Definite Matrix Research Articles

A result about the classification of quantum covariance matrices based on their eigenspectra

The set of covariance matrices of a continuous-variable quantum system with a finite number of degrees of freedom is a strict subset of the set of real positive-definite matrices (PDMs) due to Heisenberg’s uncertainty principle. This has the implication that, in general, not every orthogonal transform of a quantum covariance matrix (CM) produces a PDM that obeys the uncertainty principle. A natural question thus arises, to find the set of quantum covariance matrices consistent with a given eigenspectrum. For the special class of pure Gaussian states the set of quantum covariance matrices with a given eigenspectrum consists of a single orbit of the action of the orthogonal symplectic group. The eigenspectrum of a CM of a state in this class is composed of pairs that each multiply to one. Our main contribution is finding a non-trivial class of eigenspectra with the property that the set of quantum covariance matrices corresponding to any eigenspectrum in this class are related by orthogonal symplectic transformations. We show that all non-degenerate eigenspectra with this property must belong to this class, and that the set of such eigenspectra coincides with the class of non-degenerate eigenspectra that identify the physically relevant thermal and squeezing parameters of a Gaussian state.

Background covariance discriminative dictionary learning for hyperspectral target detection

Hyperspectral target detection (HTD) aims to identifying targets within a hyperspectral image (HSI) based on provided target spectra. In the current HTD field, representation-based detectors have attracted much attention. However, there are two prominent challenges that are particularly noteworthy. First, the background class encompasses diverse land covers, making its accurate representation challenging. Second, the detection ability can be significantly influenced by the abnormalities and noise in HSI. To tackle these concerns, we propose a novel background covariance discriminative dictionary learning (BCDDL) model for HTD. To enhance the background representation ability and overcome the sparse noise, we combine the dictionary learning with spectral covariance descriptors and undertake background reconstruction in regional scale. Specifically, the input HSI is pre-processed into superpixels, the spectral covariance of each superpixel is used to provide a compact and flexible description of local regional statistical properties. Further, a novel spatial clustering-based dictionary learning method is proposed to learn the background discriminative covariances dictionary. The collaborative representation model within symmetric positive definite (SPD) manifold is utilized to reconstruct background region and get the residual. By merging the background residual with pixel-wise target reconstruction residual, we derive final detection output. Comprehensive experiments on two public hyperspectral datasets and two novel GaoFen-5 datasets demonstrate the superiority of our BCDDL approach over 10 state-of-the-art methods, especially in terms of suppressing background.

Robust Observer-Based Proportional Derivative Fuzzy Control Approach for Discrete-Time Nonlinear Descriptor Systems with Transient Response Requirements

This paper proposes an observer-based proportional Derivative (O-BPD) fuzzy controller for uncertain discrete-time nonlinear descriptor systems (NDSs). Representing NDSs with the Takagi–Sugeno fuzzy model (T-SFM), the proportional derivative (PD) feedback method can be utilized in the fuzzy controller design via the Parallel Distributed Compensation (PDC) concept, such that the noncausal problem and impulse behavior are avoided. A fuzzy observer is proposed to obtain unmeasured states to fulfill the PD fuzzy controller. Moreover, uncertainties and transient response performances are taken into account for the NDSs. Then, a stability analysis process and corresponding stability conditions are derived from the Lyapunov theory with the robust control method and the pole constraint. Different from existing research, the Singular Value Decomposition (SVD) and the projection lemma are utilized to transfer the stability conditions into the Linear Matrix Inequation (LMI) form. Because of this reason, the conservatism of the analysis process can be reduced by eliminating the restriction on the positive definite matrix in the Lyapunov function. By giving the proper center and radius parameters of the pole constraint in the O-BPD fuzzy controller design process, the expected transient responses can be obtained for different designers and different practical applications. Finally, the effectiveness and applicability of the proposed O-BPD fuzzy controller are demonstrated by two examples of the simulation.

Spatial Variability and Source Apportionment of PM2.5 Carbon Components in Tianjin

The contents of eight carbonaceous subfractions were determined by simultaneously collecting PM2.5 samples from four sites in different functional areas of Tianjin in 2021. The results showed that the organic carbon (OC) concentration was 3.7 μg·m-3 to 4.4 μg·m-3, and the elemental carbon (EC) concentration was 1.6 μg·m-3 to 1.7 μg·m-3, with the highest OC concentration in the central urban area. There was no significant difference in EC concentration. The concentration of PM2.5 showed the distribution characteristics of the surrounding city>central city>peripheral area. The OC/EC minimum ratio method was used to estimate the concentrations of secondary organic carbon (SOC) in PM2.5, and the results showed that the secondary pollution was more prominent in the surrounding city, with SOC accounting for 48.8%. The correlation between carbon subcomponents in each functional area showed the characteristics of the peripheral area>central area>surrounding area, all showing the strongest correlation between EC1 and OC2 and EC1 and OC4. By including the carbon component concentration into the positive definite matrix factorization (PMF) model for source apportionment, the results showed that road dust sources(9.7%-23.5%), coal-combustion sources (10.2%-13.3%), diesel vehicle exhaust (12.6%-20.2%)and gasoline vehicle exhaust (18.9%-38.8%)were the main sources of carbon components in PM2.5 in Tianjin. The pollution sources of carbon components were different in different functional areas, with the central city and peripheral areas mainly affected by gasoline vehicle exhaust; the surrounding city was more prominently affected by the secondary pollution and diesel vehicle exhaust.

A new weighted spectral geometric mean and properties

In this paper, we introduce a new weighted spectral geometric mean: \begin{equation*}\label{F-mean}F_t(A,B)= (A^{-1}\sharp_t B)^{1/2} A^{2-2t} (A^{-1} \sharp_t B)^{1/2}, \quad t\in [0,1],\end{equation*} where $A$ and $B$ are positive definite matrices. We study basic properties and inequalities for $F_t(A, B)$. We also establish the Lie-Trotter formula for $F_t(A, B)$. Finally, we extend some of the results on $F_t(A, B)$ to symmetric space of noncompact types.

Modified CRI iteration methods for complex symmetric indefinite linear systems

This work investigates the iterative solution of complex symmetric linear systems with indefinite matrix term. Based on a technical equivalent reformulation of the original indefinite systems, an efficient stationary splitting iteration method is established by combining with its real and imaginary parts. In the view of the structure, the new method can be viewed as a modification of the combination method of the real and imaginary parts (CRI) in [Wang & Lu, J Comput Appl Math. 2017;325:188–197], which is originally designed for solving complex symmetric linear systems with positive semi-definite coefficient matrices. Despite that it is devoted to more difficult indefinite problems, the new method keeps the merits of unconditional convergence and parameter free implementation of the original method. Moreover, Chebyshev polynomial acceleration is considered to further improve the convergence rate of the new method. Numerical experiments on a set of model problems are tested to illustrate the efficiency of the proposed methods compared with some existing ones.

Avoiding matrix exponentials for large transition rate matrices.

Exact methods for the exponentiation of matrices of dimension N can be computationally expensive in terms of execution time (N3) and memory requirements (N2), not to mention numerical precision issues. A matrix often exponentiated in the natural sciences is the rate matrix. Here, we explore five methods to exponentiate rate matrices, some of which apply more broadly to other matrix types. Three of the methods leverage a mathematical analogy between computing matrix elements of a matrix exponential process and computing transition probabilities of a dynamical process (technically a Markov jump process, MJP, typically simulated using Gillespie). In doing so, we identify a novel MJP-based method relying on restricting the number of "trajectory" jumps that incurs improved computational scaling. We then discuss this method's downstream implications on mixing properties of Monte Carlo posterior samplers. We also benchmark two other methods of matrix exponentiation valid for any matrix (beyond rate matrices and, more generally, positive definite matrices) related to solving differential equations: Runge-Kutta integrators and Krylov subspace methods. Under conditions where both the largest matrix element and the number of non-vanishing elements scale linearly with N-reasonable conditions for rate matrices often exponentiated-computational time scaling with the most competitive methods (Krylov and one of the MJP-based methods) reduces to N2 with total memory requirements of N.

Source Apportionment and Health Risk Assessment of Groundwater Potentially Toxic Elements (PTEs) Pollution Characteristics in an Accident Site in Zhangqiu, China

Source Apportionment and Health Risk Assessment of Groundwater Potentially Toxic Elements (PTEs) Pollution Characteristics in an Accident Site in Zhangqiu, China

Petrophysical Properties Evaluation Using Well Logging of Jeribe Formation in Mansuriya Gas Field Diyala Governorate, Eastern Iraq

Al-Mansuriya gas field is regarded as one of Iraq's most important gas fields because of its good economic gas reserves. The major gas reserves of the Mansuriya field are situated in the Jeribe Formation. The present study aims to determine the petrophysical properties of the Jeribe Formation in the Mansuriya gas field by using the interpretation of different well logs for the open wells MN-1, MN-2, MN-3, and MN-4. The Jeribe Formation was divided into several reservoir units depending on the final results of CPI computer processing interpretation using the Interactive Petrophysics program (Techlog). According to the results of the CPI, the Jeribe Formation in the Mansuriya gas field was divided into seven units, which are J1, J2, J3, J4, J5, J6, and J7. Matrix definition (MID) and density-neutron cross plots indicate that the lithology of the Jeribe Formation consists of a large proportion of dolomite and limestone and a little anhydrite, also in the direction of the gas evident in the Jeribe Formation. The Gamma-ray log showed that the volume of the shale is very small in the Jeribe Formation, and the formation may be clean of shale content. The reason for this small reading is the presence of some radioactive materials in the rocks of formation. Through the readings of the RHOB and NPHI logs and the results of cutoff boundaries, we can conclude that units J1 and J6 represent non-reservoir units and can be considered a cap rock for the reservoir units, and the units J2, J3, J4, and J5 are the main reservoir units for Jeribe Formation according to CPI results and the results of high porosity in these units, as they are characterized by the porosity of up to 24% and water saturation of 40 to 60%, which makes them reservoirs of high quality. The Mn-4 well showed different results from the rest of the wells, and none of its units can be considered a reservoir due to the large water saturation volume.

HHL algorithm with mapping function and enhanced sampling for model predictive control in microgrids

This paper presents a refined quantum Harrow Hassidim Lloyd (HHL) algorithm for microgrid control. The first novelty of the developed method is that a mapping shift function enables the original HHL algorithm to handle general linear equations with non-singular and indefinite matrix. Second, a method of Matrix Extension for Amplifying Sampling Probabilities of Intended Solution (ME-ASPI) is proposed to design the reformulated linear algebraic equations, allowing for improved sampling efficiency of the quantum tomography in the refined HHL algorithm. Then, we applied the method to solve the model predictive control (MPC) problem in nonlinear dynamical microgrids. Specifically, with the ME-ASPI method, the refined HHL algorithm can effectively obtain the intended partial optimal control inputs for MPC. The optimization of quadratic programming problem in each time step of MPC is transformed into a linear system problem, which is addressed by the proposed quantum solver through using only partial information, with the time complexity improved from O(N2.37286) classically to O(N2logN×plogp) in quantum. Numerical examples have validated the effectiveness of the refined HHL algorithm with the proposed mapping function and the ME-ASPI method. By leveraging quantum properties, the proposed method provides a hybrid quantum–classical framework for microgrid control. This generic method can also potentially tackle many other challenges in analyzing and controlling general complex engineered systems.

Task-oriented analysis and visualization of correlation patterns in multi-sensor time series

Time series data are usually collected through instruments equipped with sensors. For multi-sensor time series (MSTS), it is crucial to identify the factors that affect tasks such as classification and regression. To better understand the mechanisms affecting the task, we extracted the correlation patterns between sensors and represented them as symmetric positive definite matrices. The correlation patterns were then transformed into vector form to serve as input features for regression or classification models, and the models were trained for each sensor. Finally, we leveraged the interpretability of the models to analyze and visualize the correlation patterns at both micro and macro scales. By integrating the explanatory power of the models with correlation patterns, we could interpret the task in terms of time and space, providing valuable insights for exploring the underlying data rules. We evaluated our proposed method using both synthetic and real data, and the simulation results confirmed its effectiveness.

Deep hybrid manifold for image set classification

The exponential growth of the data volume of image sets, which contain more information than a single image, has attracted increasing attention from researchers. Image set data are often described as covariance matrices or linear subspaces, and the unique geometries they span are symmetric positive definite (SPD) manifolds and Grassmann manifolds, respectively. Image set data are often described as covariance matrices or linear subspaces, and the distinctive geometries they span are symmetric positive definite (SPD) manifold and Grassmann manifold, respectively. However, most studies focus on a single manifold and ignore the useful information of the another manifold. Based on this, we propose a new Deep Hybrid Manifold Network (DHMNet).The DHMNet consists of backbone network, stackable Hybrid Manifold AutoEncoder (HMAE) and,Maximum Fusion Module (MFM). The image set data is modeled through SPD manifold and Grassmann manifold. The modeled data is input into the backbone network composed of SPDNet and GrNet for initial feature extraction, and the output manifold data are input into HMAEs. The HMAE effectively extracts and hybridizes complementary information from different manifolds and has the ability to generate deep representations with rich structural semantic information. For the three image datasets used, DHMNet with two HMAEs improves the classification accuracy by 3.83–5.76% over the classical SPDNet, and even reaches the best when compared to other models, with the best performance on the First Person Hand Action (FPHA) dataset for skeleton-based hand action recognition.

Three-stage transfer learning for motor imagery EEG recognition.

Motor imagery (MI) paradigms have been widely used in neural rehabilitation and drowsiness state assessment. The progress in brain-computer interface (BCI) technology has emphasized the importance of accurately and efficiently detecting motor imagery intentions from electroencephalogram (EEG). Despite the recent breakthroughs made in developing EEG-based algorithms for decoding MI, the accuracy and efficiency of these models remain limited by technical challenges posed by cross-subject heterogeneity in EEG data processing and the scarcity of EEG data for training. Inspired by the optimal transport theory, this study aims to develop a novel three-stage transfer learning (TSTL) method, which uses the existing labeled data from a source domain to improve classification performance on an unlabeled target domain. Notably, the proposed method comprises three components, namely, the Riemannian tangent space mapping (RTSM), source domain transformer (SDT), and optimal subspace mapping (OSM). The RTSM maps a symmetric positive definite matrix from the Riemannian space to the tangent space to minimize the marginal probability distribution drift. The SDT transforms the source domain to a target domain by finding the optimal transport mapping matrix to reduce the joint probability distribution differences. The OSM finally maps the transformed source domain and original target domain to the same subspace to further mitigate the distribution discrepancy. The performance of the proposed method was validated on two public BCI datasets, and the average accuracy of the algorithm on two datasets was 72.24% and 69.29%. Our results demonstrated the improved performance of EEG-based MI detection in comparison with state-of-the-art algorithms.

Bilevel Quadratic Programming-Based Stability and Energy Saving Control for Electric Vehicles Using Neurodynamic Optimization

Stability and energy saving are essential issues for traditional and autonomous vehicles and can be ensured through optimization and control of steering, braking, and torque distribution in electric motors. While these three control mechanisms can be optimized separately using two-stage sequential optimization, the results are often sub-optimal. As such, this study proposes simultaneous optimization and control of all three control units for the first time through Bilevel Quadratic Programming (BQP) with neurodynamic optimization. First, a BQP technique is developed by assuming the lower optimization problem to be a feasible region for the upper optimization problem. Second, the upper and lower optimization problems are integrated into a QP task to achieve simultaneous optimization and control by removing separate steps required for two-stage sequential optimization. As the formulated QP problem is more generalized, including an irreversible positive semi-definite hessian matrix and a switching subsystem, a novel neurodynamic optimization with a unique 2-norm energy function is established. It is then applied to solve the generalized BQP problem with high computational efficiency. Finally, hardware-in-the-loop experimental results confirmed the real-time effectiveness of the proposed control technique which decreased the optimization time by 43.3%, and improved control performance while reducing the braking and motor torque required by 54.6% and 33.3% respectively (compared with existing techniques).

Efficient Identification of Error-in-Variables Switched Systems Using a Riemannian Embedding

This paper considers the problem of error in variables identification for switched affine models. Since it is well known that this problem is generically NP hard, several relaxations have been proposed in the literature. However, while these approaches work well for low dimensional systems with few subsystems, they scale poorly with both the number of subsystems and their memory. To address this difficulty, we propose a computationally efficient alternative, based on embedding the data in the manifold of positive semidefinite matrices, and using a manifold metric there to perform the identification. Our main result shows that, under dwell-time assumptions, the proposed algorithm is convergent, in the sense that it is guaranteed to identify the system for suitably low noise. In scenarios with larger noise levels, we provide experimental results showing that the proposed method outperforms existing ones. The paper concludes by illustrating these results with academic examples and a non-trivial application: action video segmentation.

Joint Learning of Full-Structure Noise in Hierarchical Bayesian Regression Models.

We consider the reconstruction of brain activity from electroencephalography (EEG). This inverse problem can be formulated as a linear regression with independent Gaussian scale mixture priors for both the source and noise components. Crucial factors influencing the accuracy of the source estimation are not only the noise level but also its correlation structure, but existing approaches have not addressed the estimation of noise covariance matrices with full structure. To address this shortcoming, we develop hierarchical Bayesian (type-II maximum likelihood) models for observations with latent variables for source and noise, which are estimated jointly from data. As an extension to classical sparse Bayesian learning (SBL), where across-sensor observations are assumed to be independent and identically distributed, we consider Gaussian noise with full covariance structure. Using the majorization-maximization framework and Riemannian geometry, we derive an efficient algorithm for updating the noise covariance along the manifold of positive definite matrices. We demonstrate that our algorithm has guaranteed and fast convergence and validate it in simulations and with real MEG data. Our results demonstrate that the novel framework significantly improves upon state-of-the-art techniques in the real-world scenario where the noise is indeed non-diagonal and full-structured. Our method has applications in many domains beyond biomagnetic inverse problems.

New properties of a special matrix related to positive-definite matrices

New properties of a special matrix related to positive-definite matrices

Robust PRESB Preconditioning of a 3-Dimensional Space-Time Finite Element Method for Parabolic Problems

Abstract We present a recently developed preconditioning of square block matrices (PRESB) to be used within a parallel method to solve linear systems of equations arising from tensor-product discretizations of initial boundary-value problems for parabolic partial differential equations. We consider weak formulations in Bochner–Sobolev spaces and tensor-product finite element approximations for the heat and eddy current equations. The fast diagonalization method is employed to decouple the arising linear system of equations into auxiliary spatial complex-valued linear systems that can be solved concurrently. We prove that the real part of the system matrix is positive definite, which allows us to accelerate the flexible generalized minimal residual method (FGMRES) by the PRESB preconditioner. The action of PRESB on a vector includes two solutions with positive definite matrices. The spectrum of the preconditioned system lies between 1/2 and 1. Finally, we combine the PRESB-FGMRES method with multigrid-CG iterations and illustrate the numerical efficiency and the robustness for spatial discretizations up to 12 millions degrees of freedom.

Singular Value and Matrix Norm Inequalities between Positive Semidefinite Matrices and Their Blocks

In this paper, we obtain some inequalities involving positive semidefinite 2×2 block matrices and their blocks.

Relative operator entropy properties related to some weighted metrics

In recent decades, intensive research has been devoted to the study of various operator entropies. In this work, we investigate the properties of the parameterized relative operator entropy Sp(A | B) acting on positive definite matrices with respect to weighted Hellinger and Alpha Procrustes distances. In particular, we investigate estimation of the distance between the entropy Sp(A | B) and certain standard means.