Asymmetric Matrices Research Articles

On the undersampling strategies to accelerate time‐resolved 3D imaging using k‐t‐GRAPPA

The purpose of this study was to explore how to optimally undersample and reconstruct time-resolved 3D data using a k-t-space-based GRAPPA technique. The performance of different reconstruction strategies was evaluated using data sets with different ratios of phase (N(y)) and partition (N(z)) encoding lines (N(y) × N(z) = 64-128 × 40-64) acquired in a moving phantom. Image reconstruction was performed for different kernel configurations and different reduction factors (R = 5, 6, 8, and 10) and was evaluated using regional error quantification and SNR analysis. To analyze the temporal fidelity of the different kernel configurations in vivo, time-resolved 3D phase contrast data were acquired in the thoracic aorta of two healthy volunteers. Results demonstrated that kernel configurations with a small kernel extension yielded superior results especially for more asymmetric data matrices as typically used in clinical applications. The application of k-t-GRAPPA to in vivo data demonstrated the feasibility of undersampling of time-resolved 3D phase contrast data set with a nominal reduction factors of up to R(net) = 8, while maintaining the temporal fidelity of the measured velocity field. Extended GRAPPA-based parallel imaging with optimized multidimensional reconstruction kernels has the potential to substantially accelerate data acquisitions in time-resolved 3D MRI.

Random matrix theory of unquenched two-colour QCD with nonzero chemical potential

We solve a random two-matrix model with two real asymmetric matrices whose primary purpose is to describe certain aspects of quantum chromodynamics with two colours and dynamical fermions at nonzero quark chemical potential μ. In this symmetry class the determinant of the Dirac operator is real but not necessarily positive. Despite this sign problem the unquenched matrix model remains completely solvable and provides detailed predictions for the Dirac operator spectrum in two different physical scenarios/limits: (i) the e-regime of chiral perturbation theory at small μ, where μ2 multiplied by the volume remains fixed in the infinite-volume limit and (ii) the high-density regime where a BCS gap is formed and μ is unscaled. We give explicit examples for the complex, real, and imaginary eigenvalue densities including Nf = 2 non-degenerate flavours. Whilst the limit of two degenerate masses has no sign problem and can be tested with standard lattice techniques, we analyse the severity of the sign problem for non-degenerate masses as a function of the mass split and of μ.

Estimating reproduction numbers for adults and children from case data

We present a method for estimating reproduction numbers for adults and children from daily onset data, using pandemic influenza A(H1N1) data as a case study. We investigate the impact of different underlying transmission assumptions on our estimates, and identify that asymmetric reproduction matrices are often appropriate. Under-reporting of cases can bias estimates of the reproduction numbers if reporting rates are not equal across the two age groups. However, we demonstrate that the estimate of the higher reproduction number is robust to disproportionate data-thinning. Applying the method to 2009 pandemic influenza H1N1 data from Japan, we demonstrate that the reproduction number for children was considerably higher than that of adults, and that our estimates are insensitive to our choice of reproduction matrix.

Distributed Change Detection Based on a Consensus Algorithm

In this paper a novel distributed recursive algorithm is proposed for real time change detection using sensor networks. The algorithm is based on a combination of geometric moving average control charts generating local statistics and a global consensus strategy; it does not require any fusion center, so that the final decision is made by testing the state of any node in the network with respect to a given common threshold. The mean-square error with respect to the centralized solution defined by a weighted sum of the local statistics is analyzed in the case of constant asymmetric consensus matrices with constant and time varying forgetting factors in the underlying recursions, assuming spatially and temporally correlated data. These results are consistently extended to the case of time varying random consensus matrices, encompassing asymmetric gossip schemes, lossy networks and intermittent measurements, proving that the algorithm can be an efficient tool for practice. The given simulation results illustrate the main characteristics of the proposed algorithm, including the consensus matrix design, the mean square error with respect to the centralized solution as a function of the forgetting factor, the obtained detection quality expressed using deflection and estimation of the instant of parameter change.

Asymmetric random matrices: What do we need them for?

Complex systems are typically represented by large ensembles of observations. Correlation matrices provide an efficient formal framework to extract information from such multivariate ensembles and identify in a quantifiable way patterns of activity that are reproducible with statistically significant frequency compared to a reference chance probability, usually provided by random matrices as fundamental reference. The character of the problem and especially the symmetries involved must guide the choice of random matrices to be used for the definition of a baseline reference. For standard correlation matrices this is the Wishart ensemble of symmetric random matrices. The real world complexity however often shows asymmetric information flows and therefore more general correlation matrices are required to adequately capture the asymmetry. Here we first summarize the relevant theoretical concepts. We then present some examples of human brain activity where asymmetric time-lagged correlations are evident and hence highlight the need for further theoretical developments.

Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices

We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for the SOP and for their Cauchy transforms, given as the expectation values of traces and determinants or their inverses, respectively. Our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product. Examples for the SOP are given in terms of Laguerre polynomials for the chiral ensemble (also called the non-Hermitian real Wishart–Laguerre ensemble), both without and with the insertion of characteristic polynomials. Such characteristic polynomials play the role of mass terms in applications to complex Dirac spectra in field theory. In addition, for the elliptic real Ginibre ensemble, we recover the SOP of Forrester and Nagao in terms of Hermite polynomials.

Limiting spectral distribution of XX' matrices

Une des méthodes pour obtenir la limite des distributions spectrales (LSD) des grandes matrices aléatoires est la fameuse méthode des moments, basée sur la formule des traces. Son succès a été clairement établi pour différents types de matrices telles que les matrices de Wigner et les matrices de covariance. Dans un article récent, Bryc, Dembo et Jiang [Ann. Probab. 34 (2006) 1–38] ont obtenu la LSD pour des matrices de Toeplitz et de Hankel en utilisant cette méthode. Ils arrivent à estimer les traces des moments de telles matrices en séparant les différents termes par classes d’équivalence et en reliant les asymptotiques des dénombrements afférents avec les calculs de certains volumes. Bose et Sen [Electron. J. Probab. 13 (2008) 588–628] ont développé cette idée et ont donné un cadre général pour traiter de matrices symmétriques dont les entrées viennent d’une suite indépendante. Dans cet article, nous généralisons cette approche pour considérer des matrices de la form $A_{p}=\frac{1}{n}XX'$ où X est une matrice p×n avec des entrées réelles. Nous démontrons un résultat général d’existence de la LSD de telles matrices, correctement recentrées et rééchelonnées, quand p et n tendent vers l’infini de telle façon que p/n tende vers y∈(0, ∞). Par exemple, nous montrons l’existence de la LSD quand X est la matrice asymétrique de Hankel, de Toeplitz, circulante ou circulante inverse. En particulier, quand y=0, les limites correspondent à celles obtenues par Bryc, Dembo et Jiang [Ann. Probab. 34 (2006) 1–38]. Sinon, nous obtenons de nouvelles lois limites pour lesquelles aucune expression explicite n’est connue. Nous étudions ces lois par quelques simulations.

Averages over Ginibre's Ensemble of Random Real Matrices

AbstractWe give a method for computing the ensemble average of multiplicative class func-tions over the Gaussian ensemble of real asymmetric matrices. These averages areexpressed in terms of the Pfaffian of Gram-like antisymmetric matrices formed withrespect to a skew-symmetric inner product related to the class function. 1 Introduction In the 1960’s Ginibre introduced three statistical ensembles of matrices whose entries arechosen independently with Gaussian probability from (resp.) R, C, and Hamilton’s Quater-nions [4]. These ensembles are respectively labeled GinOE, GinUE and GinSE in analogywith their Hermitian counterparts. Ginibre introduced a physical analogy between GinUEand two-dimensional electrostatics, but gave no applications for the other two ensembles.Since their introduction, many applications have been found for GinOE and GinSE.Here we report a method for determining ensemble averages over GinOE of certainfunctions which are constant on similarity (conjugacy) classes. GinOE is the space of N×Nreal matrices R

The chiral Gaussian two-matrix ensemble of real asymmetric matrices

We solve a family of Gaussian two-matrix models with rectangular N × (N + ν) matrices, having real asymmetric matrix elements and depending on a non-Hermiticity parameter μ. Our model can be thought of as the chiral extension of the real Ginibre ensemble, relevant for Dirac operators in the same symmetry class. It has the property that its eigenvalues are either real, purely imaginary or come in complex conjugate eigenvalue pairs. The eigenvalue joint probability distribution for our model is explicitly computed, leading to a non-Gaussian distribution including K-Bessel functions. All n-point density correlation functions are expressed for finite N in terms of a Pfaffian form. This contains a kernel involving Laguerre polynomials in the complex plane as a building block which was previously computed by the authors. This kernel can be expressed in terms of the kernel for complex non-Hermitian matrices, generalizing the known relation among ensembles of Hermitian random matrices. Compact expressions are given for the density at finite N as an example, as well as its microscopic large-N limits at the origin for fixed ν at strong and weak non-Hermiticity.

Kaporin-Kon’shin’s method of parallel implementation of block preconditioners for asymmetric matrices in problems of filtration of a multicomponent mixture in a porous medium

The ILU class preconditioners (ILU(0), ILU(1), ILUT) employed for iterative algorithms for non-symmetrical linear sparse matrix systems are considered. Test matrices used in this study originate from discretization of systems of partial differential equations describing multicomponent fluid flows in porous media. A new parallel algorithm for block ILU factorization is suggested. This algorithm demonstrates a good convergence and significant speed-up in comparison with sequential algorithms. New integrated approach was tested on the wide range of matrices resulted from real hydrodynamic simulations of oil fields of Western Siberia and demonstrated significant reduction in computational time.

Random matrix theory for pseudo-Hermitian systems: Cyclic blocks

We discuss the relevance of random matrix theory for pseudo-Hermitian systems, and, for Hamiltonians that break parity P and time-reversal invariance T. In an attempt to understand the random Ising model, we present the treatment of cyclic asymmetric matrices with blocks and show that the nearest-neighbour spacing distributions have the same form as obtained for the matrices with scalar entries. We also summarize the theory for random cyclic matrices with scalar entries. We have also found that for block matrices made of Hermitian and pseudo-Hermitian sub-blocks of the form appearing in Ising model depart from the known results for scalar entries. However, there is still similarity in trends even in log-log plots.

Computing continuous core/periphery structures for social relations data with MINRES/SVD

When diagonal values are missing or excluded, MINRES is a natural continuous model for the core/periphery structure of a symmetric social network matrix. Symmetric models, however, are not so useful when dealing with asymmetric data. Singular value decomposition (SVD) is a natural choice to model asymmetry, but this method also requires the presence of diagonal values. In this paper we offer an alternative, more general, approach to continuous core/periphery structures, the minimum residual singular value decomposition (MINRES/SVD), where each node in the network receives two indices, an “in-coreness” and an “out-coreness.” The algorithm for computing these coreness vectors is a least squares computation similar to, but distinct from the SVD, again because of the missing diagonal values. And in contrast to the standard, symmetric MINRES algorithm, we can more accurately model asymmetric matrices. This allows us to distinguish, for example, countries in the world economy that are more in the exporting core than they are in the importing core. We propose two nested PRE (proportional reduction of error) measures of fit: (1) the PRE from the MINRES vector with respect to the data and (2) the PRE of the product of the two MINRES/SVD vectors. Applying the resulting method to citations between journals and to international trade in clothing, we illustrate insights gained from being able to model asymmetrical flow patterns. Finally, two permutation tests are introduced to test independently for the MINRES and MINRES/SVD results.

A nonsmooth algorithm for cone-constrained eigenvalue problems

We study several variants of a nonsmooth Newton-type algorithm for solving an eigenvalue problem of the form $$K\ni x\perp(Ax-\lambda Bx)\in K^{+}.$$ Such an eigenvalue problem arises in mechanics and in other areas of applied mathematics. The symbol K refers to a closed convex cone in the Euclidean space ℝ n and (A,B) is a pair of possibly asymmetric matrices of order n. Special attention is paid to the case in which K is the nonnegative orthant of ℝ n . The more general case of a possibly unpointed polyhedral convex cone is also discussed in detail.

Random transition-rate matrices for the master equation

Random-matrix theory is applied to transition-rate matrices in the Pauli master equation. We study the distribution and correlations of eigenvalues, which govern the dynamics of complex stochastic systems. Both the cases of identical and of independent rates of forward and backward transitions are considered. The first case leads to symmetric transition-rate matrices, whereas the second corresponds to general asymmetric matrices. The resulting matrix ensembles are different from the standard ensembles and show different eigenvalue distributions. For example, the fraction of real eigenvalues scales anomalously with matrix dimension in the asymmetric case.

Methods for the combination of kernel matrices within a support vector framework

The problem of combining different sources of information arises in several situations, for instance, the classification of data with asymmetric similarity matrices or the construction of an optimal classifier from a collection of kernels. Often, each source of information can be expressed as a similarity matrix. In this paper we propose a new class of methods in order to produce, for classification purposes, a single kernel matrix from a collection of kernel (similarity) matrices. Then, the constructed kernel matrix is used to train a Support Vector Machine (SVM). The key ideas within the kernel construction are twofold: the quantification, relative to the classification labels, of the difference of information among the similarities; and the extension of the concept of linear combination of similarity matrices to the concept of functional combination of similarity matrices. The proposed methods have been successfully evaluated and compared with other powerful classifiers and kernel combination techniques on a variety of artificial and real classification problems.

The Ginibre Ensemble of Real Random Matrices and its Scaling Limits

We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a $2 \times 2$ matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large.

STABILITY AND CONTROLLABILITY OF ASYMMETRIC COMPLEX DYNAMICAL NETWORKS: EIGENVALUE ANALYSIS

The stability and controllability of asymmetric complex dynamical networks are investigated in detail based on eigenvalue analysis. Pinning control is suggested to stabilize the homogenous stationary state of the whole coupled network. The complicated coupled problem is reduced to two independent problems: clarifying the stable regions of the coupled network and specifying the eigenvalue distribution of the asymmetric coupling and control matrices. The dependence of the controllability on both pinning density and pinning strength is studied.

Evolutionary Changes in Service Attribute Importance in a Crisis Scenario The Uruguayan Financial Crisis

This article attempts to identify and assess the evolution of consumers' differential reactions to major service attribute classes that resulted from and were propagated by a severe financial crisis. The authors perform a longitudinal analysis, using correspondence analysis of square asymmetric matrices, on data generated from three different postcrisis time periods. The first period encompasses data during the 2 weeks after the crisis occurred, the second period assesses data 1 year later, and the third period refers to 5 years later. The results from the first and second periods both indicate the growing importance of credence attributes rather than search attributes compared with during the precrisis period. The third-period analysis indicates a reversal and greater importance of the search attributes that were important in the precrisis period. The results also reveal correlations with the type of banking organization. These findings point to important managerial implications; bank management's inability to realize the shift in consumers' attitudes failed to facilitate and expedite a resolution to the crisis.

Testing realistic quark mass matrices in the custodial Randall-Sundrum model with flavor changing top decays

We study quark mass matrices in the Randall-Sundrum (RS) model with bulk symmetry $SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{R}\ifmmode\times\else\texttimes\fi{}U(1{)}_{B\ensuremath{-}L}$. The Yukawa couplings are assumed to be within an order of magnitude of each other, and perturbative. We find that quark mass matrices of the symmetrical form proposed by Koide et al. [Y. Koide, H. Nishiura, K. Matsuda, T. Kikuchi, and T. Fukuyama, Phys. Rev. D 66, 093006 (2002)] can be accommodated in the RS framework with the assumption of hierarchyless Yukawa couplings, but not the Hermitian Fritzsch-type mass matrices. General asymmetrical mass matrices are also found which fit well simultaneously with the quark masses and the Cabibbo-Kobayashi-Maskawa matrix. Both left-handed (LH) and right-handed (RH) quark rotation matrices are obtained that allow analysis of flavor changing decay of both LH and RH top quarks. At a warped down scale of 1.65 TeV, the total branching ratio of $t\ensuremath{\rightarrow}Z+\mathrm{\text{jets}}$ can be as high as $\ensuremath{\sim}5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ for symmetrical mass matrices and $\ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ for asymmetrical ones. This level of signal is within reach of the LHC.

Random matrix ensembles of time-lagged correlation matrices: derivation of eigenvalue spectra and analysis of financial time-series

We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as real, asymmetric random matrices where the time-shift superimposes some structure. We demonstrate that, for large matrices, the associated eigenvalue spectrum is circular symmetric in the complex plane. This fact allows us to exactly compute the eigenvalue density via an inverse Abel-transform of the density of the symmetrized problem. We demonstrate the validity of this approach numerically. Theoretical findings are then compared with eigenvalue densities obtained from actual high-frequency (5 min) data of the S&P 500 and the observed deviations are discussed. We identify various non-trivial, non-random patterns and find asymmetric dependencies associated with eigenvalues departing strongly from the Gaussian prediction in the imaginary part. For the same time-series, with the market contribution removed, we observe strong clustering of stocks into causal sectors. We finally comment on the stability of the observed patterns.