Definite Matrix Research Articles

On the Forsythe conjecture

Forsythe formulated a conjecture about the asymptotic behavior of the restarted conjugate gradient method in 1968. We translate several of his results into modern terms, and pose an analogous version of the conjecture (originally formulated only for symmetric positive definite matrices) for symmetric and nonsymmetric matrices. Our version of the conjecture uses a two-sided or cross iteration with the given matrix and its transpose, which is based on the projection process used in the Arnoldi (or for symmetric matrices the Lanczos) algorithm. We prove several new results about the limiting behavior of this iteration, but the conjecture still remains largely open. We hope that our paper motivates further research that eventually leads to a proof of the conjecture.

An intrinsic anisotropic feature of DTI images derived by geometric properties on the Riemannian manifold

Diffusion tensor imaging (DTI) is a promising technique for characterizing organic microstructural information by mapping a spatial point as a 3 × 3 symmetric matrix (called diffusion tensor). Analyzing DTI images directly is challenging since each element of the DTI image is a 3-order matrix rather than a scalar. It is meaningful to extract the scalar-valued feature for DTI image processing and analysis. In this paper, we propose an intrinsic anisotropic feature of DTI images using global geometric attributes of diffusion tensors on the Riemannian manifold. To measure the intrinsic similarity between diffusion tensors as symmetric matrices, we convert them into symmetric positive definite (SPD) matrices and use the normalized geodesic distance in the tensor manifold to define the visible intrinsic feature. Moreover, we extend the geodesic distance to make sure our feature can measure the intrinsic properties of diffusion tensors, and the neighborhood attributes and even the global information of a DTI image are utilized to define the proposed feature. To measure the anisotropy of DTI images, we define the average diffusion sphere at each point to measure the isotropic attributes and use the difference between the diffusion tensor and its average diffusion tensor to describe the anisotropic properties. It is experimentally demonstrated that the proposed feature performs adequately in characterizing various attributes of DTI images, including but not limited anisotropy, intrinsic geometric property, and average diffusion degree, etc. Compared with the most popular features, our anisotropy feature can represent DTI image properties more robustly and effectively since the contrast of various details in the feature is improved. Moreover, the results of fiber tracking defined by the anisotropy feature indicate that our feature performs better on representing substantial anisotropic attributes of DTI images and is potentially used in DTI image processing and analysis.

A quantitative Popoviciu type inequality for four positive semi-definite matrices

Recently, W. Berndt and S. Sra (Hlawka-Popoviciu inequalities on positive definite tensors. Linear Algebra Appl. 2015;486:317–327) proved a Popoviciu type inequality for any arbitrary finite number of positive definite matrices. In this paper, we proved a quantitative Popoviciu type inequality for four positive semi-definite matrices, which is stronger than Berndt-Sra's corresponding result and also a generalization of Hong-Qi's (Refinements of two determinantal inequalities for positive semidefinite matrices. Math Inequal Appl. 2022;25(3):673–678) result. As applications, we partially recovered our early result in F. Wang (A quantitative Popoviciu type inequality, submitted.] and obtained a quantitative Hartfiel inequality.

A stronger form of Yamamoto's theorem on singular values

For a matrix T∈Mm(C), let |T|:=T⁎T. For A∈Mm(C), we show that the matrix sequence {|An|1n}n∈N converges to a positive-semidefinite matrix H whose jth-largest eigenvalue is equal to the jth-largest eigenvalue-modulus of A (for 1≤j≤m). In fact, we give an explicit description of the spectral projections of H in terms of the eigenspaces of the diagonalizable part of A in its Jordan-Chevalley decomposition. This gives us a stronger form of Yamamoto's theorem which asserts that limn→∞⁡sj(An)1n is equal to the jth-largest eigenvalue-modulus of A, where sj(An) denotes the jth-largest singular value of An. Moreover, we also discuss applications to the asymptotic behaviour of the matrix exponential function, t↦etA.

A cross-session motor imagery classification method based on Riemannian geometry and deep domain adaptation

Recently, more and more studies have begun to use deep learning to decode and classify EEG signals. The use of deep learning has led to an increase in the classification accuracy of motor imagery (MI), but the problem of taking a long time to calibrate in brain–computer interface (BCI) applications has not been solved. To address this problem, we propose a novel Riemannian geometry and deep domain adaptation network (RGDDANet) for MI classification. Specifically, two one-dimensional convolutions are designed to extract temporal and spatial features from the EEG signals, and then the spatial covariance matrices are utilized to map the extracted features to Riemannian manifolds for processing. In order to align the source and target features’ distributions on the Riemannian manifold, we propose a Symmetric Positive Definite (SPD) matrix mean discrepancy loss (SMMDL) to minimize the distance between two domains. To analyze the feasibility of the method, we conducted extensive experiments on BCIC IV 2a and BCIC IV 2b datasets, respectively, and the results showed that the proposed method achieved better performance than some state-of-the-art methods.

Selfadhesivity in Gaussian conditional independence structures

Selfadhesivity is a property of entropic polymatroids which guarantees that the polymatroid can be glued to an identical copy of itself along arbitrary restrictions such that the two pieces are independent given the common restriction. We show that positive definite matrices satisfy this condition as well and examine consequences for Gaussian conditional independence structures. New axioms of Gaussian CI are obtained by applying selfadhesivity to the previously known axioms of structural semigraphoids and orientable gaussoids.

Square matrices with the inverse diagonal property

We identify the class of real square invertible matrices A for which the signs of the diagonal entries of A−1 match those of A, and begin their study. We say such matrices have the inverse diagonal property (IDP). This class includes many important classes: the positive definite matrices, the M-matrices, the totally positive matrices and some variants, the P-matrices, the diagonally dominant and H-matrices and their inverse classes, as well as triangular matrices. This class is closed under any real invertible diagonal multiplication on either the right or the left. So questions about this class can be reduced to the case of positive diagonal entries. Other basic properties are given. One theme is what conditions need be added to the IDP to insure membership in a familiar class. For example, the positive definite matrices are characterized as certain IDP matrices with special conditions on certain particular principal minors. The tridiagonal case is highlighted. Certain specially simple conditions on such matrices are mentioned that ensure them to be P-matrices, positive definite matrices or M-matrices. We also note that recent results about the invertibility of weakly diagonally dominant matrices are used. Examples are given throughout the paper.

Enhancing estimation accuracy of nonstationary hydrogeological fields via geodesic kernel-based Gaussian process regression

In this study, the combined application of the geodesic kernel and Gaussian process regression was explored to estimate nonstationary hydraulic conductivity fields in two-dimensional hydrogeological systems. Specifically, a semianalytical form of the geodesic distance based on the intrinsic geometry of the manifold was derived and used to define positive definite geodesic covariance matrices that were used for the Gaussian process regression. Furthermore, the proposed approach was applied to a series of synthetic hydraulic conductivity estimation problems. The results show that the incorporation of secondary information, such as interpretations of geophysical explorations or geological surveys, can considerably improve the accuracy of the estimation, especially in nonstationary fields. In addition, groundwater flow and solute transport simulations based on estimated hydraulic conductivity fields reveal that the accuracy of the simulation is strongly affected by the inclusion of secondary information. These results suggest that incorporating secondary information into manifold geometry can remarkably improve the accuracy of the estimation and provide new insights into the underlying structure of geological data. This proposed approach has critical implications for hydrogeological applications, such as groundwater resource management, safety assessments, and risk management strategies related to groundwater contamination.

Non-homogeneous Riemannian gradient equations for sum of squares of Bures–Wasserstein metric

We investigate Riemannian gradient equations on the open convex cone Pm of all m×m positive definite Hermitian matrices, for the function WA(X)=∑j=1nwjdW2(X,Aj)where dW denotes the Bures–Wasserstein metric. On the special case where the gradient of the weighted sum of squares of the Bures–Wasserstein metrics vanishes, its unique solution is known as the Wasserstein mean of A1,…,An. We discuss the existence and uniqueness of solutions for non-homogeneous Riemannian gradient equations at which vector fields are constant and congruence transformations. Furthermore, we establish some bounds for these solutions.

On the quasi-stability criteria of monic matrix polynomials

This paper is a continuation of a recent investigation by Zhan and Dyachenko (2021) on the Hurwitz stability of monic matrix polynomials with algebraic techniques. By improving an inertia formula for matrix polynomials with respect to the imaginary axis, we show that, under some conditions, the quasi-stability of a monic matrix polynomial can be tested via the Hermitian nonnegative definiteness of two block Hankel matrices built from its matricial Markov parameters. Moreover, for the so-called doubly monic matrix polynomials, the quasi-stability criteria can be formulated in a much simpler form. In particular, the relationship between Hurwitz stable matrix polynomials and Stieltjes positive definite matrix sequences established in Zhan and Dyachenko (2021) is included as a special case.

Distribution Characteristics, Source Analysis and Potential Ecological Risk Assessment of Soil Heavy Metals in Typical River Source Areas of Northeastern Hunan Province

Soil environmental quality in water source areas related to the residents' life, health and safety, has been the hotspot issues in science of ecological environment protection. A total of 87 surface soil samples were collected from typical river source areas in northeastern Hunan Province, using GIS technique and potential ecological risk assessment index to study the spatial distribution characteristics and Potential ecological risk of soil heavy metals (Cd, Pb, Cr, Hg and As), and the means of multivariate statistical analysis and positive definite matrix factor analysis models (PMF) were used to explain the possible sources and its contribution rates of heavy metals. The results show that:① The soil located in the typical river source area of northeastern Hunan Province was acidic, and the meant of ω(Cd), ω(Pb), ω(Cr), ω(Hg), and ω(As) in soil was 0.20, 41.07, 130.51, 0.29, and 11.63mg·kg-1, respectively. And except As, all of them had enrichment tendency. ② The comprehensive potential ecological risk of soil heavy metals was at a medium level, among which Cr, Pb and As were at a slight level of potential ecological risk, and Cd and Hg all reached a strong level of risk. ③ Cd and Pb of soil had significant homology, which were derived from agricultural activity, Cr and As of soil were affected by both domestic waste discharge and natural parent material, and Hg of soil was mainly derived from fossil combustion and transportation. ④ There were four pollution sources which included household waste discharge source, natural parent material source, Fossil burning and transportation sources, as well as sources of agricultural activities in typical river source area in northeastern Hunan Province, which contributed the rates of heavy metals were 21.36%, 35.92%, 19.30% and 23.42%, respectively, and the contribution rate of man-made sources was higher than that of natural sources. To sum up, this study has reference value for pollution prevention, ecological restoration and beautiful village construction in river source area of northeastern Hunan Province.

Gain margin and Lyapunov analysis of discrete‐time network synchronization via Riccati design

SummaryThis paper deals with solution analysis and gain margin analysis of a modified algebraic Riccati matrix equation, and the Lyapunov analysis for discrete‐time network synchronization with directed graph topologies. First, the structure of the solution to the Riccati equation associated with a single‐input controllable system is analyzed. The solution matrix entries are represented using unknown closed‐loop pole variables that are solved via a system of scalar quadratic equations. Then, the gain margin is studied for the modified Riccati equation for both multi‐input and single‐input systems. A disc gain margin in the complex plane is obtained using the solution matrix. Finally, the feasibility of the Riccati design for the discrete‐time network synchronization with general directed graphs is solved via the Lyapunov analysis approach and the gain margin approach, respectively. In the design, a network Lyapunov function is constructed using the Kronecker product of two positive definite matrices: one is the graph positive definite matrix solved from a graph Lyapunov matrix inequality involving the graph Laplacian matrix; the other is the dynamical positive definite matrix solved from the modified Riccati equation. The synchronizing conditions are obtained for the two Riccati design approaches, respectively.

Inapproximability of Positive Semidefinite Permanents and Quantum State Tomography

AbstractMatrix permanents are hard to compute or even estimate in general. It had been previously suggested that the permanents of Positive Semidefinite (PSD) matrices may have efficient approximations. By relating PSD permanents to a task in quantum state tomography, we show that PSD permanents are NP-hard to approximate within a constant factor, and so admit no polynomial-time approximation scheme (unless P = NP). We also establish that several natural tasks in quantum state tomography, even approximately, are NP-hard in the dimension of the Hilbert space. These state tomography tasks therefore remain hard even with only logarithmically few qubits.

Vigilance estimation using truncated l1 distance kernel-based sparse representation regression with physiological signals

BackgroundWith a large number of accidents caused by the decline in the vigilance of operators, finding effective automatic vigilance monitoring methods is a work of great significance in recent years. Based on physiological signals and machine learning algorithms, researchers have opened up a path for objective vigilance estimation. MethodsSparse representation (SR)-based recognition algorithms with excellent performance and simple models are very promising approaches in this field. This paper aims to study the adaptability and performance improvement of truncated l1 distance (TL1) kernel on SR-based algorithm in the context of physiological signal vigilance estimation. Compared with the traditional radial basis function (RBF), the TL1 kernel has good adaptiveness to nonlinearity and is suitable for the discrimination of complex physiological signals. A recognition framework based on TL1 and SR theory is proposed. Firstly, the inseparable physiological features are mapped to the reproducing kernel Kreĭn space through the infinite-dimensional projection of the TL1 kernel. Then the obtained kernel matrix is converted into the symmetric positive definite matrix according to the eigenspectrum approaches. Finally, the final prediction result is obtained through the sparse representation regression process. ResultsWe verified the performance of the proposed framework on the popular SEED-VIG dataset containing physiological signals (electroencephalogram and electrooculogram) associated with vigilance. In the experimental results, the TL1 kernel is superior to the RBF kernel in both performance and kernel parameter stability. ConclusionsThis demonstrates the effectiveness of the TL1 kernel in distinguishing physiological signals and the excellent vigilance estimation capability of the proposed framework. Moreover, the contribution of our research motivates the development of physiological signal recognition based on kernel methods.

Метод синтеза антенной решетки со сниженным уровнем боковых лепестков при наличии отказов каналов и модулей

A method of matrix amplitude-phase synthesis of a flat antenna array with module failures according to the requirements for the envelope of the side lobes is proposed. The problem is reduced to finding the extremum of the Rayleigh ratio of two Hermitian forms, one of which takes into account the requirements for the envelope of the side lobes and is positively determined. The proposed method differs from the known ones by an iterative procedure for searching for a weight function for the formation of integral coefficients of a positive-definite Hermitian matrix of the Rayleigh relation. In the proposed method, failure accounting is carried out using two identical diagonal matrices, single values of these matrices correspond to serviceable elements, and zero – faulty ones. The condition of positive definiteness of the quadratic form, taking into account the requirements for the envelope of the side lobes, is achieved by weighing the diagonal coefficients of the matrix. As a result, when changing the structure of failures of elements and modules of the antenna array, it is not necessary to adjust the dimension of the problem. It is shown that in the presence of module failures, the level of the side lobes of the volumetric directional pattern increases significantly. At the same time, the use of conventional weight signal processing of antenna array channels with a reduced level of side lobes leads to a loss of power of the signal component at the output of the antenna array with a slight decrease in the level of side lobes. It has been found that the application of the proposed method makes it possible to achieve a noticeable decrease in the level of side lobes with controllable losses of the signal-to-noise ratio at the output of the antenna array.

Proper improvements of well-known perturbation estimations for the parallel sum

Let A:B be the parallel sum of the positive semidefinite matrices A and B. In this paper, we first establish an operator inequality involving A:B and then apply it to obtain some norm upper bounds for the perturbations of A:B. Our results lead to considerable improvements of the well-known inequalities due to Anderson and Duffin (1969) [2]. Examples are also presented to illustrate the sharpness of the obtained norm upper bounds.

A Block Conjugate Gradient Method for Quaternion Linear

This study aims at the simultaneous solution of several quaternion linear systems with the same Hermitian and positive definite coefficient matrix by employing the conjugate gradient method. We consider the setting when the quaternion Hermitian positive definite coefficient matrix at hand is very large so that direct methods are not applicable. In the study, we first transform linear quaternion systems into real linear systems. The transformed real linear systems have special structure due to the fact that they are real representations of quaternion systems. Benefitting from the special structure, we further reduce the size of these linear systems. Then a block conjugate gradient method is applied to the resulting reduced real linear systems. The solution obtained after applying the conjugate gradient method is a real representation of the solution of the original quaternion problem. Thus, a conversion of this real solution to the quaternion setting is performed in the end.

Dimension selection for EEG classification in the SPD Riemannian space based on PSO

Several classification methods in the task of electroencephalogram (EEG) classification represent input features as symmetric positive definite (SPD) matrices. By translating the data into a Riemannian space, the classification methods have obtained superior performance; however, the high dimensionality of the data typically results in low computational efficiency and degraded classification performance. Therefore, this paper proposes a novel dimension selection method named dimension selection for SPD matrices on Riemannian manifold (DSSR). DSSR can eliminate redundant dimensions for SPD matrices in the Riemannian space based on the framework of particle swarm optimization (PSO). First, we treat one column of the SPD matrix and its corresponding row as a group, where upon the PSO is used to find optimal groups. Next, we use Riemannian classifiers to evaluate the goodness of the selected groups. Finally, the dimension selected SPD matrices are classified by the generalized learning Riemannian space quantization (GLRSQ). Experimental results on two motor imagery datasets show the superior performance of the proposed methods over EEG classification. Neuroscience experiments prove that the optimal dimensions selected by our proposed method are consistent with the findings of brain science research.

Hyper-reduction for Petrov–Galerkin reduced order models

Projection-based Reduced Order Models are based on the idea of minimizing the discrete residual of a “full order model” (FOM) while at the same time constraining the unknowns to live in a space of reduced dimension. For problems with symmetric positive definite (SPD) Jacobians, this minimization can be achieved optimally by projecting the full order residual onto the approximation basis (Galerkin Projection). This approach is sub-optimal for problems with non-SPD Jacobians since it only guarantees that the projection of the residual onto the chosen basis is minimized and not the residual itself. One possible alternative in such cases is to directly minimize the 2-norm of the residual. This minimization can be achieved either by using QR factorization to solve the resulting least-squares problem or by employing the method of the normal equations (LSPG) to the same end. The first approach involves constructing and factorizing a rectangular tall and skinny matrix of size proportional to the number of unknowns of the FOM. The LSPG method avoids the use of the large matrix by directly constructing the product by its transpose. Unfortunately, constructing such a product element by element is not feasible and requires the use of a complementary mesh, which adds an extra layer of complexity to the hyper-reduction process when performing mesh sampling. The main idea of this work is to propose an alternative technique based on the idea of Petrov–Galerkin minimization. Essentially, we choose a left basis so that a least-squares minimization procedure can be carried out on a reduced problem while guaranteeing that the discrete full order residual is minimized. The resulting procedure is applicable to problems with both SPD and non-SPD Jacobians. Additionally, the resulting minimization problem can be assembled element by element, avoiding the use of the complementary mesh and simplifying implementation in the context of finite elements. The resulting technique is amenable to hyper-reduction by the use of the Empirical Cubature Method and can be readily applied in the context of nonlinear reduction procedures.

Weak log-majorization between the geometric and Wasserstein means

There exist lots of distinct geometric means on the cone of positive definite Hermitian matrices such as the metric geometric mean, spectral geometric mean, log-Euclidean mean and Wasserstein mean. In this paper, we prove the log-majorization relation on the singular values of the product of given two positive definite matrices and their (metric and spectral) geometric means. We also establish the weak log-majorization between the spectra of two-variable Wasserstein mean and spectral geometric mean. In particular, we verify with certain condition on variables that two-variable Wasserstein mean converges decreasingly to the log-Euclidean mean with respect to the weak log-majorization.