Definite Matrix Research Articles

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.

A study on the combination of functional connection features and Riemannian manifold in EEG emotion recognition.

Affective computing is the core for Human-computer interface (HCI) to be more intelligent, where electroencephalogram (EEG) based emotion recognition is one of the primary research orientations. Besides, in the field of brain-computer interface, Riemannian manifold is a highly robust and effective method. However, the symmetric positive definiteness (SPD) of the features limits its application. In the present work, we introduced the Laplace matrix to transform the functional connection features, i.e., phase locking value (PLV), Pearson correlation coefficient (PCC), spectral coherent (COH), and mutual information (MI), to into semi-positive, and the max operator to ensure the transformed feature be positive. Then the SPD network is employed to extract the deep spatial information and a fully connected layer is employed to validate the effectiveness of the extracted features. Particularly, the decision layer fusion strategy is utilized to achieve more accurate and stable recognition results, and the differences of classification performance of different feature combinations are studied. What's more, the optimal threshold value applied to the functional connection feature is also studied. The public emotional dataset, SEED, is adopted to test the proposed method with subject dependent cross-validation strategy. The result of average accuracies for the four features indicate that PCC outperform others three features. The proposed model achieve best accuracy of 91.05% for the fusion of PLV, PCC, and COH, followed by the fusion of all four features with the accuracy of 90.16%. The experimental results demonstrate that the optimal thresholds for the four functional connection features always kept relatively stable within a fixed interval. In conclusion, the experimental results demonstrated the effectiveness of the proposed method.

Statistical inference for wavelet curve estimators of symmetric positive definite matrices

In this paper we treat statistical inference for a wavelet estimator of curves of symmetric positive definite (SPD) using the log-Euclidean distance. This estimator preserves positive-definiteness and enjoys permutation-equivariance, which is particularly relevant for covariance matrices. Our second-generation wavelet estimator is based on average-interpolation (AI) and allows the same powerful properties, including fast algorithms, known from nonparametric curve estimation with wavelets in standard Euclidean set-ups. The core of our work is the proposition of confidence sets for our AI wavelet estimator in a non-Euclidean geometry. We derive asymptotic normality of this estimator, including explicit expressions of its asymptotic variance. This opens the door for constructing asymptotic confidence regions which we compare with our proposed bootstrap scheme for inference. Detailed numerical simulations confirm the appropriateness of our suggested inference schemes.

Contamination Characteristics and Source Apportionment of Soil Heavy Metals in an Abandoned Pyrite Mining Area of Tongling City, China

To investigate the impact of pyrite mining on the heavy metal pollution in the surrounding soil in Tongling City, 50 surface soil and sediment samples were collected from mining fields, farmland, forests, villages, and the river. The contents of Zn, Cr, Cu, Pb, Ni, Cd, and As in soils and sediments were analyzed. Then, the spatial distribution characteristics of heavy metals in soil were analyzed, and the degree of heavy metal pollution and potential ecological risk level were assessed. Finally, the sources of soil heavy metal pollution were identified. In general, the soil in the study area was weakly acidic (average pH=6.32), and the contents of other heavy metals except Ni exceeded the background values of the soil in Tongling City. Moreover, Ni and Cd were enriched in the river sediments. According to the Nemerow pollution index, Pb and As reached heavy pollution levels, Cu and Cd reached moderate pollution levels, and other elements belonged to light or non-pollution levels. The comprehensive pollution index of different land types was ranked in the order of mining field > river > forest > farmland > village. Mining fields and the river were heavily polluted, forest land was moderately polluted, and farmland and villages were mainly mildly polluted. Pb, As, and Cd belonged to the medium ecological risk category. The contribution rates of the potential ecological risk index were 33.27%, 27.39%, and 20.22%, which were much higher than the other four elements. The ranking results of the potential ecological risk index of different land types was the same as that of the comprehensive pollution index. Mining fields and the river were at a high-risk level, forest land reached moderate risk, and the rest were at a slight risk level. The consistent results of correlation analysis, principal component analysis (PCA), and positive definite matrix factor analysis (PMF) indicated that Zn, Cu, Pb, Cd, and As were mainly derived from pyrite mining activities, Cr mainly came from the parent material and agricultural production, and Ni was mainly affected by soil-forming parent material and pyrite mining activities.

Characteristics and Source Apportionment of Carbonaceous Aerosols in the Typical Urban Areas in Chongqing During Winter

To investigate the characteristics, source apportionment, and potential source areas of carbonaceous aerosols in Chongqing during winter, PM2.5 samples were collected from January 2021 to February 2021 in the urban areas of Wanzhou (WZ), Yubei (YB), and Shuangqiao (SQ). The results showed that the average mass concentrations of PM2.5, OC, and EC in SQ were (72.6 ±33.3), (18.2 ±8.2), and (4.4 ±1.7) μg·m-3, respectively, higher than those in WZ[(67.2 ±30.3), (17.2 ±7.4), and (5.1 ±2.4) μg·m-3] and YB[(63.4 ±25.7), (15.4 ±6.3), and (4.2 ±1.9) μg·m-3]. Compared with that during the clear period, the concentration and fraction of EC in total carbon increased by 103.0% and 8.1%, respectively, in WZ compared to that in other areas during pollution period, whereas the OC/EC ratio was decreased significantly (-10.5%), indicating that the primary emission of carbonaceous aerosols increased significantly during the pollution period. The average mass concentrations of secondary organic carbon (SOC) in SQ and YB were (7.7 ±4.8) μg·m-3 and (6.9 ±2.8) μg·m-3 significantly higher, respectively, than that in WZ[(4.5 ±1.9) μg·m-3] during the campaign. This indicated that the secondary transformation had a greater influence on the carbonaceous aerosols in SQ and YB than that in WZ. Furthermore, in contrast to that in WZ, the ratios of SOC/OC were increased with the increase in PM2.5 concentrations, and significant correlations between SOC concentration and aerosol water content, NO2 concentration, and the value of NOR were observed in SQ and YB (P < 0.01), indicating that the increasing of carbonaceous aerosol concentrations might be mainly driven by the SOC with -NO2 groups produced by aqueous chemical reactions during winter in SQ and YB. The positive definite matrix factor (PMF) results in these urban areas showed that the contribution of biomass/coal combustion source in WZ (47.4%) was significantly higher than that in YB (34.2%) and SQ (38.1%), whereas the gasoline motor vehicle emission and secondary transformation impacts were more significant in YB. The results of the concentration weighted trajectory (CWT) showed that the potential sources of carbonaceous aerosols were mainly the local and northeastern parts of these urban areas (such as Changshou).

Adaptive Log-Euclidean Metrics for SPD Matrix Learning.

Symmetric Positive Definite (SPD) matrices have received wide attention in machine learning due to their intrinsic capacity to encode underlying structural correlation in data. Many successful Riemannian metrics have been proposed to reflect the non-Euclidean geometry of SPD manifolds. However, most existing metric tensors are fixed, which might lead to sub-optimal performance for SPD matrix learning, especially for deep SPD neural networks. To remedy this limitation, we leverage the commonly encountered pullback techniques and propose Adaptive Log-Euclidean Metrics (ALEMs), which extend the widely used Log-Euclidean Metric (LEM). Compared with the previous Riemannian metrics, our metrics contain learnable parameters, which can better adapt to the complex dynamics of Riemannian neural networks with minor extra computations. We also present a complete theoretical analysis to support our ALEMs, including algebraic and Riemannian properties. The experimental and theoretical results demonstrate the merit of the proposed metrics in improving the performance of SPD neural networks. The efficacy of our metrics is further showcased on a set of recently developed Riemannian building blocks, including Riemannian batch normalization, Riemannian Residual blocks, and Riemannian classifiers.