Hermitian Positive Definite Matrices Research Articles

Linearity of Cartan and Wasserstein means

The convex cone of positive definite Hermitian matrices has two important Riemannian geometries, where the parametrized weighted geometric (alternatively, Cartan) and Wasserstein means appear as the corresponding geodesics. The major problem with which this paper is concerned is linearity of Cartan and Wasserstein geodesics; when the Cartan (resp. Wasserstein) geodesic between two positive definite matrices A and B does lie in the space spanned by them and what the path in the plane to which it corresponds is. We settle the problem completely for the Cartan geometry and partially for the Wasserstein geometry. We show that their linearity problems for linearly independent A and B are equivalent to the solvability of the following equations for positive reals x and y with xy<14 respectively(xA+yB)−1=yA−1+xB−1,(AB)1/2+(BA)1/22=xA+yB.

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.

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.

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.

Ship Detection in SAR Images via Kullback-Leibler Divergence-Based Matrix Information Geometry Detector

Ship detection in synthetic aperture radar (SAR) imagery is an important application in the field of marine remote sensing. As the most common method in ship detection, classical constant false alarm rate (CFAR) methods rely on the target-to-clutter contrast and accurate estimates of distribution models, making it difficult to adapt to complex and variable sea surface backgrounds. Considering the limitations of the CFAR methods, this paper proposes a new detector for ship detection based on matrix information geometry (MIG) theory. The proposed detector models each SAR sample data as a Hermitian positive-definite (HPD) matrix, and uses the geometric mean of all HPD matrices corresponding to the background window to estimate the clutter, thus transforming the target detection problem into a measure of the Kullback-Leibler divergence (KLD) between two points on the matrix manifold. Different from the traditional CFAR detectors which use intensity information to differentiate between targets and clutter, the proposed detector utilizes the intrinsic geometry of the matrix manifold and employs KLD with good differentiation to measure the variance from targets to clutter.

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.

Derivative principles for invariant ensembles

In the present work we show that the joint probability distribution of the eigenvalues can be expressed in terms of a differential operator acting on the distribution of some other matrix quantities. Those quantities might be the diagonal or pseudo-diagonal entries as it is the case for Hermitian matrices. These representations are called derivative principles. We show them for the additive spaces of the Hermitian, Hermitian antisymmetric, Hermitian anti-self-dual, and complex rectangular matrices as well as for the two multiplicative matrix spaces of the positive definite Hermitian matrices and of the unitary matrices. In all six cases we prove the uniqueness of the derivative principles.

Joint Constellation Design for Noncoherent MIMO Multiple-Access Channels

We consider the joint constellation design problem for the noncoherent multiple-input multiple-output multiple-access channel (MAC). By analyzing the noncoherent maximum-likelihood detection error, we propose novel design criteria so as to minimize the error probability. As a baseline approach, we adapt several existing design criteria for the point-to-point channel to the MAC. Furthermore, we propose new design criteria. Our first proposed design metric is the dominating term in nonasymptotic lower and upper bounds on the pairwise error probability exponent. We give a geometric interpretation of the bound using Riemannian distance in the manifold of Hermitian positive definite matrices. From an analysis of this metric at high signal-to-noise ratio, we obtain further simplified metrics. For any given set of constellation sizes, the proposed metrics can be optimized over the set of constellation symbols. Motivated by the simplified metric, we propose a simple constellation construction consisting in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">partitioning</i> a single-user constellation. We also provide a generalization of our previously proposed construction based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">precoding</i> individual constellations of lower dimensions. For a fixed joint constellation, the design metrics can be further optimized over the per-user transmit power, especially when the users transmit at different rates. Considering unitary space-time modulation, we investigate the option of building each individual constellation as a set of truncated unitary matrices scaled by the respective transmit power. Numerical results show that our proposed metrics are meaningful, and can be used as objectives to generate constellations through numerical optimization that perform better, for the same transmission rate and power constraint, than a common pilot-based scheme and the constellations optimized with existing metrics.

Bures–Wasserstein geometry for positive-definite Hermitian matrices and their trace-one subset

In his classical argument, Rao derives the Riemannian distance corresponding to the Fisher metric using a mapping between the space of positive measures and Euclidean space. He obtains the Hellinger distance on the full space of measures and the Fisher distance on the subset of probability measures. In order to highlight the interplay between Fisher theory and quantum information theory, we extend this construction to the space of positive-definite Hermitian matrices using Riemannian submersions and quotient manifolds. The analog of the Hellinger distance turns out to be the Bures–Wasserstein (BW) distance, a distance measure appearing in optimal transport, quantum information, and optimisation theory. First we present an existing derivation of the Riemannian metric and geodesics associated with this distance. Subsequently, we present a novel derivation of the Riemannian distance and geodesics for this metric on the subset of trace-one matrices, analogous to the Fisher distance for probability measures.

The tan Θ theorem for definite matrix pairs

In this paper, we consider the perturbation of a Hermitian matrix pair , where H and M are non-singular and positive definite Hermitian matrices, respectively. A novel upper bound on a tangent of the angles between the eigenspaces of perturbed and unperturbed pairs is derived under a perturbation to the off-diagonal blocks of H. The rotation of the eigenspaces under a perturbation is measured in the matrix-dependent scalar product. We show that a bound for the standard eigenvalue problem is a special case of our new bound and that the obtained bound can be much sharper than the existing bounds.

Towards a median signal detector through the total Bregman divergence and its robustness analysis

A novel family of geometric signal detectors are proposed through medians of the total Bregman divergence (TBD), which are shown advantageous over the conventional methods and their mean counterparts. By interpreting the observation data as Hermitian positive-definite matrices, their mean or median play an essential role in signal detection. As is difficult to be solved analytically, we propose numerical solutions through Riemannian gradient descent algorithms or fixed-point algorithms.Beside detection performance, robustness of a detector to outliers is also of vital importance, which can often be analyzed via the influence functions. Introducing an orthogonal basis for Hermitian matrices, we are able to compute the corresponding influence functions analytically and exactly by solving a linear system, which is transformed from the governing matrix equation. Numerical simulations show that the TBD medians are more robust than their mean counterparts.

Unsupervised Learning Discriminative MIG Detectors in Nonhomogeneous Clutter

Principal component analysis (PCA) is a commonly used pattern analysis method that maps high-dimensional data into a lower-dimensional space maximizing the data variance, that results in the promotion of separability of data. Inspired by the principle of PCA, a novel type of learning discriminative matrix information geometry (MIG) detectors in the unsupervised scenario are developed, and applied to signal detection in nonhomogeneous environments. Hermitian positive-definite (HPD) matrices can be used to model the sample data, while the clutter covariance matrix is estimated by the geometric mean of a set of secondary HPD matrices. We define a projection that maps the HPD matrices in a high-dimensional manifold to a low-dimensional and more discriminative one to increase the degree of separation of HPD matrices by maximizing the data variance. Learning a mapping can be formulated as a two-step mini-max optimization problem in Riemannian manifolds, which can be solved by the Riemannian gradient descent algorithm. Three discriminative MIG detectors are illustrated with respect to different geometric measures, i.e., the Log-Euclidean metric, the Jensen–Bregman LogDet divergence and the symmetrized Kullback–Leibler divergence. Simulation results show that performance improvements of the novel MIG detectors can be achieved compared with the conventional detectors and their state-of-the-art counterparts within nonhomogeneous environments.

Time-varying spectral matrix estimation via intrinsic wavelet regression for surfaces of Hermitian positive definite matrices

Intrinsic wavelet transforms and denoising methods are introduced for the purpose of time-varying Fourier spectral matrix estimation. A non-degenerate time-varying spectral matrix constitutes a surface of Hermitian positive definite matrices across time and frequency and any spectral matrix estimator ideally adheres to these geometric constraints. Spectral matrix estimation of a locally stationary time series by means of linear or nonlinear wavelet shrinkage naturally respects positive definiteness at each time-frequency point, without any postprocessing. Moreover, the spectral matrix estimator enjoys equivariance in the sense that it does not nontrivially depend on the chosen basis or coordinate system of the multivariate time series. The algorithmic construction is based on a second-generation average-interpolating wavelet transform in the space of Hermitian positive definite matrices equipped with an affine-invariant metric. The wavelet coefficient decay and linear wavelet thresholding convergence rates of intrinsically smooth surfaces of Hermitian positive definite matrices are derived. Furthermore, practical nonlinear thresholding based on the trace of the matrix-valued wavelet coefficients is investigated. Finally, the time-varying spectral matrix of a nonstationary multivariate electroencephalography (EEG) time series recorded during an epileptic brain seizure is estimated.

Quantum Linear System Solver Based on Time-optimal Adiabatic Quantum Computing and Quantum Approximate Optimization Algorithm

We demonstrate that with an optimally tuned scheduling function, adiabatic quantum computing (AQC) can readily solve a quantum linear system problem (QLSP) with O (κ poly(log (κ ε))) runtime, where κ is the condition number, and ε is the target accuracy. This is near optimal with respect to both κ and ε, and is achieved without relying on complicated amplitude amplification procedures that are difficult to implement. Our method is applicable to general non-Hermitian matrices, and the cost as well as the number of qubits can be reduced when restricted to Hermitian matrices, and further to Hermitian positive definite matrices. The success of the time-optimal AQC implies that the quantum approximate optimization algorithm (QAOA) with an optimal control protocol can also achieve the same complexity in terms of the runtime. Numerical results indicate that QAOA can yield the lowest runtime compared to the time-optimal AQC, vanilla AQC, and the recently proposed randomization method.

Solving system of nonlinear matrix equations over Hermitian positive definite matrices

In this article, we first introduce the necessary and sufficient conditions for the existence of the Hermitian positive definite solution group of a system of nonlinear matrix equations Then we propose an inversion-free iteration method to find the Hermitian positive definite solution group of the system of nonlinear matrix equations. Finally, the accuracy and effectiveness of the proposed method in compare to some existing algorithms are demonstrated by various numerical examples.

Pipelined Preconditioned Conjugate Gradient Methods for real and complex linear systems for distributed memory architectures

Preconditioned Conjugate Gradient (PCG) is a popular method for solving large and sparse linear systems of equations. The performance of PCG at scale is affected due to the costly global synchronization steps that arise in dot-products on distributed memory systems. Pipelined PCG (PIPECG) removes the costly global synchronization steps from PCG by only executing a single non-blocking allreduce per iteration and overlapping it with independent computations.In our previous work, we have developed a novel pipelined PCG algorithm called PIPECG-OATI (One Allreduce per Two Iterations) for real linear systems which executes a single non-blocking allreduce per two iterations and provides a large overlap of global communication with independent computations at higher number of cores. Our method achieves this overlap by using iteration combination and by introducing new recurrence and non-recurrence computations. We implement optimizations in the PIPECG-OATI method to use cache memory efficiently.In this work, we present PIPECG-OATI-c method for linear systems with complex Hermitian positive definite and complex symmetric matrices. We compare our method with various pipelined CG methods on a variety of problems and demonstrate that our method always gives the least run times. We performed experiments with our method using 20M and 30M unknowns on up to 16K cores and obtained up to 2.48X performance improvement over PCG and 2.14X performance improvement over PIPECG methods. We also experimented with up to 1-billion unknowns on 16K cores, the largest problem size explored for the CG problem, to our knowledge, and obtained about 25% improvement over PCG.

Parallelization of Hermitian positive definite systems of equations: A hierarchical Jacobi approach

We revisit the block-Jacobi iterative solver and preconditioner for Hermitian positive definite systems of equations. In order to force convergence of the iterative scheme, the diagonal blocks are nested hierarchically in a binary fashion. The resulting parallel technique, the hierarchical binary Jacobi (hbJ), fits within the nested matrix splitting framework. The convergence criteria of the hbJ are provided and its computational complexity is studied. Comparison theorems based on the levels of the nested iteration are given. A computational study for its use as iterative solver and preconditioner is performed within the class of Wishart random generated Hermitian positive definite matrices. Counterexamples are provided for a selection of desired comparison results.

From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues

Williamson's theorem states that every real symmetric positive definite matrix $A$ of even order can be brought to diagonal form via a symplectic $T$-congruence transformation. The diagonal entries of the resulting diagonal form are called the symplectic eigenvalues of $A$. We point at an analog of this classical result related to Hermitian positive definite matrices, *-congruences, and another class of transformation matrices, namely, pseudo-unitary matrices. This leads to the concept of pseudo-unitary (or pseudo-orthogonal, in the real case) eigenvalues of positive definite matrices.

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.

A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation

The goal of this article is to investigate a new solver in the form of an iterative method to solve X+A∗X−1A=I as an important nonlinear matrix equation (NME), where A,X,I are appropriate matrices. The minimal and maximal solutions of this NME are discussed as Hermitian positive definite (HPD) matrices. The convergence of the scheme is given. Several numerical tests are also provided to support the theoretical discussions.