Definite Matrix Research Articles

Matrix decompositions in quantum optics: Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson

In this tutorial, we summarize four important matrix decompositions commonly used in quantum optics, namely the Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson decompositions. The first two of these decompositions are specialized versions of the singular-value decomposition when applied to symmetric or symplectic matrices. The third factors any symplectic matrix in a unique way in terms of matrices that belong to different subgroups of the symplectic group. The last one instead gives the symplectic diagonalization of real, positive definite matrices of even size. While proofs of the existence of these decompositions exist in the literature, we review explicit constructions to implement these decompositions using standard linear algebra packages and functionalities such as singular-value, polar, Schur, and QR decompositions, and matrix square roots and inverses.

On the matrix Heinz means and Rényi divergences

Bhatia, Lim and Yamazaki obtained results on the norm minimality of the Kubo-Ando Heinz means of positive semidefinite matrices. Namely, they proved that | | | A ♯ α B + A ♯ 1 − α B | | | ≤ | | | A α B 1 − α + A 1 − α B α | | | , for α ∈ ( 0 , 1 ) , in the case of the Schatten 1-norm or 2-norm. In this article, we refine these results via the matrix function in the definition of the Rényi divergence. We show that four different types of inequalities hold, under certain conditions, for Schatten 1- and 2-norms.

Order relations of the Wasserstein mean and the spectral geometric mean

On the space of positive definite matrices, several operator means are popular and have been studied extensively. In this paper, we investigate the near order and the Löwner order relations on the curves defined by the Wasserstein mean and the spectral geometric mean. We show that the near order $\preceq $ is stronger than the eigenvalue entrywise order and that $A\natural_t B \preceq A\diamond_t B$ for $t\in [0,1]$. We prove the monotonicity properties of the curves originated from the Wasserstein mean and the spectral geometric mean in terms of the near order. The Löwner order properties of the Wasserstein mean and the spectral geometric mean are also explored.

Riemannian Geodesic Discriminant Analysis–Minimum Riemannian Mean Distance: A Robust and Effective Method Leveraging a Symmetric Positive Definite Manifold and Discriminant Algorithm for Image Set Classification

This study introduces a novel method for classifying sets of images, called Riemannian geodesic discriminant analysis–minimum Riemannian mean distance (RGDA-MRMD). This method first converts image data into symmetric positive definite (SPD) matrices, which capture important features related to the variability within the data. These SPD matrices are then mapped onto simpler, flat spaces (tangent spaces) using a mathematical tool called the logarithm operator, which helps to reduce their complexity and dimensionality. Subsequently, regularized local Fisher discriminant analysis (RLFDA) is employed to refine these simplified data points on the tangent plane, focusing on local data structures to optimize the distances between the points and prevent overfitting. The optimized points are then transformed back into a complex, curved space (SPD manifold) using the exponential operator to enhance robustness. Finally, classification is performed using the minimum Riemannian mean distance (MRMD) algorithm, which assigns each data point to the class with the closest mean in the Riemannian space. Through experiments on the ETH-80 (Eidgenössische Technische Hochschule Zürich-80 object category), AFEW (acted facial expressions in the wild), and FPHA (first-person hand action) datasets, the proposed method demonstrates superior performance, with accuracy scores of 97.50%, 37.27%, and 88.47%, respectively. It outperforms all the comparison methods, effectively preserving the unique topological structure of the SPD matrices and significantly boosting image set classification accuracy.

A Novel Weak-Form Plane Frame Quadrature Element and its Application to Bridge Analysis

A novel weak-form plane frame quadrature element (WPFQE) is developed. A complex structure can be divided into several non-homogeneous (or homogeneous) elements with large size. The Chebyshev–Lobatto differential quadrature dealing with differentiation and the Gauss–Lobatto quadrature dealing with integration have the same integration points including the ends of each element. Based on the energy variational principle, the diagonal mass matrix and positive definite real symmetric stiffness matrix for the element can be obtained. The unknown quantities at the internal quadrature points of the element can be represented by those at the endpoints of the element based on the principle of static equal effect anterior to the development of global matrices, which significantly reduces the sizes of modified element matrices while maintaining sufficient accuracy, thereby obtaining the small-sized global matrices. Assembly of elements can be performed efficiently just as that in the finite element (FE) method. In the case study, the static and dynamic performances of a three-span high-pier rigid-frame bridge with variable cross-section are studied by applying the proposed method. The results indicate that the orders of structural matrices and calculation time obtained by the WPFQE method are much smaller than those of the FE method.

Kernel Angle Dependence Measures in Metric Spaces

Measuring and testing dependence between data in separable metric spaces is of great importance in modern statistics. Most existing work relied on the distance between random variables, which inevitably required the moment conditions to guarantee the distance is well-defined. Based on the geometry element “angle”, we develop a novel class of nonlinear dependence measures for data in metric space that can avoid such conditions. Specifically, by making use of the reproducing kernel Hilbert space equipped with Gaussian measure, we introduce kernel angle covariances that can be applied to various types of data, including low dimensional vector, high dimensional vectors, non-Euclidean data like symmetric positive definite matrices, and compositional data. We estimate kernel angle covariances based on U-statistic and establish the corresponding independence tests via gamma approximation. Our kernel angle independence tests, imposing no-moment conditions on kernels, are robust with heavy-tailed random variables. We conduct comprehensive simulation studies and apply our proposed methods to a facial recognition task. Our kernel angle covariances-based tests show remarkable performances in dealing with image data. All the codes and proofs are included in the supplementary materials.

SPD Manifold Deep Metric Learning for Image Set Classification.

By characterizing each image set as a nonsingular covariance matrix on the symmetric positive definite (SPD) manifold, the approaches of visual content classification with image sets have made impressive progress. However, the key challenge of unhelpfully large intraclass variability and interclass similarity of representations remains open to date. Although, several recent studies have mitigated the two problems by jointly learning the embedding mapping and the similarity metric on the original SPD manifold, their inherent shallow and linear feature transformation mechanism are not powerful enough to capture useful geometric features, especially in complex scenarios. To this end, this article explores a novel approach, termed SPD manifold deep metric learning (SMDML), for image set classification. Specifically, SMDML first selects a prevailing SPD manifold neural network (SPDNet) as the backbone (encoder) to derive an SPD matrix nonlinear representation. To counteract the degradation of structural information during multistage feature embedding, we construct a Riemannian decoder at the end of the encoder, trained by a reconstruction error term (RT), to induce the generated low-dimensional feature manifold of the hidden layer to capture the pivotal information about the visual data describing the imaged scene. We demonstrate through theory and experiments that it is feasible to replace the Riemannian metric with Euclidean distance in RT. Then, the ReCov layer is introduced into the established Riemannian network to regularize the local statistical information within each input feature matrix, which enhances the effectiveness of the learning process. The theoretical analysis of the activation function used in the ReCov layer in terms of continuity and conditions for generating positive definite matrices is beneficial for network design. Inspired by the fact that the single cross-entropy loss used for training is unable to effectively parse the geometric distribution of the deep representations, we finally endow the suggested model with a novel metric learning regularization term. By explicitly incorporating the encoding and processing of the data variations into the network learning process, this term can not only derive a powerful Riemannian representation but also train an effective classifier. The experimental results show the superiority of the proposed approach on three typical visual classification tasks.

Robust speckle covariance matrix estimation of sea clutter based on spectral symmetry

Speckle covariance matrix estimation plays an important role in adaptive target detection of maritime radars. Spatial heterogeneity of sea clutter incurs insufficient secondary data in estimation, which degrades detection performance. It is an effective way of estimation improvement to exploit the Doppler spectrum knowledge of sea clutter for structural dimension reduction of the speckle covariance matrix. This paper proposes a speckle covariance matrix estimation method based on the spectral symmetry knowledge of sea clutter, indicating that the spectrum of sea clutter is symmetric with respect to the Doppler offset. First of all, the spectral symmetry of sea clutter is analyzed by using the measured data from several opened databases and the results show that most of the data excluding sea spikes have symmetric Doppler spectra. Secondly, the spectral symmetry constrains the speckle covariance matrix into the Hadamard product of a special rank-one Toeplitz matrix and a real positive definite symmetric matrix. The degrees of freedom are reduced to almost half of that of a Hermitian matrix and thus fewer secondary data are required. Thirdly, a robust iterative maximum likelihood estimator is proposed to estimate the speckle covariance matrix with structural constraint and is free of the clutter texture distribution. The performance of the estimator is verified by the measured data and the results show that it is superior to traditional estimators, particularly in fewer secondary data.

Quantifying macrostructures in viscoelastic sub-diffusive flows

We present a theory to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia) for viscoelastic sub-diffusive flows, by introducing a mathematically consistent decomposition of the polymer conformation tensor, into the so-called structure tensor. Our approach bypasses an inherent problem in the standard arithmetic decomposition, namely, the fluctuating conformation tensor fields may not be positive definite and hence, do not retain their physical meaning. Using well-established results in matrix analysis, the space of positive definite matrices is transformed into a Riemannian manifold by defining and constructing a geodesic via the inner product on its tangent space. This geodesic is utilized to define three scalar invariants of the structure tensor, which do not suffer from the caveats of the regular invariants (such as trace and determinant) of the polymer conformation tensor. First, we consider the problem of formulating perturbative expansions of the structure tensor using the geodesic, which is consistent with the Riemannian manifold geometry. A constraint on the maximum time, during which the evolution of the perturbative solution can be well approximated by linear theory along the Euclidean manifold, is found. A comparison between the linear and the nonlinear dynamics, identifies the role of nonlinearities in initiating the symmetry breaking of the flow variables about the centerline. Finally, fully nonlinear simulations of the viscoelastic sub-diffusive channel flows, underscore the advantage of using these invariants in effectively quantifying the macrostructures.

On the stability of θ-methods for DDEs and PDDEs

In this paper, the stability of θ-methods for delay differential equations is studied based on the test equation y′(t)=−Ay(t)+By(t−τ), where τ is a constant delay and A is a positive definite matrix. It is mainly considered the case where the matrices A and B are not simultaneosly diagonalizable and the concept of field of values is used to prove a sufficient condition for unconditional stability of these methods and another condition which also guarantees their stability, but according to the step size. The results obtained are also simplified for the case where the matrices A and B are simultaneously diagonalizable and compared with other similar works for the general case. Several numerical examples in which the theory discussed here is applied to parabolic problems given by partial delay differential equations with a diffusion term and a delayed term are presented, too.

Disjoint sections of positive semidefinite matrices and their applications in linear statistical models

Abstract Given matrices A A and B B of the same order, A A is called a section of B B if R ( A ) ∩ R ( B − A ) = { 0 } {\mathscr{R}}\left(A)\cap {\mathscr{R}}\left(B-A)=\left\{0\right\} and R ( A T ) ∩ R ( ( B − A ) T ) = { 0 } {\mathscr{R}}\left({A}^{T})\cap {\mathscr{R}}\left({\left(B-A)}^{T})=\left\{0\right\} , where R ( . ) {\mathscr{R}}\left(.) denotes the range (column space) of a matrix argument and the superscript T T stands for the transpose of a matrix. A matrix G G is called positive semidefinite (p.s.d.) if x T G x {x}^{T}Gx is nonnegative for all real vector x x . However, we refer to a symmetric p.s.d. matrix simply as a p.s.d. matrix as the applications discussed in this article are concerned with only symmetric p.s.d. matrices. An n × n n\times n p.s.d. matrix G G admits of a symmetric section G X {G}_{X} of G G with regard to an n × k n\times k matrix X X such that R ( G X ) = R ( G ) ∩ R ( X ) {\mathscr{R}}\left({G}_{X})={\mathscr{R}}\left(G)\cap {\mathscr{R}}\left(X) . In this article, sections of the type G X {G}_{X} are used in minimization of quadratic functions under linear constraints and splitting of vector random variables into uncorrelated vector random variables. In the general Gauss-Markoff model, y = X β + ε y=X\beta +\varepsilon , with design matrix X X and a singular covariance matrix σ 2 G {\sigma }^{2}G of ε \varepsilon , y y is decomposed into four uncorrelated vector random variables as y = M 1 y + M 2 y + M 3 y + M 4 y y={M}_{1}y+{M}_{2}y+{M}_{3}y+{M}_{4}y , where M i , i = 1 , 2 , 3 {M}_{i},i=1,2,3 consist of sections of G G and X X T X{X}^{T} , and M 4 {M}_{4} is a matrix whose row space is the null space of G + X X T G+X{X}^{T} .

On the impact of spatial covariance matrix ordering on tile low‐rank estimation of Matérn parameters

AbstractSpatial statistical modeling involves processing an symmetric positive definite covariance matrix, where denotes the number of locations. However, when is large, processing this covariance matrix using traditional methods becomes prohibitive. Thus, coupling parallel processing with approximation can be an elegant solution by relying on parallel solvers that deal with the matrix as a set of small tiles instead of the full structure. The approximation can also be performed at the tile level for better compression and faster execution. The tile low‐rank (TLR) approximation has recently been used to compress the covariance matrix, which mainly relies on ordering the matrix elements, which can impact the compression quality and the efficiency of the underlying solvers. This work investigates the accuracy and performance of location‐based ordering algorithms. We highlight the pros and cons of each ordering algorithm and give practitioners hints on carefully choosing the ordering algorithm for TLR approximation. We assess the quality of the compression and the accuracy of the statistical parameter estimates of the Matérn covariance function using TLR approximation under various ordering algorithms and settings of correlations through simulations on irregular grids. Our conclusions are supported by an application to daily soil moisture data in the Mississippi Basin area.

Differentiable microstructures design via anisotropic thermal diffusion

We propose a novel method to design differentiable microstructures. Central to our algorithm is a new representation of the mapping from the parameters to microstructures, formulated as the anisotropic thermal diffusion. A metric field governs the anisotropic diffusion. The metric associated with each point is represented as a 2 × 2 symmetric positive definite matrix that becomes the design variable. To alleviate the difficulties caused by symmetric positive definite constraints, we perform the singular value decomposition of the metric matrix so that the design variable includes a rotation angle and a diagonal matrix. Then, the positive definiteness is converted to requiring the two diagonal entries of the diagonal matrix to be positive, which is easier to deal with. The effectiveness of our algorithm is demonstrated through evaluations and comparisons over various examples.

A unified formulation of geometry-aware discrete dynamic movement primitives

Learning from demonstration (LfD) is considered as an efficient way to transfer skills from humans to robots. Traditionally, LfD has been used to transfer Cartesian and joint positions and forces from human demonstrations. The traditional approach works well for some robotic tasks, but for many tasks of interest, it is necessary to learn skills such as orientation, impedance, and/or manipulability that have specific geometric characteristics. An effective encoding of such skills can be only achieved if the underlying geometric structure of the skill manifold is considered and the constrains arising from this structure are fulfilled during both learning and execution. However, typical learned skill models such as dynamic movement primitives (DMPs) are limited to Euclidean data and fail in correctly embedding quantities with geometric constraints. In this paper, we propose a novel and mathematically principled framework that uses concepts from Riemannian geometry to allow DMPs to properly embed geometric constrains. The resulting DMP formulation can deal with data sampled from any Riemannian manifold including, but not limited to, unit quaternions and symmetric and positive definite matrices. The proposed approach has been extensively evaluated both on simulated data and real robot experiments. The performed evaluation demonstrates that beneficial properties of DMPs, such as convergence to a given goal and the possibility to change the goal during operation, apply also to the proposed formulation.

Energy and reserve procurement in integrated electricity and heating system: A high-dimensional covariance matrix approach based on stochastic differential equations

The integration of stochastic renewable energy sources into energy systems presents challenges due to flexibility insufficiency and uncertainty modeling inaccuracy. To address these challenges, this paper explores a high-dimensional covariance matrix approach based on stochastic differential equations (SDEs) for energy and reserve co-dispatch in integrated electricity and heating systems. By capturing the long-term correlation of wind power forecast errors, the SDE model simultaneously models point forecasts and high-dimensional positive-definite covariance matrices, resulting in a compact uncertainty set. To ensure computational efficiency, we employ duality theory to transform the stochastic co-dispatch problem with the high-dimensional uncertainty set into deterministic convex programming. Additionally, a virtual heat storage model is introduced for the district heating network, leveraging the thermal inertia of the network to enhance reserve capacity. Simulations validate the calibration and sharpness of the proposed high-dimensional uncertainty set. The results demonstrate that co-dispatch using the SDE-based uncertainty set reduces operational costs by 4.9% compared to a set that does not consider the long-term correlation of wind power, under a 95% coverage rate. Moreover, the storage capabilities of district heating pipelines and smart buildings provide cost-free reserves, saving the operational costs of the two test systems by 2.8% and 1.3%, respectively.

Further properties of accretive matrices

To better understand the algebra \(\mathcal{M}_n\) of all \(n\times n\) complex matrices, we explore the class of accretive matrices. This class has received renowned attention in recent years due to its role in complementing those results known for positive definite matrices. Among many results, we present order-preserving results, Choi–Davis-type inequalities, mean-convex inequalities, sub-multiplicative results for the real part, and new bounds of the absolute value of accretive matrices. These results will be compared with the existing literature. In the end, we quickly pass through related entropy results for accretive matrices.

On the convergence of orthogonalization-free conjugate gradient method for extreme eigenvalues of Hermitian matrices: A Riemannian optimization interpretation

In many applications, it is desired to obtain extreme eigenvalues and eigenvectors of large Hermitian matrices by efficient and compact algorithms. In particular, orthogonalization-free methods are preferred for large-scale problems for finding eigenspaces of extreme eigenvalues without explicitly computing orthogonal vectors in each iteration. For the top p eigenvalues, the simplest orthogonalization-free method is to find the best rank-p approximation to a positive semi-definite Hermitian matrix by algorithms solving the unconstrained Burer–Monteiro formulation. We show that the nonlinear conjugate gradient method for the unconstrained Burer–Monteiro formulation is equivalent to a Riemannian conjugate gradient method on a quotient manifold with the Bures–Wasserstein metric, thus its global convergence to a stationary point can be proven. Numerical tests suggest that it is efficient for computing the largest k eigenvalues for large-scale matrices if the largest k eigenvalues are nearly distributed uniformly.

Matrix Pencil Optimal Iterative Algorithms and Restarted Versions for Linear Matrix Equation and Pseudoinverse

We derive a double-optimal iterative algorithm (DOIA) in an m-degree matrix pencil Krylov subspace to solve a rectangular linear matrix equation. Expressing the iterative solution in a matrix pencil and using two optimization techniques, we determine the expansion coefficients explicitly, by inverting an m×m positive definite matrix. The DOIA is a fast, convergent, iterative algorithm. Some properties and the estimation of residual error of the DOIA are given to prove the absolute convergence. Numerical tests demonstrate the usefulness of the double-optimal solution (DOS) and DOIA in solving square or nonsquare linear matrix equations and in inverting nonsingular square matrices. To speed up the convergence, a restarted technique with frequency m is proposed, namely, DOIA(m); it outperforms the DOIA. The pseudoinverse of a rectangular matrix can be sought using the DOIA and DOIA(m). The Moore–Penrose iterative algorithm (MPIA) and MPIA(m) based on the polynomial-type matrix pencil and the optimized hyperpower iterative algorithm OHPIA(m) are developed. They are efficient and accurate iterative methods for finding the pseudoinverse, especially the MPIA(m) and OHPIA(m).

Permanents of block matrices

In this paper, we provide a formula to compute the permanent of block matrices depending on entries of each block. As a consequence, a generalized Lieb permanent inequality on positive semi-definite block matrices is given.

Continuous Time Algebraic Riccati Equation Solution for Second Order Systems

Abstract The continuous time algebraic Riccati equation (ARE) is often utilized in control, estimation, and optimization. For a linear system with a second order structure, the ARE required to be solved to get the control values in standard control problems results in complex sub-equations in terms of the second order system matrices. The computational costs of solving the algebraic Riccati equation through standard methods such as the Hamiltonian matrix pencil approach increase substantially as matrix sizes increase for a second order system, due to the eigendecomposition of the 2n x 2n system matrices involved. This work introduces a new solution that does not require the eigendecomposition of the 2n x 2n system matrices, while satisfying all of the requirements of the solution to the Riccati equation (e.g., detectability, stabilizability, positive semi-definite solution matrix).