Definite Matrix Research Articles

A Modified q-BFGS Algorithm for Unconstrained Optimization

This paper presents a modification of the q-BFGS method for nonlinear unconstrained optimization problems. For this modification, we use a simple symmetric positive definite matrix and propose a new q-quasi-Newton equation, which is close to the ordinary q-quasi-Newton equation in the limiting case. This method uses only first order q-derivatives to build an approximate q-Hessian over a number of iterations. The q-Armijo-Wolfe line search condition is used to calculate step length, which guarantees that the objective function value is decreasing. This modified q-BFGS method preserves the global convergence properties of the q-BFGS method, without the convexity assumption on the objective function. Numerical results on some test problems are presented, which show that an improvement has been achieved. Moreover, we depict the numerical results through the performance profiles.

Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics

A framework to study the eigenvalue probability density function for products of unitary random matrices with an invariance property is developed. This involves isolating a class of invariant unitary matrices, to be referred to as cyclic Pólya ensembles, and examining their properties with respect to the spherical transform on $$\mathrm U(N)$$ . Included in the cyclic Pólya ensemble class are Haar invariant unitary matrices, the circular Jacobi ensemble, known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz determinants, as well as the heat kernel for Brownian motion on the unitary group. We define cyclic Pólya frequency functions and show their relation to the cyclic Pólya ensembles, and give a uniqueness statement for the corresponding weights. The natural appearance of bilateral hypergeometric series is highlighted, with this special function playing the role of the Meijer G-function in the transform theory of unitary invariant product of positive definite matrices. We construct a family of functions forming bi-orthonormal pairs which underly the correlation kernel of the corresponding determinantal point processes, and furthermore obtain an integral formula for the correlation kernel involving just two of these functions.

Characteristics of Soil Heavy Metal Pollution and Health Risk Assessment in Urban Parks at a Megacity of Central China

In this study, we compared the concentrations of the heavy metals Cd, Cr, Cu, Zn, Ni, and Pb in the surface soils of urban parks in Wuhan, Hubei Province, with those in the surface soils of urban parks worldwide. The soil contamination data were assessed using enrichment factors and spatial analysis of heavy metals using inverse distance weighting and quantitative analysis of heavy metal sources with a positive definite matrix factor (PMF) receptor model. Further, a probabilistic health risk assessment of children and adults using Monte Carlo simulation was performed. The average Cd, Cr, Cu, Zn, Ni, and Pb concentrations in the surface soils of urban parks were 2.52, 58.74, 31.39, 186.28, 27.00, and 34.89 mg·kg-1, respectively, which exceeded the average soil background values in Hubei. From the inverse distance spatial interpolation map, heavy metal contamination was primarily observed to be present to the southwest of the main urban area. The PMF model resolved four sources: mixed traffic and industrial emission, natural, agricultural, and traffic sources, with relative contributions of 23.9%, 19.3%, 23.4%, and 33.4%, respectively. The Monte Carlo health risk evaluation model demonstrated negligible noncancer risks for both adult and child populations, whereas the health effects of Cd and Cr on children were a concern for cancer risks.

Low Complex Analog Beamforming Design in Multi-User mmWave Non-Orthogonal Multiple Access (NOMA)

In this paper, we investigate non-orthogonal multiple access (NOMA) scheme in millimeter wave (mmWave) communication to serve nonclustered multiple users. We explore a low complex design of an analog beamforming weight vector and the power requirement of users aiming to minimize the total power targeting to satisfy the spectral efficiency (SE) requirements of all users. We propose a low complex constant modulus analog beamforming (CMAB) algorithm, where we first reduce the number of signal to interference plus noise ratio (SINR) constraints, which is attained from the order of equivalent channel gain of users. Then, the nonconvex constraint of constant modulus (CM) is relaxed and semi-definite programming (SDP) is used to solve the problem. Obtained weight vector and power for all users are optimal since the rank of positive semi-definite (PSD) matrix is one. Later, CM constraint is included. Simulation results show that the proposed algorithm requires less power with minimum complexity compared to the existing research, digital beamforming NOMA and time division multiple access (TDMA) for the same SE requirements.

Geometric means of quasi-Toeplitz matrices

We study means of geometric type of quasi-Toeplitz matrices, that are semi-infinite matrices $$A=(a_{i,j})_{i,j=1,2,\ldots }$$ of the form $$A=T(a)+E$$ , where E represents a compact operator, and T(a) is a semi-infinite Toeplitz matrix associated with the function a, with Fourier series $$\sum _{k=-\infty }^{\infty } a_k e^{{{\mathfrak {i}}}k t}$$ , in the sense that $$(T(a))_{i,j}=a_{j-i}$$ . If a is real valued and essentially bounded, then these matrices represent bounded self-adjoint operators on $$\ell ^2$$ . We prove that if $$a_1,\ldots ,a_p$$ are continuous and positive functions, or are in the Wiener algebra with some further conditions, then matrix geometric means, such as the ALM, the NBMP and the weighted mean of quasi-Toeplitz positive definite matrices associated with $$a_1,\ldots ,a_p$$ , are quasi-Toeplitz matrices associated with the function $$(a_1\cdots a_p)^{\frac{1}{p}}$$ , which differ only by the compact correction. We introduce numerical algorithms for their computation and show by numerical tests that these operator means can be effectively approximated numerically.

A limited memory Quasi-Newton approach for multi-objective optimization

In this paper, we deal with the class of unconstrained multi-objective optimization problems. In this setting we introduce, for the first time in the literature, a Limited Memory Quasi-Newton type method, which is well suited especially in large scale scenarios. The proposed algorithm approximates, through a suitable positive definite matrix, the convex combination of the Hessian matrices of the objectives; the update formula for the approximation matrix can be seen as an extension of the one used in the popular L-BFGS method for scalar optimization. Equipped with a Wolfe type line search, the considered method is proved to be well defined even in the nonconvex case. Furthermore, for twice continuously differentiable strongly convex problems, we state global and R-linear convergence to Pareto optimality of the sequence of generated points. The performance of the new algorithm is empirically assessed by a thorough computational comparison with state-of-the-art Newton and Quasi-Newton approaches from the multi-objective optimization literature. The results of the experiments highlight that the proposed approach is generally efficient and effective, outperforming the competitors in most settings. Moreover, the use of the limited memory method results to be beneficial within a global optimization framework for Pareto front approximation.

On the role of tensor interpolation in solving high-WI viscoelastic fluid flow

The high Weissenberg number (Wi) problem (HWNP) has long been a challenge of viscoelastic fluid flow simulation. This Letter points out that the tensor interpolation method during solving the differential constitutive equations is the main origin of the loss of symmetric positive-definite (SPD) property of the conformation tensor, which is the trigger of the HWNP. Instead of component-based interpolation, we propose a tensor-based interpolation method for the conformation tensor, which is essentially SPD, and the results show that this method is very effective in dealing with the HWNP by significantly improving the numerical accuracy on the invariants of conformation tensor as well as greatly improving the SPD property of the conformation tensor. Moreover, the high-order total variation diminishing schemes can also be easily constructed and applied to solve high-Wi viscoelastic fluid flow under the proposed framework without adding artificial diffusion.

Local additivity revisited

We make a number of simplifications in Gour and Friedland’s proof of local additivity of the minimum output entropy of a quantum channel. We follow them in reframing the question as one about the entanglement entropy of bipartite states associated with a dB × dE matrix. We use a different approach to reduce the general case to that of a square positive definite matrix. We use the integral representation of the log to obtain expressions for the first and second derivatives of the entropy, and then exploit the modular operator and functional calculus to streamline the proof following their underlying strategy. We also extend this result to the maximum relative entropy with respect to a fixed reference state, which has important implications for studying the superadditivity of the capacity of a quantum channel to transmit classical information.

Remarks on a Question of Bourin for Positive Semidefinite Matrices

Remarks on a Question of Bourin for Positive Semidefinite Matrices

Computing the Square Root of a Low-Rank Perturbation of the Scaled Identity Matrix

We consider the problem of computing the square root of a perturbation of the scaled identity matrix, , where and are matrices with . This problem arises in various applications, including computer vision and optimization methods for machine learning. We derive a new formula for the th root of that involves a weighted sum of powers of the th root of the matrix . This formula is particularly attractive for the square root, since the sum has just one term when . We also derive a new class of Newton iterations for computing the square root that exploit the low-rank structure. We test these new methods on random matrices and on positive definite matrices arising in applications. Numerical experiments show that the new approaches can yield a much smaller residual than existing alternatives and can be significantly faster when the perturbation has low rank.

A permanent inequality for positive semidefinite matrices

In this paper, we prove an inequality involving the permanent of a positive semidefinite matrix and its leading submatrices. We obtain a result in the similar spirit of Bapat-Sunder per-max conjecture.

New lower bounds on the minimum singular value of a matrix

The study of constraining the eigenvalues of the sum of two symmetric matrices, say P+Q, in terms of the eigenvalues of P and Q, has a long tradition. It is closely related to estimating a lower bound on the minimum singular value of a matrix, which has been discussed by a great number of authors. To our knowledge, no study has yielded a positive lower bound on the minimum eigenvalue, λmin(P+Q), when P+Q is symmetric positive definite with P and Q singular positive semi-definite. We derive two new lower bounds on λmin(P+Q) in terms of the minimum positive eigenvalues of P and Q. The bounds take into account geometric information by utilizing the Friedrichs angles between certain subspaces. The basic result is when P and Q are two non-zero singular positive semi-definite matrices such that P+Q is non-singular, then λmin(P+Q)≥(1−cos⁡θF)min⁡{λmin(P),λmin(Q)}, where λmin represents the minimum positive eigenvalue of the matrix, and θF is the Friedrichs angle between the range spaces of P and Q. Such estimates lead to new lower bounds on the minimum singular value of full rank 1×2, 2×1, and 2×2 block matrices. Some examples provided in this paper further highlight the simplicity of applying the results in comparison to some existing lower bounds.

Downward quantum learning from element 118: Automated generation of Fermi-Löwdin orbitals for all atoms.

A new algorithm based on a rigorous theorem and quantum data computationally mined from element 118 guarantees automated construction of initial Fermi-Löwdin-Orbital (FLO) starting points for all elements in the Periodic Table. It defines a means for constructing a small library of scalable FLOs for universal use in molecular and solid-state calculations. The method can be systematically improved for greater efficiency and for applications to excited states such as x-ray excitations and optically silent excitations. FLOs were introduced to recast the Perdew-Zunger self-interaction correction (PZSIC) into an explicit unitarily invariant form. The FLOs are generated from a set of N quasi-classical electron positions, referred to as Fermi-Orbital descriptors (FODs), and a set of N-orthonormal single-electron orbitals. FOD positions, when optimized, minimize the PZSIC total energy. However, creating sets of starting FODs that lead to a positive definite Fermi orbital overlap matrix has proven to be challenging for systems composed of open-shell atoms and ions. The proof herein guarantees the existence of a FLOSIC solution and further guarantees that if a solution for N electrons is found, it can be used to generate a minimum of N - 1 and a maximum of 2N - 2 initial starting points for systems composed of a smaller number of electrons. Applications to heavy and super-heavy atoms are presented. All starting solutions reported here were obtained from a solution for element 118, Oganesson.

The double-relaxation modulus-based matrix splitting iteration method for linear complementarity problems

For the large sparse linear complementarity problems (LCP), by adopting the relaxation and relaxation two-sweep techniques for the modulus-based matrix splitting (MMS) iteration method, we establish a double-relaxation two-sweep modulus-based matrix splitting (DRTMMS) iteration method. This new method contains some known ones developed recently. Some sufficient conditions for guaranteeing the convergence of the DRTMMS method are presented when the system matrices are positive definite matrices and H+-matrices, which generalize some existing results. And the parameter selection strategy of the DRTMMS iteration method is given. Besides, the convergence analyses of the DRTM accelerated overrelaxation (DRTMAOR) method are given. Finally, numerical experiments illustrate that the DRTMMS method is efficient and has better performance than several existing MMS-like methods, and outperforms the projected splitting methods in some cases.

Extrinsic Bayesian Optimization on Manifolds

We propose an extrinsic Bayesian optimization (eBO) framework for general optimization problems on manifolds. Bayesian optimization algorithms build a surrogate of the objective function by employing Gaussian processes and utilizing the uncertainty in that surrogate by deriving an acquisition function. This acquisition function represents the probability of improvement based on the kernel of the Gaussian process, which guides the search in the optimization process. The critical challenge for designing Bayesian optimization algorithms on manifolds lies in the difficulty of constructing valid covariance kernels for Gaussian processes on general manifolds. Our approach is to employ extrinsic Gaussian processes by first embedding the manifold onto some higher dimensional Euclidean space via equivariant embeddings and then constructing a valid covariance kernel on the image manifold after the embedding. This leads to efficient and scalable algorithms for optimization over complex manifolds. Simulation study and real data analyses are carried out to demonstrate the utilities of our eBO framework by applying the eBO to various optimization problems over manifolds such as the sphere, the Grassmannian, and the manifold of positive definite matrices.

On the convergence of iterative schemes for solving a piecewise linear system of equations

This paper is devoted to studying the global and finite convergence of the semi-smooth Newton method for solving a piecewise linear system that arises in cone-constrained quadratic programming problems and absolute value equations. We first provide a negative answer via a counterexample to a conjecture on the global and finite convergence of the Newton iteration for symmetric and positive definite matrices. Additionally, we discuss some surprising features of the semi-smooth Newton iteration in low dimensions and its behavior in higher dimensions. Secondly, we present two iterative schemes inspired by the classical Jacobi and Gauss-Seidel methods for linear systems of equations for finding a solution to the problem. We study sufficient conditions for the convergence of both proposed procedures, which are also sufficient for the existence and uniqueness of solutions to the problem. Lastly, we perform some computational experiments designed to illustrate the behavior (in terms of CPU time) of the proposed iterations versus the semi-smooth Newton method for dense and sparse large-scale problems. Moreover, we included the numerical solution of a discretization of the Boussinesq PDE modeling a two-dimensional flow in a homogeneous phreatic aquifer.

Observer-Based Robust Fuzzy Controller Design for Uncertain Singular Fuzzy Systems Subject to Passivity Criterion

This paper discusses an observer-based control problem for uncertain Takagi–Sugeno Fuzzy Singular Systems (T-SFSS) subject to passivity performance constraints. Through the Parallel Distributed Compensation (PDC) approach and the Proportional Derivative (PD) control scheme, an observer-based fuzzy controller is constructed to achieve the stability of the considered system. An unlimited positive definite matrix is utilized to construct the Lyapunov function and derive sufficient stability conditions to develop a relaxed design method. Moreover, some technologies, such as the Schur complement, projection lemma, and Singular Value Decomposition (SVD), are applied to convert the conditions to Linear Matrix Inequality (LMI) form. Therefore, the convex optimization algorithm is used to solve the LMI conditions to find feasible solutions. The observer-based fuzzy controller is established with the obtained solutions to guarantee stability and passivity performance for the uncertain nonlinear singular systems. Finally, two examples are provided to verify the availability of the proposed fuzzy control approach.

New Hilbert–Schmidt norm inequalities for positive semidefinite matrices

New Hilbert–Schmidt norm inequalities for positive semidefinite matrices

Phase space reconstruction, geometric filtering based Fisher discriminant analysis and minimum distance to the Riemannian means algorithm for epileptic seizure classification

BackgroundMost clinicians observe the changes of the electroencephalograms (EEGs) via the visual inspection of the recorded signals of the patients. However, the visual inspection is time consuming. Also, it is subjective and empirical. Therefore, designing an accurate and an automatic epilepsy classification system has a profound significance for reducing the workload of the medical officers. MethodFirst, this paper applies the phase space reconstruction (PSR) method for constructing the covariance matrix for characterizing the chaotic states of the EEGs. The covariance matrices are symmetric positive definite (SPD) matrices used as the feature descriptors of the EEGs. This set of the covariance matrices is in fact a kind of the Riemannian manifold. Denote this Riemannian manifold as M. Secondly, the tangent space at the Riemannian geometric mean is computed via the logarithmic mapping operator. Denote this tangent space as TM. This tangent space TM is a kind of the Euclidean space containing a point in the Riemannian manifold M. That is, the SPD matrices in the Riemannian manifold M are mapped to the matrices in the tangent space TM at the Riemannian geometric mean via the logarithmic mapping operator. Then, these matrices in the tangent space TM are further mapped to the tangent feature vectors. The set of the tangent feature vectors is a kind of the vector space. Denote this vector space as VT. Thirdly, the transformation matrix based on the Fisher discrimination analysis (FDA) is used to map the high dimensional vectors in the vector space VT to the low dimensional vectors in another vector space. Then, the discriminative vectors are projected back to the Riemannian manifold M via the exponential mapping operator. Finally, the minimum distance to the Riemannian means (MDRM) algorithm is applied for performing the epileptic seizure classification. ResultsThe computer numerical simulations are conducted based on three well known EEGs epileptic seizure datasets. Overall, our proposed method achieves the best performances for three datasets compared to the state of the art methods. Also, the computational complexity of our proposed method is low. ConclusionOur proposed method is effective and robust for performing the seizure classification. Also, our proposed method can be implemented in the real time.

Restricted Riemannian geometry for positive semidefinite matrices

We introduce the manifold of restrictedn×n positive semidefinite matrices of fixed rank p, denoted S(n,p)⁎. The manifold itself is an open and dense submanifold of S(n,p), the manifold of n×n positive semidefinite matrices of the same rank p, when both are viewed as manifolds in Rn×n. This density is the key fact that makes the consideration of S(n,p)⁎ statistically meaningful. We furnish S(n,p)⁎ with a convenient, and geodesically complete, Riemannian geometry, as well as a Lie group structure, that permits analytical closed forms for endpoint geodesics, parallel transports, Fréchet means, exponential and logarithmic maps. This task is done partly through utilizing a reduced Cholesky decomposition, whose algorithm is also provided. We produce a second algorithm from this framework to estimate principal eigenspaces and demonstrate its superior performance over other existing algorithms.