, to avoid the computation of pseudo-inverse and accelerate reconstruction speed, we perform QR decomposition on the measurement matrix. We conduct the MRI experiments to evaluate the superiority and effectiveness of ATOMP-QR in peak signal-to-noise ratio (PSNR), structure similarity index measure (SSIM), and running time. Specifically, for the T1-w image with a sparsity of 10, the PSNR was improved from 24 to 32 dB; the SSIM was increased from 0.87 to 0.99. The maximum time consumed decreased from 0.2276 to 0.0107 s. [ABSTRACT FROM AUTHOR]

The two-sided Jacobi method for singular value decomposition (SVD) has the advantage of obtaining singular vectors quickly and accurately. In previous research, fast and accurate implementations of the two-sided Jacobi method have been achieved for real matrices. In this study, we implemented SVD for complex matrices using the two-sided Jacobi method. In SVD, given rectangular matrices can be converted into upper-triangular matrices by conducting QR decomposition as a preprocessing step. Then, the upper-triangular matrices are decomposed. In the case where the given matrices are complex, the upper-triangular matrices are complex. Two implementations are proposed: the first requires SVD for complex matrices, whereas the second requires SVD for only real matrices. [ABSTRACT FROM AUTHOR]

QR decomposition is widely used for solving the least squares problem. However, existing materials about it may be too abstract for non-mathematicians, especially STEM students, and/or require serious background in linear algebra. The paper describes theoretical background and examples of GNU Octave compatible MATLAB scripts that give relatively simple but complete explanations about how to use QR decomposition for the least squares problem solution. Only basic knowledge of linear algebra and calculus are required. Both Givens rotations and Householder reflections usage for the linear least squares problem were considered. It was shown that the algorithm based on Givens rotations is even easier to program than explicit formation of the normal equations with subsequent usage of Gaussian elimination. [ABSTRACT FROM AUTHOR]

The diffusive viscous (DV) model is a useful tool for interpreting low-frequency seismic attenuation and the influence of fluid saturation on frequency-dependent reflections. Among present methods for the numerical solution of the corresponding DV wave equation, the finite-difference frequency-domain (FDFD) method with complex-valued adaptive coefficients (CVAC) has the advantage of efficiently suppressing both numerical dispersion and numerical attenuation. In this research, the FDFD method with CVAC is first generalized to a 3D DV equation. In addition, the current calculation of CVAC involves the numerical integration of propagation angles, conjugate gradient (CG) iterative optimization, and the sequential selection of initial values, which is difficult and inefficient for implementation. An improved method is developed for calculating CVAC, in which a complex-valued least-squares problem is constructed by substituting the 3D complex-valued plane-wave solutions into the FDFD scheme. The QR-decomposition method is used to efficiently solve the least-squares problem. Numerical dispersion and attenuation analyses reveal that the FDFD method with CVAC requires ∼2.5 spatial points in a wavelength within a dispersion deviation of 1% and an attenuation deviation of 10% for the 3D DV equation. An analytic solution for 3D DV wave equation in homogeneous media is proposed to verify the effectiveness of the proposed method. Numerical examples also demonstrate that the FDFD method with CVAC can obtain accurate wavefield modelling results for 3D DV models with a limited number of spatial points in a wavelength, and the FDFD method with QR-based CVAC requires less computational time than the FDFD method with CG-based CVAC. [ABSTRACT FROM AUTHOR]

This paper introduces two robust and optimized watermarking techniques designed to protect the copyright of color digital images. These techniques involve embedding a biometric-based digital watermark, generated from an iris image, within the redistributed invariant discrete wavelet transform (RIDWT) domain. In the proposed algorithms, the Y channel of the color image in the YCbCr color space is initially decomposed using RIDWT. Then, the lowest frequency sub-band is partitioned into non-overlapping blocks. Blocks with lower standard deviations are selected to embed each bit of the watermark derived from the iris image. To conceal these watermark bits, two distinct approaches are used. In the first approach, selected blocks undergo a transformation using QR decomposition, while the second approach utilizes Schur decomposition for the same purpose. Before the embedding step, we optimize the embedding strength of the watermarking using the firefly algorithm, striking a balance between robustness and perceptual transparency. Experimental evaluations demonstrate that the proposed algorithms not only deliver good perceptual quality but also effectively resist various geometric and common image processing attacks. Our results underscore the superior robustness of these techniques compared to existing studies. Additionally, both methods maintain an exceptionally low error rate during ownership verification. [ABSTRACT FROM AUTHOR]

In order to improve the blind separation performance of approximate joint diagonalization of real matrix sets and to avoid trivial solutions, a Jacobi‑like joint diagonalization algorithm based on QR decomposition is proposed. Using the numerical stability of QR decomposition, the Jacobi rotation matrix is used to decompose the separation matrix into the product of several elementary triangular matrices and orthogonal matrices. The structure of Jacobi rotation matrix and the related elements of the target matrix transformation are used to obtain the optimal parameters. The high-dimensional minimization problem is iteratively transformed into a series of low-dimensional sub-problems, which enhances the recovery accuracy of the source signal. The algorithm complexity is reduced by solving the simplified Frobenius-norm objective function. The simulation results of mixed electrocardiogram (ECG) signals show that compared with QRJ2D, LUCJD and EGJLUD, the proposed algorithm has certain advantages in separation accuracy and convergence speed. [ABSTRACT FROM AUTHOR]

Open Radio Access Networks (Open RANs), realized fully in software, require excessive computing resources to support time-sensitive signal-processing algorithms in the physical layer. Among them, multiple-input-multiple-output (MIMO) processing is a key functionality used to drive higher connectivity in the uplink, but it is computationally intensive, triggering the need for hardware acceleration to overcome the processing inefficiency of software-based solutions. Additionally, energy efficiency is becoming a key focus in Open RAN to enable sustainable deployments that utilize available resources efficiently. Because channel-inversion complexity increases polynomially with the number of users in linear detectors, such as zero-forcing (ZF) and minimum-mean-square-error (MMSE), acceleration based on channel-inverse approximations has gained significant attention. However, they unnecessarily multiply the number of base station (BS) antennas to ensure accurate detection, leading to a drastic increase in power consumption owing to the additional radio frequency (RF) chains employed. In contrast, linear detectors achieve a sufficiently good performance with only twice the number of BS antennas as users. This work introduces an exact-MMSE and soft-output hardware accelerator that includes an inversion-free, highly-parallel QR decomposition (QRD) architecture and a low-complexity detector stage with per-cycle soft-output generation, significantly improving the processing latency and throughput. The proposed architecture is fully scalable to support diverse MIMO configurations. Implementation evaluations on a Xilinx Virtex Ultrascale+ field-programmable gate array (FPGA) demonstrate that the proposed exact solution can achieve more than $2\times $ improvement in hardware throughput over existing approximate designs. Moreover, the peak throughput can be increased around 10-fold in slowly fading channels.

In this paper, we consider the variable selection for a class of semiparametric spatial autoregressive models. By using orthogonal projection technique, we propose a new orthogonality-based variable selection procedure, which can select important covariates, and can identify the significance of spatial effects simultaneously. The consistency of the proposed variable selection procedure and the convergence rate of the resulting estimators are derived under some regular conditions. Furthermore, some simulation studies are carried out to examine the finite sample performance of the proposed method. [ABSTRACT FROM AUTHOR]

The t-distributed stochastic neighbor embedding (t-SNE) is a powerful technique for visualizing high-dimensional datasets. By reducing the dimensionality of the data, t-SNE transforms it into a format that can be more easily understood and analyzed. The existing approach is to visualize high-dimensional data but not deeply visualize. This paper proposes a model that enhances visualization and improves the accuracy. The proposed model combines the non-linear embedding technique t-SNE, the linear dimensionality reduction method principal component analysis (PCA), and the QR decomposition algorithm for discovering eigenvalues and eigenvectors. In Addition, we quantitatively compare the proposed model QRPCA-t-SNE with PCA-t-SNE using the following criteria: data visualization with different perplexity and different principal components, confusion matrix, model score, mean square error (MSE), training, testing accuracy, receiver operating characteristic curve (ROC) score, and AUC score. [ABSTRACT FROM AUTHOR]

In hidden Markov chain (HMC) models, widely used for target tracking, the process noise and measurement noise are in general assumed to be independent and Gaussian for mathematical simplicity. However, the independence and Gaussian assumptions do not always hold in practice. For instance, in a typical radar tracking application, the measurement noise is correlated over time as the sampling frequency of a radar is generally much higher than the bandwidth of the measurement noise. In addition, target maneuvers and measurement outliers imply that the process noise and measurement noise are non-Gaussian. To solve this problem, we resort to triplet Markov chain (TMC) models to describe stochastic systems with correlated noise and derive a new filter under the maximum correntropy criterion to deal with non-Gaussian noise. By stacking the state vector, measurement vector, and auxiliary vector into a triplet state vector, the TMC model can capture the complete dynamics of stochastic systems, which may be subjected to potential parameter uncertainty, non-stationarity, or error sources. Correntropy is used to measure the similarity of two random variables; unlike the commonly used minimum mean square error criterion, which uses only second-order statistics, correntropy uses second-order and higher-order information, and is more suitable for systems in the presence of non-Gaussian noise, particularly some heavy-tailed noise disturbances. Furthermore, to reduce the influence of round-off errors, a square-root implementation of the new filter is provided using QR decomposition. Instead of the full covariance matrices, corresponding Cholesky factors are recursively calculated in the square-root filtering algorithm. This is more numerically stable for ill-conditioned problems compared to the conventional filter. Finally, the effectiveness of the proposed algorithms is illustrated via three numerical examples. [ABSTRACT FROM AUTHOR]

This paper presents a QR decomposition based low-rank approximation algorithm for training and prediction in Gaussian process regression. Conventional low-rank methods like FITC, Nyström, etc., rely on low-rank approximations to the full kernel matrix that are derived from a set of representative data points. Training with FITC involves the selection of these representative points and is computationally intensive when the dimension of the dataset is large. Prediction accuracy of Nyström suffers when the number of representative points is small or when the length scale is small. The algorithm proposed here uses the thin QR decomposition of the low-rank matrix used in the Nyström approximation. We show that the marginal likelihood and its derivatives needed for training can be split into the subspace belonging to that approximation and its orthogonal complement. During prediction, we further restrict the target vectors into this subspace and thus eliminate the numerical error caused by very low noise variance. Use of the QR factorization improves the training and prediction performance. Experiments on real and synthetic data show that our approach is more accurate and faster than conventional methods. Our algorithm performs well at low length scales and achieves the prediction accuracy comparable to the full kernel matrix approach even for low rank approximations, without the need to optimize the representative points. This results in a simpler and faster approximation that could be used to scale Gaussian process regression to large datasets. [ABSTRACT FROM AUTHOR]

Block Landweber scheme plays an important role in linear image reconstruction field. In this paper, we propose a block partition strategy based on oblique QR decomposition so as to improve the computational efficiency of block Landweber scheme. And then we provide sensitivity analysis of subproblem from the linear imaging system. Furthermore, in order to prevent excessive memory usage in the process of oblique QR decomposition, we design pseudo code to update these two partition criteria. Finally, we test the performance of our strategy by the image reconstruction of block simultaneous algebra reconstruction technique which is a well known special case of Landweber scheme. Compared with sequential partition, our block partition strategy provides reconstructed images with smaller root mean square errors, but with fewer iteration cycles. [ABSTRACT FROM AUTHOR]

The finite-difference frequency domain (FDFD) method is widely applied for simulating seismic wavefields, and a key to achieving successful FDFD simulation is to construct FDFD coefficients that can effectively suppress numerical dispersion. Among the existing FDFD coefficients for seismic wavefield simulation, adaptive FDFD coefficients that vary with the number of wavelengths per grid can suppress numerical dispersion to the maximum extent. The current methods for calculating adaptive FDFD coefficients involve numerical integration, conjugate gradient (CG) optimization, sequential initial value selection, and smooth regularization, which are difficult to implement and inefficient in calculations. To simplify the calculation of adaptive FDFD coefficients and improve the corresponding computational efficiency, this paper proposes a new method for calculating adaptive FDFD coefficients. First, plane-wave solutions with different discrete propagation angles are substituted in the FDFD scheme, and the corresponding least-squares problem is constructed. As this problem is ill-conditioned and obtaining smooth adaptive FDFD coefficients by the conventional solving method based on normal equations is difficult, this paper proposes solving the least-squares problem by solving the corresponding overdetermined linear system of equations through QR matrix decomposition. Compared with the existing methods for calculating adaptive FDFD coefficients based on numerical integration, CG optimization, and sequential initial value selection, the proposed method allows for a simplified computational process and considerably higher computational efficiency. Numerical wavefield simulation results show that the adaptive-coefficient FDFD method based on QR matrix decomposition can achieve the same accuracy as those based on numerical integration, CG optimization, and sequential initial value selection while requiring less computation time. [ABSTRACT FROM AUTHOR]

Medical video watermarking is one of the beneficial and efficient tools to prohibit important patients' data from illicit enrollment and redistribution. In this paper, a new blind watermarking scheme has been proposed to improve the confidentiality, integrity, authenticity, and perceptual quality of a medical video with minimum distortion. The proposed scheme is based on 2D-DWT and dual Hessenberg-QR decomposition, where the input medical video is initially processed into frames. Then, the processed frames are transformed into sub-bands using 2D-DWT, followed by applying Hessenberg-QR decomposition on the selected wavelet HL2 sub-band. The watermark is scrambled via Arnold cat map to raise confidentiality and then concealed in the modified selected features. The watermark is extracted in a fully blind mode without referencing the original video, which reduces the extraction time. The proposed scheme maintained a fundamental tradeoff between robustness and visual imperceptibility compared to existing methods against many commonly encountered attacks. The visual imperceptibility has been evaluated using well-known metrics PSNR, SSIM, Q-index, and histogram analysis. The proposed scheme achieves a high PSNR value of (70.6899 dB) with minimal distortion and a high robustness level with an average NC value of (0.9998) and BER value of (0.0023) while conserving a large payload capacity. The obtained results show superior performance over similar video watermarking methods. The limitation of this scheme is the elapsed time during the embedding process since we utilized dual Hessenberg-QR decomposition. One possible solution to reduce time consumption is simple decompositions like bound-constrained SVM or similar decompositions.

This paper presents a robust empirical likelihood procedure based on the exponential squared loss (ESL) function and leverage-based weights for the partially linear varying coefficient model with longitudinal data. The proposed method simultaneously solves the problems of correlation structure of longitudinal data and the existence of outliers, and achieves robustness and efficiency by introducing an appropriate data-driven tuning parameter. More importantly, profit from the QR decomposition technique, our method allows the parametric and nonparametric parts of the models to be estimated separately, which can avoid the mutual influence between them and make the implementation easier. Under some mild conditions, the large sample theoretical properties of the robust empirical likelihood approach are established. Simulation studies and a real data analysis are also carried out to assess and illustrate the finite sample performance. [ABSTRACT FROM AUTHOR]

A linear transformation from vector space to another vector space can be represented as a matrix. This close relationship between the matrix and the linear transformation is helpful for the study of matrices. In this paper, the tensor is regarded as a generalization of the matrix from the viewpoint of the linear transformation instead of the quadratic form in matrix theory; we discuss some operations and present some definitions and theorems related to tensors. For example, we provide the definitions of the triangular form and the eigenvalue of a tensor, and the theorems of the tensor QR decomposition and the tensor singular value decomposition. Furthermore, we explain the significance of our definitions and their differences from existing definitions. [ABSTRACT FROM AUTHOR]

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Low-rank matrix completion, which aims to recover a matrix with many missing values, has attracted much attention in many fields of computer science. A low-rank matrix fitting (LMaFit) method has been proposed for fast matrix completion recently. However, this method cannot converge accurately on matrices of real-world images. For improving the accuracy of LMaFit method, an improved low-rank matrix fitting (ILMF) method based on the weighted L 1 , p norm minimization is proposed in this paper, where the L 1 , p norm is the summation of the p -power (0 < p < 2) of L 1 norms of rows in a matrix. In the proposed method, i.e. the ILMF method, the incomplete matrix that may be corrupted by noises is decomposed into the summation of a low-rank matrix and a noise matrix at first. Then, a weighted L 1 , p norm minimization problem is solved by using an alternating direction method for improving the accuracy of matrix completion. Experimental results on real-world images show that the ILMF method has much better performances in terms of both the convergence accuracy and convergence speed than the compared methods. [ABSTRACT FROM AUTHOR]

By combining penalized GMM estimation method with the QR decomposition technique, we propose an orthogonal projection-based regularization estimation method for a class of partially linear spatial autoregressive models. The proposed method can select important covariates in the parametric component, and can also identify the significance of spatial effects. Under some conditions, some theoretical properties are studied, such as the consistency of the proposed variable selection procedure and the oracle property of the resulting estimators for parametric and nonparametric components. Furthermore, some simulation studies are carried out to examine the finite sample performances of the proposed regularization estimation method. [ABSTRACT FROM AUTHOR]

For the active safety control of the vehicle, it is extremely important to estimate the vehicle state in real-time and accurately during the driving process. A joint state and parameter estimation method based on QR decomposition and receding horizon estimation (RHE) is proposed. Firstly, by introducing the receding horizon strategy, the authors optimized the state and parameter estimation with a fixed number of variables, which can better deal with the estimation problem of time-varying parameters. Then, based on the principle of forward dynamic programming, the calculation of arrival cost is transformed into a least square equation, which is solved by QR decomposition. At the same time, an update method of arrival cost based on QR decomposition is given. In this way, the whole receding horizon estimation method is based on the optimization, and the feedback mechanism is introduced to improve the estimation accuracy and speed. The simulation results show that the accuracy of receding horizon estimation is obviously better than that of unscented Kalman filter (UKF), and the arrival cost update method based on QR decomposition is more convenient than the traditional arrival cost update method based on error covariance estimation. [ABSTRACT FROM AUTHOR]

This paper presents a parallel approach to the Levenberg-Marquardt algorithm (LM). The use of the Levenberg-Marquardt algorithm to train neural networks is associated with significant computational complexity, and thus computation time. As a result, when the neural network has a big number of weights, the algorithm becomes practically ineffective. This article presents a new parallel approach to the computations in Levenberg-Marquardt neural network learning algorithm. The proposed solution is based on vector instructions to effectively reduce the high computational time of this algorithm. The new approach was tested on several examples involving the problems of classification and function approximation, and next it was compared with a classical computational method. The article presents in detail the idea of parallel neural network computations and shows the obtained acceleration for different problems. [ABSTRACT FROM AUTHOR]

In recent years,with the rapid development of cloud computing and artificial intelligence,digital images are more and more widely used in multimedia,medical and military fields.Among them,remote sensing technology can detect and process remote targets through the theory of electromagnetic wave,which makes the transmission security of remote sensing images increasingly significant.In order to solve the problems of digital image information security and privacy protection,this paper proposes a reversible remote sensing image hiding algorithm based on novel Hachimoji deoxyribonucleic acid (DNA) and QR decomposition.Firstly,the system parameters and initial values of the coupled map lattice(CML) are updated by using the information entropy of remote sensing image and host image,and the generated chaotic sequences are used as the one-time key stream of the encryption process,which increases resistance to known/selected plain-text attacks.Then,Hachimoji DNA technology is used to encode the remote sensing image with 8-bit base,and key streams are used to perform XOR operation and cyclic shift operation on the image matrix.Finally,the host image is decomposed into blocks,and the encrypted remote sensing image is embedded into the host block in the form of DNA bases.In particular,the host image still visually meaningful after embedding remote sensing information,and the embedded image information can also be extracted from it without loss.Simulation results show that the proposed algorithm has good embedding effect,high security and strong robustness.

The paper presents a rigorous perturbation analysis of the QR decomposition A = Q R of an n × m matrix A using the method of splitting operators. New asymptotic componentwise perturbation bounds are derived for the elements of Q and R and the subspaces spanned by the first p ≤ m columns of A. The new bounds are less conservative than the known bounds and are significantly better than the normwise bounds. An iterative scheme is proposed to determine global componentwise bounds in the case of perturbations for which such bounds are valid. Several numerical results are given that illustrate the analysis and the quality of the bounds obtained. [ABSTRACT FROM AUTHOR]

Response functions completely define the constitutive equations for a hyperelastic material. A strain measure providing an orthogonal stress response, grants response functions directly from experimental curves. One of these strain measures is the Laplace stretch based on QR-decomposition of the deformation gradient. Such a recovery of response functions from experimental data fits the paradigm of data-driven modeling. The set of independent conjugate stress–strain base pairs were proposed as a simple alternative for constitutive modeling and thus might be efficient for data-driven modeling. In the present paper we explore applicability of the conjugate pairs approach for data-driven modeling. The analysis is based on representation of the conjugate pairs in terms of the response functions due to the Laplace stretch. Our analysis shows that one can not guarantee independence of these pairs except in the case of infinitesimal strain. [ABSTRACT FROM AUTHOR]

In this study, we address the selection of both fixed and random effects in partial linear mixed effects models. By combining B-spline and QR decomposition techniques, we propose a double-penalized likelihood procedure for both estimating and selecting these effects. Furthermore, we introduce an orthogonality-based method to estimate the non-parametric component, ensuring that the fixed and random effects are separated without any mutual interference. The asymptotic properties of the resulting estimators are investigated under mild conditions. Simulation studies are conducted to evaluate the finite sample performance of the proposed method. Finally, we demonstrate the practical applicability of our methodology by analyzing a real data. [ABSTRACT FROM AUTHOR]

Dual number matrix decompositions have played an important role in fields such as kinematics and computer graphics in recent years. In this paper, we present a QR decomposition algorithm for dual number matrices. When dealing with large-scale problems, we present the thin QR decomposition of dual number matrices, along with its algorithm with column pivoting. In numerical experiments, we discuss the suitability of different QR algorithms when confronted with various large-scale dual matrices, providing their respective domains of applicability. Finally, we employ the QR decomposition of dual matrices to compute the DMPGI, attaining results of higher precision. [ABSTRACT FROM AUTHOR]

In hidden Markov chain (HMC) models, widely used for target tracking, the process noise and measurement noise are in general assumed to be independent and Gaussian for mathematical simplicity. However, the independence and Gaussian assumptions do not always hold in practice. For instance, in a typical radar tracking application, the measurement noise is correlated over time as the sampling frequency of a radar is generally much higher than the bandwidth of the measurement noise. In addition, target maneuvers and measurement outliers imply that the process noise and measurement noise are non-Gaussian. To solve this problem, we resort to triplet Markov chain (TMC) models to describe stochastic systems with correlated noise and derive a new filter under the maximum correntropy criterion to deal with non-Gaussian noise. By stacking the state vector, measurement vector, and auxiliary vector into a triplet state vector, the TMC model can capture the complete dynamics of stochastic systems, which may be subjected to potential parameter uncertainty, non-stationarity, or error sources. Correntropy is used to measure the similarity of two random variables; unlike the commonly used minimum mean square error criterion, which uses only second-order statistics, correntropy uses second-order and higher-order information, and is more suitable for systems in the presence of non-Gaussian noise, particularly some heavy-tailed noise disturbances. Furthermore, to reduce the influence of round-off errors, a square-root implementation of the new filter is provided using QR decomposition. Instead of the full covariance matrices, corresponding Cholesky factors are recursively calculated in the square-root filtering algorithm. This is more numerically stable for ill-conditioned problems compared to the conventional filter. Finally, the effectiveness of the proposed algorithms is illustrated via three numerical examples.

Matrix decomposition is one of the important operations in matrix inversion, which is widely used in neural networks, digital signal processing, wireless communication technology and other fields. Based on a column-vector optimized QR decomposition algorithm, this study proposes a one-dimensional linear matrix decomposition structure and completes the ASIC implementation of the structure to address the shortcomings of the traditional decomposition algorithm operations that are not conducive to hardware implementation. The matrix decomposer supports matrix decomposition operations of order 2～32 and operates at 700 MHz at TSMC 28 nm process. Simulation and FPGA test results show that the relative error between the decomposer and MATLAB results is less than 10-12. When performing matrix decomposition of more than 12-orders, the operation cycle of the decomposer has a speedup ratio of 2.3 times compared with the traditional one-dimensional linear structure. When performing 32-order matrix decomposition, the operation cycle of the decomposer has a speedup ratio of 22.8 times compared with NIVIDA RTX2070. [ABSTRACT FROM AUTHOR]

For protecting the digital copyright of color image, a color image watermarking method should have strong robustness and short running-time at same time. To achieve these two goals, an effective color image watermarking method is presented in this paper. Firstly, the element in the first row and first column of the R matrix of QR decomposition is obtained by the presented method in spatial domain. Secondly, the obtained element of R matrix is selected to protect the copyright of color image in spatial domain without the true QR decomposition. Finally, the processes of the presented watermarking technique are performed in spatial domain. In the compared experimentations, the imperceptibility of the presented method is about 40 dB, its average NC is 0.9435, its average running-time is 0.703613 s, its embedding capacity is 0.03125 bpp, and its key space is 2138, which proves the proposed new method has better performances in terms of the robustness and real-time feature. [ABSTRACT FROM AUTHOR]