Arbitrary Matrices Research Articles

The limiting distributions of large heavy Wigner and arbitrary random matrices

A heavy Wigner matrix XN is defined similarly to a classical Wigner one. It is Hermitian, with independent sub-diagonal entries. The diagonal entries and the non-diagonal entries are identically distributed. Nevertheless, the moments of the entries of NXN tend to infinity with N, as for matrices with truncated heavy tailed entries or adjacency matrices of sparse Erdös–Rényi graphs. Consider a family XN of independent heavy Wigner matrices and an independent family YN of arbitrary random matrices with a bound condition and converging in ⁎-distribution in the sense of free probability. We characterize the possible limiting joint ⁎-distributions of (XN,YN), giving explicit formulas for joint ⁎-moments. We find that they depend on more than the ⁎-distribution of YN and that in general XN and YN are not asymptotically ⁎-free. We use the traffic distributions and the associated notion of independence [21] to encode the information on YN and describe the limiting ⁎-distribution of (XN,YN). We develop this approach for related models and give recurrence relations for the limiting ⁎-distribution of heavy Wigner and independent diagonal matrices.

Uncertainty characterization of the orthogonal Procrustes problem with arbitrary covariance matrices

This paper addresses the weighted orthogonal Procrustes problem of matching stochastically perturbed point clouds, formulated as an optimization problem with a closed-form solution. A novel uncertainty characterization of the solution of this problem is proposed resorting to perturbation theory concepts, which admits arbitrary transformations between point clouds and individual covariance and cross-covariance matrices for the points of each cloud. The method is thoroughly validated through extensive Monte Carlo simulations, and particularly interesting cases where nonlinearities may arise are further analyzed.

Less-conservative robust adaptive control of neutral systems with mixed time-delays

ABSTRACTIn this paper, the problem of robust stabilisation of a class of neutral-type uncertain linear systems via adaptive control approach is investigated. The neutral systems are assumed to be subjected to model uncertainties, disturbances and mixed time-delays. In comparison to the previous work in the literature, with the assumption that the states of the system and the delayed-states of the system with its derivatives are completely accessible, a new and less-conservative robust adaptive control algorithm is proposed. Based on Lyapunov–Krasovskii function, the model uncertainties and the disturbances are compensated and the stability of the uncertain neutral-type system is guaranteed. The tracking problem via model reference adaptive control is also considered. The most important characteristic of the proposed approach is to make the overall system stability constraints depend on desired arbitrary matrices, and not on the practical system matrices. This is achieved by means of adaptation techniques. Finally, three illustrative examples with numerical simulations are carried out to indicate the significant improvements over some existing results.

New Nonsingularity Conditions for General Matrices and the Associated Eigenvalue Inclusion Sets

The paper suggests generalizations of some known sufficient nonsingularity conditions for matrices with constant principal diagonal and the corresponding eigenvalue inclusion sets to the cases of arbitrary matrices and matrices with nonzero diagonal entries. Bibliography: 11 titles.

Some matrix completions over integral domains

We characterize 3×3 nilpotent matrices which are completions of 2×2 arbitrary matrices and 3×3 idempotent matrices which are completions of 2×2 arbitrary matrices over integral domains. As an application we show that a nil-clean element of a ring which belongs to a corner of the ring, may not be nil-clean in this corner.

Generalization of the Kohn-Sham system that can represent arbitrary one-electron density matrices

Density functional theory is currently the most widely applied method in electronic structure theory. The Kohn-Sham method, based on a fictitious system of non-interacting particles, is the work horse of the theory. The particular form of the Kohn-Sham wavefunction admits only idem-potent one electron density matrices whereas wavefunctions of correlated electrons in post-Hartree-Fock methods invariably have fractional occupation numbers. Here we show that by generalizing the orbital concept, and introducing a suitable dot-product as well as a probability density a non-interacting system can be chosen that can represent the one-electron density matrix of any system, even one with fractional occupation numbers. This fictitious system ensures that the exact electron density is accessible within density functional theory. It can also serve as the basis for reduced density matrix functional theory. Moreover, to aid the analysis of the results the orbitals may be assigned energies from a mean-field Hamiltonian. This produces energy levels that are akin to Hartree-Fock orbital energies such that conventional analyses based on Koopmans theorem are available. Finally, this system is convenient in formalisms that depend on creation and annihilation operators as they are trivially applied to single determinant wavefunctions.

Sampled-data filtering of Takagi–Sugeno fuzzy neural networks with interval time-varying delays

This paper is concerned with sample-data filtering of T–S fuzzy neural networks with interval time-varying delays, which is formed by a fuzzy plant with time delay and a sampled-data fuzzy controller connected in a closed loop. A Takagi–Sugeno (T–S) fuzzy model is adopted for the neural networks and the sampled-data fuzzy controller is designed for a T–S fuzzy system. To develop the guaranteed cost control, a new stability condition of the closed-loop system is guaranteed in the continuous-time Lyapunov sense, and its sufficient conditions are formulated in terms of linear matrix inequalities. By using a descriptor representation, the sampled-data fuzzy control system with time delay can be reduced to ease the stability analysis, which effectively leads to a smaller number of LMI-stability conditions. Information of the membership functions of both the fuzzy plant model and fuzzy controller are considered, which allows arbitrary matrices to be introduced, to ease the satisfaction of the stability conditions. By a newly proposed inequality bounding technique, the fuzzy sampled-data filtering performance analysis is carried out such that the resultant neural networks is asymptotically stable. Numerical example and simulation result are given to illustrate the usefulness and effectiveness of the proposed theoretical results.

Perturbation bounds for eigenvalues of diagonalizable matrices and singular values

Perturbation bounds for eigenvalues of diagonalizable matrices are derived. Perturbation bounds for singular values of arbitrary matrices are also given. We generalize some existing results.

Kernels of Discrete Convolutions and Application to Stationary Subdivision Operators

We consider kernels of discrete convolution operators or, equivalently, homogeneous solutions of a system of partial difference operators. We show that whenever the space of homogeneous solutions of a system of partial difference operators is finite dimensional, it always has to consist of exponential polynomials which can be described exactly. We give an explicit relationship between the polynomial part of theses spaces in a direct though somewhat intricate relation to the multiplicity of the common zeros of certain multivariate polynomials. This concept was introduced by Grobner in his characterization of the kernels of partial differential operators with constant coefficients. Based on the characterization of the kernels of convolutions, we can then also determine the kernels of stationary subdivision operators based on arbitrary scaling matrices.

Common Information and Unique Disjointness

We provide an information-theoretic framework for establishing strong lower bounds on the nonnegative rank of matrices by means of common information, a notion previously introduced in Wyner (IEEE Trans Inf Theory 21(2):163---179, 1975). The framework is a generalization of the one in Braverman and Moitra (Proceedings of the forty-fifth annual ACM symposium on theory of computing, pp 161---170, 2013) for the shifted uniqe disjointness (UDISJ) matrix to arbitrary nonnegative matrices. Common information is a natural lower bound for the nonnegative rank of a matrix and by combining it with Hellinger distance estimations we compute the (almost) exact common information of UDISJ partial matrix. The bounds are obtained very naturally. We also establish robustness of this estimation under random or adversarial removal of rows and columns of the UDISJ partial matrix. This robustness translates, via a variant of Yannakakis's factorization theorem, to lower bounds on the average case and adversarial approximate extension complexity of removals. We present the first family of polytopes, the hard pair introduced in Braun et al. (Math Oper Res 40(3):756---772, 2015) related to the CLIQUE problem, with high average case and adversarial approximate extension complexity of removals. The framework relies on a strengthened version of the link between information theory and Hellinger distance from Bar-Yossef (J Comput Syst Sci 68(4):702---732, 2004). We also provide an information theoretic variant of the fooling set method that allows us to extend fooling set lower bounds from extension complexity to approximate extension complexity.

Construction of Quantum LDPC Codes From Classical Row-Circulant QC-LDPCs

Classical row-circulant quasi-cyclic (QC) low-density parity check (LDPC) matrices are known to generate efficient high-rate short and moderate-length QC-LDPC codes, while the comparable random structures exhibit numerous short cycles of length-4. Therefore, we conceive a general formalism for constructing nondual-containing Calderbank–Shor–Steane (CSS)-type quantum low-density parity check (QLDPC) codes from arbitrary classical row-circulant QC-LDPC matrices. We apply our proposed formalism to the classical balanced incomplete block design (BIBD)-based row-circulant QC-LDPC codes for demonstrating that our designed codes outperform their dual-containing CSS-type counterparts as well as the entanglement-assisted (EA)-QLDPC codes.

Online PCA with optimal regret

We investigate the online version of Principle Component Analysis (PCA), where in each trial t the learning algorithm chooses a k-dimensional subspace, and upon receiving the next instance vector xt, suffers the loss, which is the squared Euclidean distance between this instance and its projection into the chosen subspace. When viewed in the right parameterization, this compression loss is linear, i.e. it can be rewritten as tr(Wtxtxt⊤), where Wt is the parameter of the algorithm and the outer product xtxt⊤ (with ||xt|| ≤ 1) is the instance matrix. In this paper generalize PCA to arbitrary positive definite instance matrices Xt with the linear loss tr(WtXt). We evaluate online algorithms in terms of their worst-case regret, which is a bound on the additional total loss of the online algorithm on all instances matrices over the compression loss of the best k-dimensional subspace (chosen in hindsight). We focus on two popular online algorithms for generalized PCA: the Gradient Descent (GD) and Matrix Exponentiated Gradient (MEG) algorithms. We show that if the regret is expressed as a function of the number of trials, then both algorithms are optimal to within a constant factor on worst-case sequences of positive definite instances matrices with trace norm at most one (which subsumes the original PCA problem with outer products). This is surprising because MEG is believed be suboptimal in this case. We also show that when considering regret bounds as a function of a loss budget, then MEG remains optimal and strictly outperforms GD when the instance matrices are trace norm bounded. Next, we consider online PCA when the adversary is allowed to present the algorithm with positive semidefinite instance matrices whose largest eigenvalue is bounded (rather than their trace which is the sum of their eigenvalues). Again we can show that MEG is optimal and strictly better than GD in this setting.

First Order Asymptotic Expansions for Eigenvalues of Multiplicatively Perturbed Matrices

Given an arbitrary square matrix $A$, we obtain the leading terms of the asymptotic expansions in the small, real parameter $\varepsilon$ of multiplicative perturbations $\widehat{A}(\varepsilon)=(I+\varepsilon\,B)A(I+\varepsilon\,C)$ of $A$ for arbitrary matrices $B$ and $C$. The analysis is separated into two rather different cases, depending on whether the unperturbed eigenvalue is zero or not. It is shown that in either case the leading exponents are obtained from the partial multiplicities of the eigenvalue of interest, and the leading coefficients generically involve only appropriately normalized left and right eigenvectors of $A$ associated with that eigenvalue, with no need of generalized eigenvectors. Similar results are obtained for multiplicative perturbation of singular values as well.

A model with two inert scalar doublets

We consider an extension of the standard model (SM) with three SU(2) scalar doublets and discrete S3⊗Z2 symmetries. The irreducible representation of S3 has a singlet and a doublet, and here we show that the singlet corresponds to the SM-like Higgs and the two additional SU(2) doublets forming a S3 doublet are inert. In general, in a three scalar doublet model, with or without S3 symmetry, the diagonalization of the mass matrices implies arbitrary unitary matrices. However, we show that in our model these matrices are of the tri-bimaximal type. We also analyzed the scalar mass spectra and the conditions for the scalar potential is bounded from below at the tree level. We also discuss some phenomenological consequences of the model.

Pinning adaptive hybrid synchronization of two general complex dynamical networks with mixed coupling

In this paper, the hybrid synchronization problem has been investigated for two delayed dynamical networks with time-varying delay and mixed coupling by the pinning adaptive control strategy. The Barbalat lemma and linear matrix inequality technique are employed to obtain some synchronization criteria less-conservative and delay-dependent. Here, the coupling configuration matrices are not required to be symmetric or irreducible. Moreover the inner linking matrices are arbitrary real matrices. Finally, numerical examples are given to demonstrate the effectiveness of the proposed synchronous criteria.

Markov Bases and Generalized Lawrence Liftings

Minimal Markov bases of configurations of integer vectors correspond to minimal binomial generating sets of the assocciated lattice ideal. We give necessary and sufficient conditions for the elements of a minimal Markov basis to be (a) inside the universal Gr{\" o}bner basis and (b) inside the Graver basis. We study properties of Markov bases of generalized Lawrence liftings for arbitrary matrices $A\in\mathcal{M}_{m\times n}(\Bbb{Z})$ and $B\in\mathcal{M}_{p\times n}(\Bbb{Z})$ and show that in cases of interest the {\em complexity} of any two Markov bases is the same.

On biunimodular vectors for unitary matrices

A biunimodular vector of a unitary matrix A∈U(n) is a vector v∈Tn⊂Cn such that Av∈Tn as well. Over the last 30 years, the sets of biunimodular vectors for Fourier matrices have been the object of extensive research in various areas of mathematics and applied sciences. Here, we broaden this basic harmonic analysis perspective and extend the search for biunimodular vectors to arbitrary unitary matrices. This search can be motivated in various ways. The main motivation is provided by the fact, that the existence of biunimodular vectors for an arbitrary unitary matrix allows for a natural understanding of the structure of all unitary matrices.

Confronting predictive texture zeros in lepton mass matrices with current data

Several popular Ans\"atze of lepton mass matrices that contain texture zeros are confronted with current neutrino observational data. We perform a systematic ${\ensuremath{\chi}}^{2}$ analysis in a wide class of schemes, considering arbitrary Hermitian charged-lepton mass matrices and symmetric mass matrices for Majorana neutrinos or Hermitian mass matrices for Dirac neutrinos. Our study reveals that several patterns are still consistent with all the observations at the 68.27% confidence level, while some others are disfavored or excluded by the experimental data. The well-known Frampton-Glashow-Marfatia two-zero textures, hybrid textures, and parallel structures (among others) are considered.

Computation of Gaussian orthant probabilities in high dimension

We study the computation of Gaussian orthant probabilities, i.e. the probability that a Gaussian variable falls inside a quadrant. The Geweke---Hajivassiliou---Keane (GHK) algorithm (Geweke, Comput Sci Stat 23:571---578 1991, Keane, Simulation estimation for panel data models with limited dependent variables, 1993, Hajivassiliou, J Econom 72:85---134, 1996, Genz, J Comput Graph Stat 1:141---149, 1992) is currently used for integrals of dimension greater than 10. In this paper, we show that for Markovian covariances GHK can be interpreted as the estimator of the normalizing constant of a state-space model using sequential importance sampling. We show for an AR(1) the variance of the GHK, properly normalized, diverges exponentially fast with the dimension. As an improvement we propose using a particle filter. We then generalize this idea to arbitrary covariance matrices using Sequential Monte Carlo with properly tailored MCMC moves. We show empirically that this can lead to drastic improvements on currently used algorithms. We also extend the framework to orthants of mixture of Gaussians (Student, Cauchy, etc.), and to the simulation of truncated Gaussians.

On computing of positive integer powers for r-circulant matrices

In this paper, we give a method of computing arbitrary positive integer powers for arbitrary order r-circulant matrices by using the basis expression of r-circulant matrices in matrix spaces and the Multinomial Theorem.