Hermitian Positive Definite Matrix Research Articles

Matrix nearness problems with off-block-diagonal rank constraints

This paper explores the problem of finding a rank-structured approximation to a given Hermitian positive definite matrix. We first study a fundamental matrix nearness problem which seeks an approximation whose off-diagonal block must have low rank. Our investigation focuses on three objective functions: the spectral norm, Frobenius divergence, and log-determinant divergence. We derive closed-form solutions for the fundamental nearness problem, extending our analysis to include problems with a range restriction on the off-diagonal block. Building upon these results, we propose a practical, scalable algorithm to compute a positive definite hierarchical off-diagonal low-rank approximation to a Hermitian positive definite matrix. Through numerical experiments, we explore the use of our approximation as a general-purpose preconditioner for the conjugate gradient method. Our results demonstrate that the log-determinant divergence-based approximation exhibits good performance across a diverse set of test problems.

Estimation of clutter covariance matrix in stap based on knowledge-aided and geometric methods

The insufficient number of available samples can cause inaccurate estimation of the clutter covariance matrix (CCM) in space-time adaptive processing (STAP), resulting in degraded clutter suppression performance. To tackle this problem, a CCM estimation approach based on knowledge-aided (KA) and geometric methods is proposed in this paper. A combination of environmental as well as structural (Persymmetric or Symmetric structure) knowledge information is utilized to model the covariance matrix of each sample as a knowledge-aided Hermitian positive definite (KA-HPD) covariance matrix. The estimation problem is introduced into the Riemannian manifold composed of the KA-HPD covariance matrices, and the geometric method is used for nonlinear processing. Based on the Kullback-Leibler (KL) divergence and the KL mean, the final estimated CCM is designed as a weighted combination of each KA-HPD covariance matrix. Experiment results show that the two designed structural covariance matrix estimators possess superior clutter suppression performance.

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.

A new fixed point theorem in [formula omitted]-metric space and its application to solve a class of nonlinear matrix equations

In this article we extend the notions of G-metric and b-metric and define a new metric called Gb-metric with coefficient b≥1. A fixed point theorem is proved in this metric space. We obtain parallel results of several existing fixed point theorems such as that of Banach, Geraghty and Boyd–Wong in Gb-metric space using our theorem. As an application of our fixed point theorem we provide a fixed point iteration to solve a class of nonlinear matrix equations of the form Xs+A∗G(X)A=Q, where s≥1, A is an n×n matrix, G is a continuous function from the set of all Hermitian positive definite matrices to the set of all Hermitian positive semi-definite matrices and Q is an n×n Hermitian positive definite matrix. It is noted that the error in estimated solution we get by following our method is lesser than the error we get with Ćirić’s fixed point iteration.

A mixed precision LOBPCG algorithm

The locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm is a popular approach for computing a few smallest eigenvalues and the corresponding eigenvectors of a large Hermitian positive definite matrix A. In this work, we propose a mixed precision variant of LOBPCG that uses a (sparse) Cholesky factorization of A computed in reduced precision as the preconditioner. To further enhance performance, a mixed precision orthogonalization strategy is proposed. To analyze the impact of reducing precision in the preconditioner on performance, we carry out a rounding error and convergence analysis of PINVIT, a simplified variant of LOBPCG. Our theoretical results predict and our numerical experiments confirm that the impact on convergence remains marginal. In practice, our mixed precision LOBPCG algorithm typically reduces the computation time by a factor of 1.4--2.0 on both CPUs and GPUs.

On Riemann type relations for theta functions on bounded symmetric domains of type I

We provide a practical technique to obtain plenty of algebraic relations for theta functions on the bounded symmetric domains of type I. In our framework, each theta relation is controlled by combinatorial properties of a pair (T,P) of a regular matrix T over an imaginary quadratic field and a positive-definite Hermitian matrix P over the complex number field.

LDA-MIG Detectors for Maritime Targets in Nonhomogeneous Sea Clutter

This paper deals with the problem of detecting maritime targets embedded in nonhomogeneous sea clutter, where limited number of secondary data is available due to the heterogeneity of sea clutter. A class of linear discriminant analysis (LDA)-based matrix information geometry (MIG) detectors is proposed in the supervised scenario. As customary, Hermitian positive-definite (HPD) matrices are used to model the observational sample data, and the clutter covariance matrix of received dataset is estimated as geometric mean of the secondary HPD matrices. Given a set of training HPD matrices with class labels, that are elements of a higher-dimensional HPD matrix manifold, the LDA manifold projection learns a mapping from the higher-dimensional HPD matrix manifold to a lower-dimensional one subject to maximum discrimination. In the current study, the LDA manifold projection, with the cost function maximizing between-class distance while minimizing within-class distance, is formulated as an optimization problem in the Stiefel manifold. Four robust LDA-MIG detectors corresponding to different geometric measures are proposed. Numerical results based on both simulated radar clutter with interferences and real IPIX radar data show the advantage of the proposed LDA-MIG detectors against their counterparts without using LDA as well as the state-of-art maritime target detection methods in nonhomogeneous sea clutter.

The relaxed MGHSS-like method for absolute value equations

Based on the matrix splitting techniques and the ideas of the GPHSS-like method, we proposed the relaxed modified generalized HSS-like method (RMGHSS-like), which is more efficient and more robust than the RPHSS-like, the MBAS, the NI and the NHSS-like methods for the absolute value equation. Furthermore, the RMGHSS-like method is the general form of the relaxed PHSS-like method. The convergence of the RMGHSS-like iterative method is proved by theoretical analysis, and the relationships between the parameters are rigorously discussed when the coefficient matrix E is a Hermitian positive definite matrix under the minimum spectral radius. Numerical experiments had been given to recognize the effectiveness of the RMGHSS-like method.

Positive solutions of nonlinear matrix equations via fixed point results in relational metric spaces with w-distance

We consider the non-linear matrix equation (NME) of the form U = Q + ?ki =1A*i ?(U)Ai, where Q is an n?n Hermitian positive definite matrix,A1,A2,...,Am are n?n matrices, and ? is a non-linear self mapping of the set of all Hermitian matrices which are continuous in the trace norm. We discuss sufficient conditions ensuring the existence of a unique positive definite solution of the given NME. In order to do this, we introduce?w-contractive conditions involving modified simulation functions in relational metric spaces and derive fixed points results based on them, followed by two suitable examples. In order to demonstrate the obtained conditions, we consider three different sets of matrices. Three different types of examples (including randomly generated matrix and a complex matrix) are given, together with convergence and error analysis, as well as average CPU time analysis with different dimensions bar graphs, and visualization of solutions in surface plot.

Geodesic Normal Coordinate-Based Manifold Filtering for Target Detection

Recently, the matrix information geometry (MIG) detector, which characterizes sample data as a Hermitian positive definite (HPD) matrix located on the HPD manifold, was rapidly developed and demonstrated extraordinary performance in numerous applications, especially in heterogeneous clutter backgrounds. In this paper, the geodesic normal coordinate (GNC)-based manifold filter is proposed to improve the detection performance of the MIG detector in strong clutter backgrounds. Using the GNC system, the distribution of target echoes and clutter on the high-dimensional manifold can be visualized and analyzed. Moreover, by exploiting the information concerning the distribution of matrices, the manifold filter is proposed to enhance target echoes and suppress strong clutter. Then, the manifold-filter-based MIG detector is designed, and its superiority is theoretically analyzed. The actual clutter data is utilized to verify the effectiveness of the proposed method. The results show that the proposed manifold filter achieves a signal-to-clutter ratio improvement of more than 5 dB over the existing MIG detectors.

Riemannian Nearest-Regularized Subspace Classification for Polarimetric SAR Images

Nearest-regularized subspace (NRS) algorithm is a kind of effective representation learning method, which can obtain both accuracy and speed for PolSAR image classification. However, existing NRS methods use the polarimetric feature vector instead of the PolSAR original coherency matrix (known as Hermitian positive definite (HPD) matrix) as the input. This will destroy the matrix structure, and miss the instinct correlation among channels. How to utilize the original coherency matrix to NRS method is a key problem. To address this limitation, a Riemannian NRS method is proposed, which considers the HPD matrices endowed in a Riemannian space. First, to utilize the PolSAR original data, a Riemannian NRS method (RNRS) is proposed by constructing HPD dictionary and HPD distance metric. Then, a new Tikhonov regularization term is designed to reduce the differences within the same class. Finally, the optimization method is developed to resolve the proposed model, and the first-order derivative is inferred. Besides, only coherency matrix T is used as the input in the proposed method, while multiple features are utilized for compared methods in the experiments. Experimental results demonstrate the proposed method can outperform the state-of-the-art algorithms even with fewer features.

CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data

This paper investigates the central limit theorem for linear spectral statistics of high dimensional sample covariance matrices of the form \({\mathbf {B}}_n=n^{-1}\sum _{j=1}^{n}{\mathbf {Q}}{\mathbf {x}}_j{\mathbf {x}}_j^{*}{\mathbf {Q}}^{*}\) under the assumption that \(p/n\rightarrow y>0\), where \({\mathbf {Q}}\) is a \(p\times k\) nonrandom matrix and \(\{{\mathbf {x}}_j\}_{j=1}^n\) is a sequence of independent k-dimensional random vector with independent entries. A key novelty here is that the dimension \(k\ge p\) can be arbitrary, possibly infinity. This new model of sample covariance matrix \({\mathbf {B}}_n\) covers most of the known models as its special cases. For example, standard sample covariance matrices are obtained with \(k=p\) and \({\mathbf {Q}}={\mathbf {T}}_n^{1/2}\) for some positive definite Hermitian matrix \({\mathbf {T}}_n\). Also with \(k=\infty \) our model covers the case of repeated linear processes considered in recent high-dimensional time series literature. The CLT found in this paper substantially generalizes the seminal CLT in Bai and Silverstein (Ann Probab 32(1):553–605, 2004). Applications of this new CLT are proposed for testing the AR(1) or AR(2) structure for a causal process. Our proposed tests are then used to analyze a large fMRI data set.

Some Fixed Point Results on Relational Quasi Partial Metric Spaces and Application to Non-Linear Matrix Equations

We introduce a qϱ-implicit contractive condition by an implicit relation on relational quasi partial metric spaces and establish new (unique) fixed point results and periodic point results based on it. We justify the results by two suitable examples and compare with them related work. We discuss sufficient conditions ensuring the existence of a unique positive definite solution of the non-linear matrix equation U=B+∑i=1mAi*G(U)Ai, where B is an n×n Hermitian positive definite matrix, A1, A2, … Am are n×n matrices, and G is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Two examples (with randomly generated matrices and complex matrices, respectively) are given, together with convergence and error analysis, as well as average CPU time analysis and visualization of solution in surface plot.

MIG median detectors with manifold filter

In this paper, we propose a class of median-based matrix information geometry (MIG) detectors with a manifold filter and apply them to signal detection in nonhomogeneous environments. As customary, the sample data is assumed to be modeled as Hermitian positive-definite (HPD) matrices, and the geometric median of a set of HPD matrices is interpreted as an estimate of the clutter covariance matrix (CCM). Then, the problem of signal detection can be reformulated as discriminating two points on the manifold of HPD matrices, one of which is the HPD matrix in the cell under test while the other represents the CCM. By manifold filter, we map a set of HPD matrices to another set of HPD matrices by weighting them, that consequently improves the discriminative power by reducing the intra-class distances while increasing the inter-class distances. Three MIG median detectors are designed by resorting to three geometric measures on the matrix manifold, and the corresponding geometric medians are shown to be robust to outliers. Numerical simulations show the advantage of the proposed MIG median detectors in comparison with their state-of-the-art counterparts as well as the conventional detectors in nonhomogeneous environments.

On Hermitian positive definite solutions of a nonlinear matrix equation

In this paper, the Hermitian positive definite solutions of the matrix equation $$X^s +A^* X^{ - t}A = Q$$ , where A is an $$n \times n$$ nonsingular complex matrix, Q is an $$n \times n$$ Hermitian positive definite matrix and $$s, t> 0$$ , are discussed. Some conditions for the existence of Hermitian positive definite solutions of this equation are derived. In addition, two iterative methods to obtaining the maximum or minimum Hermitian positive definite solutions of this equation are proposed. In addition, a necessary and sufficient condition for the existence of these solutions is presented. Theoretical results are illustrated by some numerical examples.

Representations of the necklace braid group {{mathcal {N}}{mathcal {B}}}_n of dimension 4 (n=2,3,4)

We consider the irreducible representations each of dimension 2 of the necklace braid group {mathcal {N}}{mathcal {B}}_n (n=2,3,4). We then consider the tensor product of the representations of {mathcal {N}}{mathcal {B}}_n (n=2,3,4) and determine necessary and sufficient condition under which the constructed representations are irreducible. Finally, we determine conditions under which the irreducible representations of {mathcal {N}}{mathcal {B}}_n (n=2,3,4) of degree 2 are unitary relative to a hermitian positive definite matrix.

Investigation of positive definite solution of nonlinear matrix equation $$X^{p}=Q +\sum \nolimits _{i=1}^m A_i^*X^{\delta }A_i$$

In this paper, we consider the nonlinear matrix equation $$X^{p}=Q +\sum \nolimits _{i=1}^m A_i^*X^{\delta }A_i,$$ where $$A_i (i=1,2,\ldots ,m)$$ are $$n\times n$$ nonsingular complex matrices, Q is a $$n\times n$$ Hermitian positive definite (HPD) matrix, $$p\ge 1, m\ge 1$$ are positive integers, and $$\delta \in (0,1)$$ . We discuss the solution of this equation via properties of Thompson metric and two fixed point theorems in ordered Banach spaces and estimate the bounds of the HPD solution. Furthermore, perturbation analysis is investigated.

Target Detection Within Nonhomogeneous Clutter Via Total Bregman Divergence-Based Matrix Information Geometry Detectors

Information divergences are commonly used to measure the dissimilarity of two elements on a statistical manifold. Differentiable manifolds endowed with different divergences may possess different geometric properties, which can result in totally different performances in many practical applications. In this paper, we propose a total Bregman divergence-based matrix information geometry (TBD-MIG) detector and apply it to detect targets emerged into nonhomogeneous clutter. In particular, each sample data is assumed to be modeled as a Hermitian positive-definite (HPD) matrix and the clutter covariance matrix is estimated by the TBD mean of a set of secondary HPD matrices. We then reformulate the problem of signal detection as discriminating two points on the HPD matrix manifold. Three TBD-MIG detectors, referred to as the total square loss, the total log-determinant and the total von Neumann MIG detectors, are proposed, and they can achieve great performances due to their power of discrimination and robustness to interferences. Simulations show the advantage of the proposed TBD-MIG detectors in comparison with the geometric detector using an affine invariant Riemannian metric as well as the adaptive matched filter in nonhomogeneous clutter.

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.