Square Root Of Matrix Research Articles

On maps preserving square roots of idempotent and rank-one nilpotent matrices

We characterize bijective linear maps on [Formula: see text] that preserve the square roots of an idempotent matrix (of any rank). Every such map can be presented as a direct sum of a map preserving involutions and a map preserving square-zero matrices. Next, we consider bijective linear maps that preserve the square roots of a rank-one nilpotent matrix. These maps do not have standard forms when compared to similar linear preserver problems.

Square roots of H-nonnegative matrices

Roots of matrices are well-studied. The conditions for their existence are understood: The block sizes of nilpotent Jordan blocks, arranged in pairs, have to satisfy some simple algebraic property.More interesting are structured roots of structured matrices. Probably the best known example is the existence and uniqueness of positive definite square roots of a positive definite matrix. If one drops the requirement of positive definiteness of the square root, it turns out that there exists an abundance of square roots. Here a description of all canonical forms of all square roots is possible and is straightforward.H-nonnegative matrices are H-selfadjoint and are nonnegative with respect to an indefinite inner product with Gramian H. An H-nonnegative matrix B allows a decomposition in a negative definite, a nilpotent H-nonnegative, and a positive definite matrix, B=B−⊕B0⊕B+. The interesting part is B0, as only Jordan blocks of size one and two occur. Determining a square root of B reduces to determining a square root of each of B−, B0, and B+. Here we investigate for an H-nonnegative matrix: its square roots without additional structure, as well as its structured square roots that are H-nonnegative or H-selfadjoint.For these three classes of square roots of H-nonnegative matrices we show a simple criterion for their existence and describe all possible canonical forms. This is based mainly on known results but an important new part is that in all three cases we describe all possible square roots of the nilpotent H-nonnegative matrix B0 explicitly. Moreover, we show how our results can be applied to the conditional and unconditional stability of H-nonnegative square roots of H-nonnegative matrices, where the explicit description of the square roots of B0 is used.

Robust and high-order correlation alignment for unsupervised domain adaptation

How to measure the domain discrepancy is of significant importance in the field of unsupervised domain adaptation. Among them, Correlation Alignment (CORAL), aligning second-order statistics of source and target domains, has become one of the most widely used discrepancy-based methods. However, the performance of CORAL is limited by: (1) aligning covariance with usual Euclidean metric is suboptimal, and (2) second-order statistics have limited expression for the non-Gaussian distribution. To address these limitations, we propose a Robust Correlation Alignment as well as a High-order Correlation Alignment method. The Robust Correlation Alignment exploits the geometric structure of covariance with matrix square-root normalization. To circumvent unstable and time-consuming properties of the Singular Value Decomposition, we employ the variant of Newton iteration to compute the matrix square-root. Besides, we also propose a High-order Correlation Alignment method, which exploits the third-order statistics for domain alignment. We show that the High-order CORAL can be generalized to Maximum Mean Discrepancy, CORAL and arbitrary-order statistics. Specifically, we propose group matching to reduce space complexity and improve the feasibility in real-world application. Extensive experiments on standard benchmark datasets demonstrate that our proposed methods outperform previous methods by a large margin.

Efficient Scalar Bidirectional Beam Propagation Analysis for Photonic Devices With Circular Symmetry

In this letter, a scalar non-iterative bi-directional finite element based beam propagation method (NI-BiBPM) is proposed for the analysis of light propagation in photonic devices with circular symmetry. We adopt the Blocked Schur algorithm (BSA) for calculating the square root of the characteristic matrix in order to achieve fast and stable results and to avoid the complications and limitations of other computational modeling techniques such as Taylor or Pade approximations. The suggested NI-BiBPM approach does not only offer reduction in execution time and memory requirements but also guarantees the numerical stability.

Pulay forces in density-functional theory with extended Hubbard functionals: From nonorthogonalized to orthogonalized manifolds

We present a derivation of the exact expression for Pulay forces in density-functional theory calculations augmented with extended Hubbard functionals and arising from the use of orthogonalized atomic orbitals as projectors for the Hubbard manifold. The derivative of the inverse square root of the orbital overlap matrix is obtained as a closed-form solution of the associated Lyapunov (Sylvester) equation. The expression for the resulting contribution to the forces is presented in the framework of ultrasoft pseudopotentials and the projector-augmented-wave method and using a plane-wave basis set. We have benchmarked the present implementation with respect to finite differences of total energies for the case of NiO, finding excellent agreement. Owing to the accuracy of Hubbard-corrected density-functional theory calculations---provided the Hubbard parameters are computed for the manifold under consideration---the present work paves the way for systematic studies of solid-state and molecular transition-metal and rare-earth compounds.

A Variational Method for Analyzing Vortex Flows in Radar-Scanned Tornadic Mesocyclones. Part I: Formulations and Theoretical Considerations

AbstractA variational method is formulated with theoretical considerations for analyzing vortex flows in Doppler radar–scanned tornadic mesocyclones. The method has the following features. (i) The vortex center axis (estimated as a continuous function of time and height in the four-dimensional space) is used as the vertical coordinate, so the coordinate system used for the analysis is slantwise curvilinear and nonorthogonal in general. (ii) The vortex flow (VF), defined by the three-dimensional vector wind minus the horizontal moving velocity of vortex center axis, is expressed in terms of the covariant basis vectors (tangent to the coordinate curves), so its axisymmetric part can be properly defined in that slantwise-curvilinear coordinate system. (iii) To satisfy the mass continuity automatically, the axisymmetric part is expressed by the scalar fields of azimuthally averaged tangential velocity and cylindrical streamfunction and the remaining asymmetric part is expressed by the scalar fields of streamfunction and vertically integrated velocity potential. (iv) VF-dependent covariance functions are formulated for these scalar variables and then deconvoluted to construct the square root of background error covariance matrix analytically with the latter used to transform the control vector to precondition the cost function. (v) The deconvoluted covariance functions and their transformed control variables satisfy two required boundary conditions (i.e., zero vertical velocity at the lower rigid boundary and zero cross-axis velocity along the vortex center axis), so the analyzed VF satisfies not only the mass continuity but also the two boundary conditions automatically.

Efficient Krylov subspace methods for uncertainty quantification in large Bayesian linear inverse problems

SummaryUncertainty quantification for linear inverse problems remains a challenging task, especially for problems with a very large number of unknown parameters (e.g., dynamic inverse problems) and for problems where computation of the square root and inverse of the prior covariance matrix are not feasible. This work exploits Krylov subspace methods to develop and analyze new techniques for large‐scale uncertainty quantification in inverse problems. In this work, we assume that generalized Golub‐Kahan‐based methods have been used to compute an estimate of the solution, and we describe efficient methods to explore the posterior distribution. In particular, we use the generalized Golub‐Kahan bidiagonalization to derive an approximation of the posterior covariance matrix, and we provide theoretical results that quantify the accuracy of the approximate posterior covariance matrix and of the resulting posterior distribution. Then, we describe efficient methods that use the approximation to compute measures of uncertainty, including the Kullback‐Liebler divergence. We present two methods that use the preconditioned Lanczos algorithm to efficiently generate samples from the posterior distribution. Numerical examples from dynamic photoacoustic tomography demonstrate the effectiveness of the described approaches.

Constrained expectation maximisation algorithm for estimating ARMA models in state space representation

This paper discusses the fitting of linear state space models to given multivariate time series in the presence of constraints imposed on the four main parameter matrices of these models. Constraints arise partly from the assumption that the models have a block-diagonal structure, with each block corresponding to an ARMA process, that allows the reconstruction of independent source components from linear mixtures, and partly from the need to keep models identifiable. The first stage of parameter fitting is performed by the expectation maximisation (EM) algorithm. Due to the identifiability constraint, a subset of the diagonal elements of the dynamical noise covariance matrix needs to be constrained to fixed values (usually unity). For this kind of constraints, so far, no closed-form update rules were available. We present new update rules for this situation, both for updating the dynamical noise covariance matrix directly and for updating a matrix square-root of this matrix. The practical applicability of the proposed algorithm is demonstrated by a low-dimensional simulation example. The behaviour of the EM algorithm, as observed in this example, illustrates the well-known fact that in practical applications, the EM algorithm should be combined with a different algorithm for numerical optimisation, such as a quasi-Newton algorithm.

Improving formulas for the eigenvalues of finite block-Toeplitz tridiagonal matrices

Abstract After a short overview, improvements (based on the Kronecker product) are proposed for the eigenvalues of (N × N) block-Toeplitz tridiagonal (block-TT) matrices with (K × K) matrix-entries, common in applications. Some extensions of the spectral properties of the Toeplitz-tridiagonal matrices are pointed-out. The eigenvalues of diagonalizable symmetric and skew-symmetric block-TT matrices are studied. Besides, if certain matrix square-root is well-defined, it is proved that every block-TT matrix with commuting matrix-entries is isospectral to a related symmetric block-TT one. Further insight about the eigenvalues of hierarchical Hermitian block-TT matrices, of use in the solution of PDEs, is also achieved.

Multi-Sensor Combined Measurement While Drilling Based on the Improved Adaptive Fading Square Root Unscented Kalman Filter.

In the process of the attitude measurement for a steering drilling system, the measurement of the attitude parameters may be uncertain and unpredictable due to the influence of server vibration on bits. In order to eliminate the interference caused by vibration on the measurement and quickly obtain the accurate attitude parameters of the steering drilling tool, a new method for multi-sensor dynamic attitude combined measurement is presented. Firstly, by using a triaxial accelerometer and triaxial magnetometer measurement system, the nonlinear model based on the quaternion is established. Then, an improved adaptive fading square root unscented Kalman filter is proposed for eliminating the vibration disturbance signal. In this algorithm, the square root of the state covariance matrix is used to replace the covariance matrix in the classical unscented Kalman filter (UKF) to avoid the filter divergence caused by the negative definite state covariance matrix. The fading factor is introduced into UKF to adjust the filter gain in real-time and improve the adaptive ability of the algorithm to mutation state. Finally, the computational method of the fading factor is optimized to ensure the self-adaptability of the algorithm and reduce the computational complexity. The results of the laboratory test and the field-drilling data show that the proposed method can filter out the interference noise in the attitude measurement sensor effectively, improve the solution accuracy of attitude parameters of drilling tools in the case of abrupt changes in the measuring environment, and thus ensuring the dynamic stability of the well trajectory.

Affine Invariant Interacting Langevin Dynamics for Bayesian Inference

We propose a computational method (with acronym ALDI) for sampling from a given target distribution based on first-order (overdamped) Langevin dynamics which satisfies the property of affine invariance. The central idea of ALDI is to run an ensemble of particles with their empirical covariance serving as a preconditioner for their underlying Langevin dynamics. ALDI does not require taking the inverse or square root of the empirical covariance matrix, which enables application to high-dimensional sampling problems. The theoretical properties of ALDI are studied in terms of nondegeneracy and ergodicity. Furthermore, we study its connections to diffusion on Riemannian manifolds and Wasserstein gradient flows. Bayesian inference serves as a main application area for ALDI. In case of a forward problem with additive Gaussian measurement errors, ALDI allows for a gradient-free approximation in the spirit of the ensemble Kalman filter. A computational comparison between gradient-free and gradient-based ALDI is provided for a PDE constrained Bayesian inverse problem.

Synthetic Aperture Radar Scene Classification Using Multiview Cross Correlation Attention Network

The second-order pooling manner, exploring higher feature statistics than the first-order pooling, has achieved impressive performance in scene classification. However, the object not only presents similarity but also exhibits diversified singularity on the synthetic aperture radar (SAR) image. These make the second-order pooling approaches to explore the single-view second-order feature statistics less adaptable for SAR scene classification. To solve this issue, an end-to-end training framework based on the multiview cross correlation attention network (MCAN) is proposed. The spatial and channelwise self-attention modules are first employed to model the interdependences between the spatial and channel dimensions of their convolutional features. Subsequently, the global spatial and channelwise covariance pooling layers are drawn into the MCAN, and then, they learn the spatial and channel cross correlations within the feature statistics, respectively. Finally, an iterative matrix square-root normalization layer, which owns the capability to fast computing approximate square root of the covariance matrix, is introduced for making the feature representation more discriminative. Experiments on the SAR data set from the TerraSAR-X images for scene classification demonstrate that the MCAN performs better than other related works.

Recursive Variable Span Linear Filter for Noise Reduction

The design of variable span linear filters for noise reduction involves a generalized eigenvalue decomposition problem that is of high computational complexity. In order to address this issue, this work proposes a recursive algorithm that computes the filter weights with streaming signal data. Specifically, the inverse square root of the noise covariance matrix is recursively computed with a rank-one update strategy, and the generalized eigenvalues and eigenvectors are approached with the projection approximation subspace tracking method. Numerical simulations show that the proposed recursive method is able to achieve satisfactory performance with significantly lower complexity as compared to the batch algorithm.

Analysis of Missile Longitudinal Autopilot Based on the State-Dependent Riccati Equation Method

An analytic solution of the state-dependent Riccati equation (SDRE) method is presented for two-dimensional nonlinear systems, and the closed-loop system of the missile longitudinal autopilot designed by the SDRE method is analyzed using the analytic solution. The analytic solution is obtained using the matrix sign function of the Hamiltonian matrix corresponding to the SDRE, where the matrix sign function is represented using the principal square root of the Hamiltonian matrix. The missile longitudinal acceleration autopilot is designed based on the output feedback SDRE method for command tracking. The asymptotic stability of the closed-loop system and the nonsingularity of the output feedback control command are shown using the analytic solution to the SDRE method. A numerical example of the main results is provided, including the effects of aerodynamic uncertainty. Additionally, a six-degrees-of-freedom simulation for the skid-to-turn missile demonstrates the performance of the proposed acceleration autopilot with the roll stabilization control.

Full-vectorial analysis of optical waveguide discontinuities using a propagation operator method based on the finite element scheme

We propose an efficient numerical method for the full-vectorial analysis of three-dimensional (3-D) optical waveguide discontinuities. In this method, the finite element method with higher adaptability and flexibility is employed to discretize the waveguide cross section. In order to calculate the square root of the characteristic matrix, the Denman-Beavers iterative scheme is used. Applying this method to 3-D strongly guiding waveguide discontinuity problems, the modal reflectivities of the fundamental TE-like and TM-like modes are calculated. These results show unique vector properties and significantly differ from those of scalar analysis because various mode couplings between the field components occur at the discontinuity facet and they cannot be ignored.

Zolotarev Iterations for the Matrix Square Root

We construct a family of iterations for computing the principal square root of a square matrix $A$ using Zolotarev's rational minimax approximants of the square root function. We show that these rational functions obey a recursion, allowing one to iteratively generate optimal rational approximants of $\sqrt{z}$ of high degree using compositions and products of low-degree rational functions. The corresponding iterations for the matrix square root converge to $A^{1/2}$ for any input matrix $A$ having no nonpositive real eigenvalues. In special limiting cases, these iterations reduce to known iterations for the matrix square root: the lowest-order version is an optimally scaled Newton iteration, and for certain parameter choices, the principal family of Padé iterations is recovered. Theoretical results and numerical experiments indicate that the iterations perform especially well on matrices having eigenvalues with widely varying magnitudes.

An Improved IMM Algorithm Based on STSRCKF for Maneuvering Target Tracking

To overcome the IMM algorithm is easy divergence and low tracking accuracy when dealing with complex maneuvering situations, this paper proposes an improved interactive multiple model strong tracking square room cubature Kalman filter (IIMM-STSRCKF) algorithm under the idea of real-time dynamic adjustment of gain matrix and transition probability matrix. The algorithm has been improved in two aspects: on the one hand, the algorithm uses the idea of a strong tracking filter to deduce a new method for time-varying fading factor and introduce it into the square root of the state error covariance matrix of the SRCKF, which improves the tracking accuracy for strong maneuver; on the other hand, the probability difference between two consecutive time points in the IMM submodel is used to adjust the Markov probability transfer matrix to adaptively improve the switching speed of the submodel and the rationality of the allocation. By comparing with IMM-CKF algorithm by maneuvering target tracking case and results show that the IIMM-STSRCKF algorithm has better tracking performance in nonmaneuvering, weak maneuvering, and strong maneuvering cases.

Sparse common component analysis for multiple high-dimensional datasets via noncentered principal component analysis

There is currently much discussion about the analysis of multiple datasets from different groups, among which especially identifying a common basic structure of multiple groups has drawn a large amount of attention. In order to identify a common basic structure, common component analysis (CCA) was proposed by generalizing techniques for principal component analysis (PCA); i.e., CCA becomes standard PCA when applied to only one dataset. Although CCA can identify the common structure of multiple datasets, which cannot be extracted by standard PCA, CCA suffers from the following drawbacks. The common components are estimated as linear combinations of all variables, and thus it is difficult to interpret the identified common components. The fully dense loadings lead to erroneous results in CCA, because noisy features are inevitably included in datasets. To address these issues, we incorporate sparsity into CCA, and propose a novel strategy for sparse common component analysis based on $$L_{1}$$ -type regularized regression modeling. We focus CCA which is formulated as the eigenvalue decomposition (EVD) of a Gram matrix (i.e., common loadings of multiple datasets can be estimated by EVD of a Gram matrix), and it can be performed by Singular value decomposition of a square root of the Gram matrix. We then propose sparse common component analysis based on sparse PCA to estimate sparse common loadings of multiple datasets. We also propose an algorithm to estimate sparse common loadings of multiple datasets. The proposed method can not only identify a common subspace but also select crucial common-features for multiple groups. Monte Carlo simulations and real-data analysis are conducted to examine the efficiency of the proposed sparse CCA. We observe from the numerical studies that our strategies can incorporate sparsity into the common loading estimation and efficiently recover a sparse common structure efficiently in multiple dataset analysis.

An approximate empirical Bayesian method for large-scale linear-Gaussian inverse problems

We study Bayesian inference methods for solving linear inverse problems, focusing on hierarchical formulations where the prior or the likelihood function depend on unspecified hyperparameters. In practice, these hyperparameters are often determined via an empirical Bayesian method that maximizes the marginal likelihood function, i.e. the probability density of the data conditional on the hyperparameters. Evaluating the marginal likelihood, however, is computationally challenging for large-scale problems. In this work, we present a method to approximately evaluate marginal likelihood functions, based on a low-rank approximation of the update from the prior covariance to the posterior covariance. We show that this approximation is optimal in a minimax sense. Moreover, we provide an efficient algorithm to implement the proposed method, based on a combination of the randomized SVD and a spectral approximation method to compute square roots of the prior covariance matrix. Several numerical examples demonstrate good performance of the proposed method.

Fast random field generation with H-matrices

We use the H-matrix technology to compute the approximate square root of a covariance matrix in linear cost. This allows us to generate normal and log-normal random fields on general point sets with optimal cost. We derive rigorous error estimates which show convergence of the method. Our approach requires only mild assumptions on the covariance function and on the point set. Therefore, it might be also a nice alternative to the circulant embedding approach which applies only to regular grids and stationary covariance functions.