QR Decomposition Research Articles

A user selection algorithm for downlink MU-MIMO systems using product of singular values of effective channels

User selection plays a crucial role when aiming to improve channel throughput in multiuser multiple- input multiple-output downlink systems with block diagonalization linear precoding. While an optimal group of users can be found by exhaustively searching all possible combinations of users, the process becomes prohibitively complex when there is a large number of users. In this paper, we propose a greedy suboptimal user selection algorithm, which iteratively selects a user to maximize the product of the singular values of the effective channels from a set of unselected users. The proposed algorithm is based on the iterative precoder design method, whereby interuser interference is perfectly cancelled with lower complexity than with the singular value decomposition-based precoding technique. Moreover, to avoid frequent use of singular value decomposition, the proposed algorithm applies QR decomposition to the selected channel matrices in order to obtain singular values, which are equivalent to the product of diagonal elements in the upper triangular matrix. Simulation results show that, under a high SNR regime, the proposed algorithm outperforms several other greedy algorithms and can provide the same performance as a capacity-based algorithm for both correlated and uncorrelated channels with significantly reduced complexity.

A least-squares Fourier frame method for nonlocal diffusion models on arbitrary domains

We introduce a least-squares Fourier frame method for solving nonlocal diffusion models with Dirichlet volume constraint on arbitrary domains. The mathematical structure of a frame rather than a basis allows using a discrete least-squares approximation on irregular domains and imposing non-periodic boundary conditions. The method has inherited the one-dimensional integral expression of Fourier symbols of the nonlocal diffusion operator from Fourier spectral methods for any d spatial dimensions. High precision of its solution can be achieved via a direct solver such as pivoted QR decomposition even though the corresponding system is extremely ill-conditioned, due to the redundancy in the frame. The extension of AZ algorithm improves the complexity of solving the rectangular linear system to O(Nlog2⁡N) for 1d problems and O(N2log2⁡N) for 2d problems, compared with O(N3) of the direct solvers, where N is the number of degrees of freedom. We present ample numerical experiments to show the flexibility, fast convergence and asymptotical compatibility of the proposed method.

Trace Ratio Based Manifold Learning With Tensor Data

ABSTRACTIn this article, we propose an extension of trace ratio based Manifold learning methods to deal with multidimensional data sets. Based on recent progress on the tensor‐tensor product, we present a generalization of the trace ratio criterion by using the properties of the t‐product. This will conduct us to introduce some new concepts such as Laplacian tensor and we will study formally the trace ratio problem by discussing the conditions for the existence of solutions and optimality. Next, we will present a tensor Newton QR decomposition algorithm for solving the trace ratio problem. Manifold learning methods such as Laplacian eigenmaps, linear discriminant analysis and locally linear embedding will be formulated in a tensor representation and optimized by the proposed algorithm. Finally, we will evaluate the performance of the different studied dimension reduction methods on several synthetic and real world data sets.

A tensor completion method based on tensor QR decomposition with truncated nuclear norm and sparse regularization

A tensor completion method based on tensor QR decomposition with truncated nuclear norm and sparse regularization

Kinematic calibration of a 5-DOF double-driven parallel mechanism with sub-closed loop on limbs

Abstract This paper proposes a kinematic calibration method of a novel 5-degree-of-freedom double-driven parallel mechanism with the sub-closed loop on limbs. At first, considering the introduction of a sub-closed loop significantly increased the complexity and difficulty of kinematic error modeling, an equivalent transformation method is proposed for the limb with a sub-closed loop. Then kinematic error model of the parallel mechanism is established based on the closed-loop vector method and parasitic motion analysis, which is verified by virtual prototype technology. Because the full kinematic error model is generally redundant, error parameter identifiability analysis is carried out by QR decomposition of the identification Jacobian matrix, and the redundant parameters are removed. Additionally, the Sequence Forward Floating Search algorithm is utilized to optimize measurement configurations to reduce the influence of measurement noise. Finally, with a laser tracker as the measuring device, numerical simulations and experiments are implemented to verify the proposed kinematic calibration method. The experiment results show that average position and orientation errors are reduced from 2.778 mm and 1.115° to 0.263 mm and 0.176°, respectively, within the prescribed workspace.

A State Estimation Method for Rectangular Singular Systems Based on State Decomposition

This paper proposes a state estimation method of linear discrete rectangular singular systems. The system is observable and regular, and the system matrix is rectangular without full column rank. To give the estimation of the state, the state is decomposed into two parts based on QR factorization, and the weighted least squares method is used to obtain the prediction of one part of the state. Then, the partial measurement equation is used to obtain the prediction of the other part of the state, and the projection theorem is used to obtain the state estimation value. Combined with the data-driven idea, a Kalman filtering algorithm based on historical data modeling is established. Finally, the feasibility and effectiveness of our approach is discussed and verified through performance analysis and numerical simulation perspectives.

A stochastic perturbation analysis of the QR decomposition and its applications

A stochastic perturbation analysis of the QR decomposition and its applications

Data Reconstruction Using Smart Sensor Placement.

This paper deals with spatio-temporal field estimation with efficient sensor placement based on the QR decomposition. The proposed method also identifies the optimal number of sensors needed for field estimation that captures the most relevant features of the field of interest. To address the uncertainties inherent in spatio-temporal field estimation, a robust data-driven control method is utilized, providing resilience against unpredictable environmental and model changes. In particular, the approach uses the Kriged Kalman Filter (KKF) for uncertainty-aware field reconstruction. Unlike other reconstruction methods, the positional uncertainty originating from the data acquisition platform is integrated into the KKF estimator. Numerical results are presented to show the efficacy of the proposed dynamic sensor placement strategy together with the KKF field estimator.

Complex Palais matrix and a new unitary transform with bounded component norms

Abstract The Palais matrix represents an n n -dimensional rotation between two vectors that is functionally equivalent to the Householder reflection. This study introduces a one-parameter family of unitary transforms, termed the θ \theta transform, which encompasses the transform by the Palais matrix, the Householder reflection, and their unitary extensions. Furthermore, we define the θ ∠ {\theta }_{\angle } transform, a variant of the θ \theta transform featuring bounded component norms. It is demonstrated that the θ ∠ {\theta }_{\angle } transform is computationally efficient and backward stable when one of the vectors has the “one-hot” structure, making it highly valuable for matrix decompositions such as the QR decomposition. In addition, the θ \theta transform exhibits additional characteristics, including its convergence to the identity and the rowwise structure of its backward error.

Ecg signal watermarking using QR decomposition.

This study introduces a novel watermarking technique for electrocardiogram (ECG) signals. Watermarking embeds critical information within the ECG signal, enabling data origin authentication, ownership verification, and ensuring the integrity of research data in domains like telemedicine, medical databases, insurance, and legal proceedings. Drawing inspiration from image watermarking, the proposed method transforms the ECG signal into a two-dimensional format for QR decomposition. The watermark is then embedded within the first row of the resulting R matrix. Three implementation scenarios are proposed: one in the spatial domain and two in the transform domain utilizing discrete wavelet transform (DWT) for improved watermark imperceptibility. Evaluation on real ECG signals from MIT-BIH Arrhythmia database and comparison to existing methods demonstrate that the proposed method achieves: (1) higher Peak Signal-to-Noise Ratio (PSNR) indicating minimal alterations to the watermarked signal, (2) lower bit error rates (BER) in robustness tests against external modifications such as AWGN noise (additive white Gaussian noise), line noise and down-sampling, and (3) lower computational complexity. These findings emphasize the effectiveness of the proposed QR decomposition-based watermarking method, achieving a balance between robustness and imperceptibility. The proposed approach has the potential to improve the security and authenticity of ECG data in healthcare and legal contexts, while its lower computational complexity enhances its practical applicability.

Robust estimation for semiparametric spatial autoregressive models via weighted composite quantile regression

In this article, the robust estimation for a class of semiparametric spatial autoregressive models has been investigated. By combining the QR decomposition technique for matrix and the weighted composite quantile regression method, we propose a robust estimation procedure for the parametric and non parametric components. Under certain regularity conditions, asymptotic properties of the resulting estimators are proved. Several simulation analyses have been conducted for further illustrating the performance of the proposed method, and the simulation results demonstrate that the proposed method improve the robustness of the models.

Performance evaluation of symbol detection algorithms in massive MIMO communication systems

Abstract Massive MIMO (mMIMO) is the essential technique for attaining the exponential increase in the data rate needed for future communication systems. These advancements have been significantly supported by progress in VLSI technology, which enables the integration of an enormous number of antennas and the complex signal processing essential for massive MIMO systems on a single chip. This work conducts a comprehensive analysis of the intricacy of seven matrix decomposition techniques for symbol detection in future mMIMO communication systems: ADMM-based infinity norm (ADMIN), Neumann series (NS), Newton iteration (NI), Jacobi iteration (Ja), improved Gauss-Seidel (IGS), conjugate gradient (CG), and QR decomposition (QR), against linear and near-optimal minimum mean square error (MMSE). QR, GS, Ja, and CG belong to linear algebraic methods based on matrix decomposition. MMSE and ADMIN are nonlinear optimization methods; NS and NI are iterative methods. The analysis examines into the complexity of these detection algorithms, considering the symbol error rate, the convergence rate, the initial solution vector, and the correlation factor. Performance evaluations are conducted on 8 and 16-user mMIMO systems with 64 and 128 base station antennas, modulation schemes (32-QAM and 64-QAM), iteration counts (p = 1, 2, 3, 4), correlation factors (α = 0.2, 0.4, and 0.6), and initial solution vectors (zero vector, D −1 and W 2 −1 yMF ). The result clearly shows that if the initial solution and number of iterations are chosen properly, the IGS-based linear detector achieves near-optimal performance with a lower computational complexity of O(K 2) compared to the nonlinear MMSE-based detector with a computational complexity of O(K 3).

A novel tensor-based modal decomposition method for reduced order modeling and optimal sparse sensor placement

Data-driven modal decomposition methods have become fundamental algorithms in constructing reduced order models (ROMs), which serve as a critical tool in the digital twin modeling of large complex systems. Classical modal decomposition methods are particularly adept at handling matrix-form input data that contains columnized snapshots without retaining spatial structure and neighborhood information. Focusing on addressing this problem, this paper proposed a Tensor-Based Modal Decomposition (TBMD) method to deal with the high-order tensor-form input data. TBMD method is effective in finding the potential low-order manifolds of a given high-order dynamical system with low energy loss within fewer decomposited modes. Moreover, TBMD is extended to be used in the field of optimal sensor placement via the development of a tensor-based QR factorization method and a tensor-based compressive sensing algorithm. The tensor-based QR factorization method is able to consider the spatial structure and neighborhood information of the basis matrices data in tensor form. This is the first time that a compressive sensing model has been discussed and solved for the frontal slice-wise product of a tensor and a vector. Subsequently, comprehensive case studies are conducted based on the cylinder wake dataset, airfoil vortex dataset, sea surface temperature dataset, eigenfaces dataset and turbulent channel flow dataset. The experiment results show that TBMD can accurately capture lower-order manifolds, and the tensor-based QR factorization is effective in optimizing sensor placement with fewer sensors while maintaining high reconstruction accuracy. The proposed methods in this paper is promising to be utilized in the construction of large-scale complex system digital twins, in which TBMD can serve as a fundamental algorithm in constructing ROM.

Probabilistic perturbation bounds of matrix decompositions

AbstractIn 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.

On rank‐revealing QR factorizations of quaternion matrices

On rank‐revealing QR factorizations of quaternion matrices

Dynamics Parameter Identification of Articulated Robot

Dynamics parameter identification in the establishment of a multiple degree-of-freedom (DOF) robot’s dynamics model poses significant challenges. This study employs a non-symbolic numerical method to establish a dynamics model based on the Newton–Euler formula and then derives a proper dynamics model through decoupling. Initially, a minimum inertial parameter set is acquired by using QR decomposition, with the inclusion of a friction model in the robot dynamics model. Subsequently, the least squares method is employed to solve for the minimum inertial parameters, forming the basis for a comprehensive robot dynamics parameter identification system. Then, after the optimization of the genetic algorithm, the Fourier series trajectory function is used to derive the trajectory function for parameter identification. Validation of the robot’s dynamics parameter identification is performed through simulation and experimentation on a 6-DOF robot, leading to a precise identification value of the robot’s inertial parameters. Furthermore, two methods are employed to verify the inertia parameters, with analysis of experimental errors demonstrating the effectiveness of the robot dynamics parameter identification method. Overall, the effectiveness of the entire calibration system is verified by experiments, which can provide valuable insights for practical engineering applications, and a complete and effective robot dynamics parameter identification scheme for a 6-DOF robot is established and improved.

An improved reversible watermarking scheme using embedding optimization and quaternion moments

Digital watermarking of images is an essential method for copyright protection and image security. This paper presents an innovative, robust watermarking system for color images based on moment and wavelet transformations, algebraic decompositions, and chaotic systems. First, we extended classical Charlier moments to quaternary Charlier moments (QCM) using quaternion algebra. This approach eliminates the need to decompose color images before applying the discrete wavelet transform (DWT), reducing the computational load. Next, we decompose the resulting DWT matrix using QR and singular value decomposition (SVD). To enhance the system's security and robustness, we introduce a modified version of Henon's 2D chaotic map. Finally, we integrate the arithmetic optimization algorithm to ensure dynamic and adaptive watermark insertion. Our experimental results demonstrate that our approach outperforms current color image watermarking methods in security, storage capacity, and resistance to various attacks, while maintaining a high level of invisibility.

A modified Levenberg–Marquardt algorithm for low order-value optimization problem

AbstractIn this paper, we consider a modified Levenberg–Marquardt algorithm for Low Order Value Optimization problems(LOVO). In the algorithm, we obtain the search direction by a combination of LM steps and approximate LM steps, and solve the subproblems therein by QR decomposition or cholesky decomposition. We prove the global convergence of the algorithm theoretically and discuss the worst-case complexity of the algorithm. Numerical results show that the algorithm in this paper is superior in terms of number of iterations and computation time compared to both LM-LOVO and GN-LOVO algorithm.

Matrix decompositions in quantum optics: Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson

In this tutorial, we summarize four important matrix decompositions commonly used in quantum optics, namely the Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson decompositions. The first two of these decompositions are specialized versions of the singular-value decomposition when applied to symmetric or symplectic matrices. The third factors any symplectic matrix in a unique way in terms of matrices that belong to different subgroups of the symplectic group. The last one instead gives the symplectic diagonalization of real, positive definite matrices of even size. While proofs of the existence of these decompositions exist in the literature, we review explicit constructions to implement these decompositions using standard linear algebra packages and functionalities such as singular-value, polar, Schur, and QR decompositions, and matrix square roots and inverses.

Double penalized variable selection for high-dimensional partial linear mixed effects models

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.