Hermitian Positive Definite Matrix Research Articles

Target Detection Through Riemannian Geometric Approach With Application to Drone Detection

Radar detection of small drones in presence of noise and clutter is considered from a differential geometry viewpoint. The drone detection problem is challenging due to low radar cross section (RCS) of drones, especially in cluttered environments and when drones fly low and slow in urban areas. This paper proposes two detection techniques, the Riemannian-Brauer matrix (RBM) and the angle-based hybrid-Brauer (ABHB), to improve the probability of drone detection under small sample size and low signal-to-clutter ratio (SCR). These techniques are based on the regularized Burg algorithm (RBA), the Brauer disc (BD) theorem, and the Riemannian mean and distance. Both techniques exploit the RBA to obtain a Toeplitz Hermitian positive definite (THPD) covariance matrix from each snapshot and apply the BD theorem to cluster the clutter-plus-noise THPD covariance matrices. The proposed Riemannian-Brauer matrix technique is based on the Riemannian distance between the Riemannian mean of clutter-plus-noise cluster and potential targets. The proposed angle-based hybrid-Brauer technique uses the Euclidean tangent space and the Riemannian geodesical distances between the Riemannian mean, the Riemannian median and the potential target point. The angle at the potential target on the manifold is computed using the law of cosines on the manifold. The proposed detection techniques show advantage over the fast Fourier transform, the Riemannian distance-based matrix and the Kullback-Leibler (KLB) divergence detectors. The validity of both proposed techniques are demonstrated with real data.

A Novel Approach to Data Encryption Based on Matrix Computations

In this paper, we provide a new approach to data encryption using generalized inverses. Encryption is based on the implementation of weighted Moore–Penrose inverse A†MN(nxm) over the nx8 constant matrix. The square Hermitian positive definite matrix N8x8 p is the key. The proposed solution represents a very strong key since the number of different variants of positive definite matrices of order 8 is huge. We have provided NIST (National Institute of Standards and Technology) quality assurance tests for a random generated Hermitian matrix (a total of 10 different tests and additional analysis with approximate entropy and random digression). In the additional testing of the quality of the random matrix generated, we can conclude that the results of our analysis satisfy the defined strict requirements. This proposed MP encryption method can be applied effectively in the encryption and decryption of images in multi-party communications. In the experimental part of this paper, we give a comparison of encryption methods between machine learning methods. Machine learning algorithms could be compared by achieved results of classification concentrating on classes. In a comparative analysis, we give results of classifying of advanced encryption standard (AES) algorithm and proposed encryption method based on Moore–Penrose inverse.

A remark on approximating permanents of positive definite matrices

Let A be an n×n positive definite Hermitian matrix with all eigenvalues between 1 and 2. We represent the permanent of A as the integral of some explicit log-concave function on R2n. Consequently, there is a fully polynomial randomized approximation scheme (FPRAS) for perA.

Detection of maneuvering target with motion compensation errors based on α log-determinant divergence and symmetrized α log-determinant divergence

Long-term coherent integration is an effective method for target detection. However, the inevitable motion compensation errors and complex background clutter seriously limit the detection performance of a cell-averaging constant false alarm rate (CFAR) detector. A generalized CFAR detector on the Riemannian manifold of Hermitian positive-definite matrix is proposed. Based on the α log-determinant divergence (α-LDD) and symmetrized α-LDD, the mean matrix of the reference cells is first estimated. Then the dissimilarity between the covariance matrix of the cell under test and the mean matrix is measured to determine the presence of a target. Numerical experiments show that the proposed detectors have better detection performance than other representative methods when detecting a target with motion compensation errors in complex background.

Solution of a Class of Nonlinear Matrix Equations

In this article, we present several necessary and sufficient conditions for the existence of Hermitian positive definite solutions of nonlinear matrix equations of the form $$X^s+A^*X^{-t}A+B^*X^{-p}B=Q$$ , where $$s,t,p \ge 1$$ , A, B are nonsingular matrices and Q is a Hermitian positive definite matrix. We derive some iterations to compute the solutions followed by some examples. In this context, we also discuss about the maximal and the minimal Hermitian positive definite solution of this particular nonlinear matrix equation.

Two Structure-Preserving-Doubling Like Algorithms to Solve the Positive Definite Solution of the Equation $$X-{A^{\rm{H}}}{\overline{X}}^{-1}A=Q$$

In this paper, we study the nonlinear matrix equation $$X-{A^{\rm{H}}}{\overline{X}}^{-1}A=Q$$ , where $$A,Q \in {{\mathbb {C}}}^{n\times n}$$ , Q is a Hermitian positive definite matrix and $$X \in {{\mathbb {C}}}^{n\times n}$$ is an unknown matrix. We prove that the equation always has a unique Hermitian positive definite solution. We present two structure-preserving-doubling like algorithms to find the Hermitian positive definite solution of the equation, and the convergence theories are established. Finally, we show the effectiveness of the algorithms by numerical experiments.

Application of gradient descent algorithms based on geodesic distances

In this paper, the Riemannian gradient algorithm and the natural gradient algorithm are applied to solve descent direction problems on the manifold of positive definite Hermitian matrices, where the geodesic distance is considered as the cost function. The first proposed problem is control for positive definite Hermitian matrix systems whose outputs only depend on their inputs. The geodesic distance is adopted as the difference of the output matrix and the target matrix. The controller to adjust the input is obtained such that the output matrix is as close as possible to the target matrix. We show the trajectory of the control input on the manifold using the Riemannian gradient algorithm. The second application is to compute the Karcher mean of a finite set of given Toeplitz positive definite Hermitian matrices, which is defined as the minimizer of the sum of geodesic distances. To obtain more efficient iterative algorithm compared with traditional ones, a natural gradient algorithm is proposed to compute the Karcher mean. Illustrative simulations are provided to show the computational behavior of the proposed algorithms.

Perturbation theory for Hermitian quadratic eigenvalue problem – damped and simultaneously diagonalizable systems

The main contribution of this paper is a novel approach to the perturbation theory of a structured Hermitian quadratic eigenvalue problems (λ2M+λD+K)x=0. We propose a new concept without linearization, considering two structures: general quadratic eigenvalue problems (QEP) and simultaneously diagonalizable quadratic eigenvalue problems (SDQEP). Our first two results are upper bounds for the difference |∥X2*MX˜1∥F2−∥X2*MX1∥F2|, and for ∥X2*MX˜1−X2*MX1∥F, where the columns of X1=[x1,…,xk] and X2=[xk+1,…,xn] are linearly independent right eigenvectors and M is positive definite Hermitian matrix. As an application of these results we present an eigenvalue perturbation bound for SDQEP. The third result is a lower and an upper bound for ∥sinΘ(X1,X˜1)∥F, where Θ is a matrix of canonical angles between the eigensubspaces X1 and X˜1,X1 is spanned by the set of linearly independent right eigenvectors of SDQEP and X˜1 is spanned by the corresponding perturbed eigenvectors. The quality of the mentioned results have been illustrated by numerical examples.

Matrix Information Geometry for Signal Detection via Hybrid MPI/OpenMP

The matrix information geometric signal detection (MIGSD) method has achieved satisfactory performance in many contexts of signal processing. However, this method involves many matrix exponential, logarithmic, and inverse operations, which result in high computational cost and limits in analyzing the detection performance in the case of a high-dimensional matrix. To address these problems, in this paper, a high-performance computing (HPC)-based MIGSD method is proposed, which is implemented using the hybrid message passing interface (MPI) and open multiple processing (OpenMP) techniques. Specifically, the clutter data are first modeled as a Hermitian positive-definite (HPD) matrix and mapped into a high-dimensional space, which constitutes a complex Riemannian manifold. Then, the task of computing the Riemannian distance on the manifold between the sample data and the geometric mean of these HPD matrices is assigned to each MPI process or OpenMP thread. Finally, via comparison with a threshold, the signal is identified and the detection probability is calculated. Using this approach, we analyzed the effect of the matrix dimension on the detection performance. The experimental results demonstrate the following: (1) parallel computing can effectively optimize the MIGSD method, which substantially improves the practicability of the algorithm; and (2) the method achieves superior detection performance under a higher dimensional HPD matrix.

An extension of the positive-definite and skew-Hermitian splitting method for preconditioning of generalized saddle point problems

We extend the positive-definite and skew-Hermitian splitting (PSS) iterative method to EPSS (for extended PSS) for solving non-Hermitian positive-definite systems of linear equations. In this kind of extension, the shift matrix is replaced by a Hermitian positive-definite matrix Σ. It is proved that the method is unconditionally convergent. Then, the EPSS method with a 2 × 2-block diagonal matrix Σ as the shift matrix is applied to generalized saddle point problems and the convergence of the method is verified. Naturally, the induced preconditioner can be applied to generalized saddle point problems. In some special cases, this preconditioner coincides with some existing preconditioners. A special case of the EPSS preconditioner, say SEPSS, is used to accelerate the convergence of GMRES. Effectiveness of this preconditioner is investigated by some numerical experiments.

On the Unitary Representations of the Braid Group B6

We consider a non-abelian leakage-free qudit system that consists of two qubits each composed of three anyons. For this system, we need to have a non-abelian four dimensional unitary representation of the braid group B 6 to obtain a totally leakage-free braiding. The obtained representation is denoted by ρ . We first prove that ρ is irreducible. Next, we find the points y ∈ C * at which the representation ρ is equivalent to the tensor product of a one dimensional representation χ ( y ) and μ ^ 6 ( ± i ) , an irreducible four dimensional representation of the braid group B 6 . The representation μ ^ 6 ( ± i ) was constructed by E. Formanek to classify the irreducible representations of the braid group B n of low degree. Finally, we prove that the representation χ ( y ) ⊗ μ ^ 6 ( ± i ) is a unitary relative to a hermitian positive definite matrix.

Application of joint domain localised matrix CFAR detector for HFSWR

Target detection is one of the most important parts of high-frequency surface wave radar (HFSWR) signal processing, to find targets in noise or clutter and obtain targets’ information. However, some factors, such as the influence of clutter and the increase of detection range, will degrade the signal-to-clutter ratio (SCR). In the low SCR scenario, the classical detector of HFSWR, which uses the amplitude of receipt signal only, could hardly detect targets. To solve this problem, this study proposes a novel detector called joint domain localised matrix constant false-alarm rate (CFAR) detector based on receipt signal's multi-dimensional information. This detector employs joint domain localised algorithm to get signal information in angle and Doppler domain, and use information geometry method to map them to Hermitian positive-definite (HPD) matrix space which can be depicted as Riemannian manifold. Then, based on HPD matrix, in the range domain the matrix CFAR detector is built to detect targets. The experiments’ results verify that the detector can improve radar detection performance effectively.

Mathematical analysis of intensity distribution of the optical image in various degrees of coherence of illumination (representation of intensity by Hermitian matrices)

This is a historical translation of the seminal paper by H. Gamo, originally published in Oyo Buturi ( Applied Physics , a journal of The Japan Society of Applied Physics) Vol. 25, pp. 431–443, 1956. English translation by Kenji Yamazoe, with further editing by the translator and Anthony Yen. Since optical systems have distinctive features as compared to electrical communication systems, some formulation should be prepared for the optical image in order to use it in information theory of optical systems. In this paper the following formula for the intensity distribution of the image by an optical system having a given aperture constant α in the absence of both aberration and defect in focusing is obtained by considering the nature of illumination, namely coherent, partially coherent, and incoherent: I ( y ) = ∑ n ∑ m a n m u n ( y ) u m * ( y ) , where u n(y) = sin 2πα/λ (y − nλ/2α) / 2πα/λ (y − nλ/2α) and a nm = (2α/λ)2 ∬ Γ12( x 1 − x 2) E( x 1) E * ( x 2) | A( x 1) || A * ( x 2) | u n( x 1) u m( x 2)d x 1 d x 2. I ( y ) is the intensity of the image at a point of coordinate y , Γ12 the phase coherence factor introduced by H. H. Hopkins et al., E ( x ) the complex transmission coefficient of the object and A ( x ) the complex amplitude of the incident waves at the object, and the integration is taken over the object plane. The above expression has some interesting features, namely the “intensity matrix” composed of the element a nm mentioned above is a positive-definite Hermitian matrix, and the diagonal elements are given by the intensities sampled at every point of the image plane separated by a distance λ / 2α, and the trace of the matrix or the sum of diagonal elements is equal to the total intensity integrated over the image plane. Since a Hermitian matrix can be reduced to diagonal form by a unitary transformation, the intensity distribution of the image can be expressed as I ( y ) = λ 1 | ∑ S <mml:mrow

On (Λ,Υ,ℜ)-Contractions and Applications to Nonlinear Matrix Equations

In this paper, we study the behavior of Λ , Υ , ℜ -contraction mappings under the effect of comparison functions and an arbitrary binary relation. We establish related common fixed point theorems. We explain the significance of our main theorem through examples and an application to a solution for the following nonlinear matrix equations: X = D + ∑ i = 1 n A i ∗ X A i − ∑ i = 1 n B i ∗ X B i X = D + ∑ i = 1 n A i ∗ γ X A i , where D is an Hermitian positive definite matrix, A i , B i are arbitrary p × p matrices and γ : H ( p ) → P ( p ) is an order preserving continuous map such that γ ( 0 ) = 0 . A numerical example is also presented to illustrate the theoretical findings.

Detecting Changes in Fully Polarimetric SAR Imagery With Statistical Information Theory

Images obtained from coherent illumination processes are contaminated with speckle. A prominent example of such imagery systems is the polarimetric synthetic aperture radar (PolSAR). For such remote sensing tool the speckle interference pattern appears in the form of a positive definite Hermitian matrix, which requires specialized models and makes change detection a hard task. The scaled complex Wishart distribution is a widely used model for PolSAR images. Such distribution is defined by two parameters: the number of looks and the complex covariance matrix. The last parameter contains all the necessary information to characterize the backscattered data and, thus, identifying changes in a sequence of images can be formulated as a problem of verifying whether the complex covariance matrices differ at two or more takes. This paper proposes a comparison between a classical change detection method based on the likelihood ratio and three statistical methods that depend on information-theoretic measures: the Kullback-Leibler distance and two entropies. The performance of these four tests was quantified in terms of their sample test powers and sizes using simulated data. The tests are then applied to actual PolSAR data. The results provide evidence that tests based on entropies may outperform those based on the Kullback-Leibler distance and likelihood ratio statistics.

Inequalities for the eigenvalues of the positive definite solutions of the nonlinear matrix equation

In this paper, the Hermitian positive definite solutions of the matrix equation Xs + A*X-tA = Q are considered, where Q is a Hermitian positive definite matrix, s and t are positive integers. Bounds for the sum of eigenvalues of the solutions to the equation are given. The equivalent conditions for solutions of the equation are obtained. The eigenvalues of the solutions of the equation with the case AQ = QA are investigated.

The Bauer-Type Factorization of Matrix Polynomials Revisited and Extended

For a Laurent polynomial $$a(\lambda )$$ , which is Hermitian and positive definite on the unit circle, the Bauer method provides the spectral factorization $$a(\lambda ) = p(\lambda )p{\kern 1pt} {\text{*}}({{\lambda }^{{ - 1}}})$$ , where $$p(\lambda )$$ is a polynomial having all its roots outside the unit circle. Namely, as the size of the banded Hermitian positive definite Toeplitz matrix associated with the Laurent polynomial increases, the coefficients at the bottom of its Cholesky lower triangular factor tend to the coefficients of $$p(\lambda )$$ . We study extensions of the Bauer method to the non-Hermitian matrix case. In the Hermitian case, we give new convergence bounds with computable coefficients.

Geometric target detection based on total Bregman divergence

This paper develops a geometric detection approach based upon the total Bregman divergence on the Riemannian manifold of Hermitian Positive-Definite (HPD) matrices to realize target detection in a clutter. First of all, the radar received clutter data in each range cell in one coherent processing interval is modeled and mapped into an HPD matrix space, which can be described as a complex Riemannian manifold. Each point of this manifold is an HPD matrix. Then, a class of total Bregman divergences are presented to measure the closeness between HPD matrices. Based on these divergences, the medians for a finite collection of HPD matrices are derived. Furthermore, the three divergences, namely the total square loss, the total log-determinant divergence, and the total von Neumann divergence are deduced, and their corresponding geometric detection methods are designed. The principle of detection is that if a location has enough dissimilarity from the median estimated by its neighboring locations, targets are supposed to appear at this location. Numerical experiments and real clutter data are given to demonstrate the effectiveness of the proposed geometric detection methods.

Noncrossing partitions, noncrossing graphs, and q-permanental equations

We first prove a few simple results to illustrate some algebraic combinatorial features of the q-permanent. This is followed by a characterization of a noncrossing permutation in terms of the numbers of inversions of its cycles. Then we use a family of derivative formulas for the q-permanent of a square matrix A to characterize several structures of noncrossing kind. Each such formula f characterizes a set Df of digraphs, in the sense that D∈Df iff f is valid for all matrices A with digraph D. In this way we characterize, among others, digraphs with noncrossing permutation subdigraphs, noncrossing graphs, noncrossing forests. We use the derivative formulas to prove two particular cases of a conjecture on the q-monotonicity of the q-permanent of a Hermitian positive definite matrix.

A two-grid discretization scheme of non-conforming finite elements for transmission eigenvalues

In this paper, for the Helmholtz transmission eigenvalue problem, we propose a two-grid discretization scheme of non-conforming finite elements. With this scheme, the solution of the transmission eigenvalue problem on a fine grid πh is reduced to the solution of the primal and dual eigenvalue problem on a much coarser grid πH and the solutions of two linear algebraic systems with the same positive definite Hermitian and block diagonal coefficient matrix on the fine grid πh. We prove the resulting solution still maintains an asymptotically optimal accuracy, and we report some numerical examples in two dimension and three dimension on the modified-Zienkiewicz element to validate the efficiency of our approach for solving transmission eigenvalues.