Generalized Hankel Matrix Research Articles

The geometry of the modular bootstrap

We explore the geometry behind the modular bootstrap and its image in the space of Taylor coefficients of the torus partition function. In the first part, we identify the geometry as an intersection of planes with the convex hull of moment curves on R+⊗ℤ, with boundaries characterized by the total positivity of generalized Hankel matrices. We phrase the Hankel constraints as a semi-definite program, which has several advantages, such as the validity of bounds irrespective of spin truncation. We derive bounds on the gap, twist-gap, and the space of Taylor coefficients themselves. We find that if the gap is above ∆gap∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Delta }_{\ extrm{gap}}^{\\ast } $$\\end{document}, where c−112<Δgap∗<c12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{c-1}{12}<{\\Delta}_{\ extrm{gap}}^{\\ast }<\\frac{c}{12} $$\\end{document}, all coefficients become bounded on both sides and kinks develop in the space. In the second part, we propose an analytic method of imposing the integrality condition for the degeneracy number in the spinless bootstrap, which leads to a non-convex geometry. We find that even at very low derivative order this condition rules out regions otherwise allowed by bootstraps at high derivative order.

Approximate greatest common divisor of several polynomials from Hankel matrices

This paper is devoted to present a new method for computing the approximate Greatest Common Divisor (GCD) of several polynomials (not pairwise) from the generalized Hankel matrix. Our approach based on the calculation of cofactors is tested for several sets of polynomials.

Detecting optimality and extracting solutions in polynomial optimization with the truncated GNS construction

A basic closed semialgebraic subset of {mathbb {R}}^{n} is defined by simultaneous polynomial inequalities p_{1}ge 0,ldots ,p_{m}ge 0. We consider Lasserre’s relaxation hierarchy to solve the problem of minimizing a polynomial over such a set. These relaxations give an increasing sequence of lower bounds of the infimum. In this paper we provide a new certificate for the optimal value of a Lasserre relaxation to be the optimal value of the polynomial optimization problem. This certificate is to check if a certain matrix has a generalized Hankel form. This certificate is more general than the already known certificate of an optimal solution being flat. In case we have detected optimality we will extract the potential minimizers with a truncated version of the Gelfand–Naimark–Segal construction on the optimal solution of the Lasserre relaxation. We prove also that the operators of this truncated construction commute if and only if the matrix of this modified optimal solution is a generalized Hankel matrix. This generalization of flatness will enable us to prove, with the use of the GNS truncated construction, a result of Curto and Fialkow on the existence of quadrature rule if the optimal solution is flat and a result of Xu and Mysovskikh on the existence of a Gaussian quadrature rule if the modified optimal solution is a generalized Hankel matrix . At the end, we provide a numerical linear algebraic algorithm for detecting optimality and extracting solutions of a polynomial optimization problem.

On the Realization of Hidden Markov Models and Tensor Decomposition

The minimum realization problem of hidden Markov models (HMM’s) is a fundamental question of stationary discrete-time processes with a finite alphabet. It was shown in the literature that tensor decomposition methods give the hidden Markov model with the minimum number of states generically. However, the tensor decomposition approach does not solve the minimum HMM realization problem when the observation is a deterministic function of the state, which is an important class of HMM’s not captured by a generic argument. In this paper, we show that the reduction of the number of rank-one tensors necessary to decompose the third-order tensor constructed from the probabilities of the process is possible when the reachable subspace is not the whole space or the null space is not the zero space. In fact, the rank of the tensor is not greater than the dimension of the effective subspace or the rank of the generalized Hankel matrix.

Cyclostationary Signals Analysis Methods Based on High-Dimensional Space Transformation Under Impulsive Noise

Cyclostationary signals analysis is a highly widespread tool for non-stationary signals processing. In practical transmission, signals may be contaminated by non-stationary non-Gaussian impulse noises. However, cyclostationary is weak in dealing with such non-Gaussian distributions. In this paper, a cyclic mean kernel function is proposed for cyclostationary signal analysis under impulse noise, based on a high-dimensional space transformation. Then, the generalized cyclic mean kernel function is proposed based on the constructed generalized Hankel matrix. The simulation results show that the cyclostationary signal frequency estimation methods based on these two functions have advantages in performance and robustness over the existing methods under impulsive noise.

Parameter Estimation of Three-dimensional Scattering Centers Based on State Space and ESPRIT Method

For the problem of radar target three-dimensional scattering centers parameter estimation, a new method that based on state space method and spectrum estimation method (3D-SS-ESPRIT) is proposed. According to the 3D-GTD model theory, we obtained the high-frequency radar target parameter signal model, via constructing generalized Hankel matrix, we estimated 3D scattering centers parameters based on the state space method and reconstructed the high-frequency radar target characteristic data. Numerical simulations show that the proposed method is an efficient and reliable method to estimate these parameters, compared to convention 3D spectrum estimation method, the proposed method gain better estimation, reconstruction property and has better stability.

A time-domain system identification numerical procedure for obtaining linear dynamical models of multibody mechanical systems

This paper is focused on the development of a numerical procedure for solving the system identification problem of linear dynamical models that mathematically describe multibody mechanical systems. To this end, an input–output representation of the time evolution of a general mechanical system based on a sequence of matrices referred to as Markov parameters is employed. The set of Markov parameters incorporate the state-space matrices that allow for describing the dynamic behavior of a general mechanical system considering the assumption of structural linearity. The system Markov parameters are defined by means of a discretization process applied to the analytical description of a mechanical system, and therefore, they are difficult to obtain directly from observable measurements. However, a state observer can be introduced in order to define a set of observer Markov parameters that can be readily recovered from input–output experimental data. The observer Markov parameters obtained by using a least-square approach allow for computing in a recursive manner the system Markov parameters as well as another discrete sequence of matrices referred to as observer gain Markov parameters. Subsequently, the system and observer gain Markov parameters identified from observable input–output data are used for constructing a sequence of generalized Hankel matrices from which a state-space model of the mechanical system of interest can be extracted. This fundamental step of the identification procedure is performed in the algorithm elaborated in this work employing a numerical procedure which relies on the use of the Moore–Penrose pseudoinverse matrix obtained by means of the singular value decomposition. In the paper, the principal analytical and numerical aspects of the proposed identification algorithm are described in detail. Furthermore, a numerical example based on a simple vehicle model is discussed in order to verify by means of numerical experiments the effectiveness of the identification procedure developed in this work.

Deterministic Bilinear System Identification

A unified identification method is proposed for system realization of a deterministic continuous-time/discrete-time bilinear models from input and output measurement data. A generalized Hankel matrix is formed with the output measurements obtained by applying a set of repeated input sequences to a bilinear system. A computational procedure is developed to extract a time varying discrete-time state-space model from the generalized Hankel matrix. The bilinear system models are realized by transforming the identified time varying discrete-time model to the bilinear models. Numerical simulations are given to show the effectiveness of the proposed identification method.

Time-Varying Eigensystem Realization Algorithm

An identification algorithm called the time-varying eigensystem realization algorithm is proposed to realize discrete-time-varying plant models from input and output experimental data. It is shown that this singular value decomposition based method is a generalization of the eigensystem realization algorithm developed to realize time invariant models from pulse response sequences. Using the results from discrete-time identification theory, the generalized Markov parameter and the generalized Hankel matrix sequences are computed via a least squares problem associated with the input-output map. The computational procedure presented in the paper outlines a methodology to extract a state space model from the generalized Hankel matrix sequence in different time-varying coordinate systems. The concept of free response experiments is suggested to identify the subspace of the unforced system response. For the special case of systems with fixed state space dimension, the free response subspace is used to construct a uniform coordinate system for the realized models at different time steps. Numerical simulation results on general systems discuss the details and effectiveness of the algorithms.

Generalized Eigenvalue Decomposition in Time Domain Modal Parameter Identification

This paper is intended to point out the relationship among current time domain modal analysis methods by employing generalized eigenvalue decomposition. Ibrahim time domain (ITD), least-squares complex exponential (LSCE) and eigensystem realization algorithm (ERA) methods are reviewed and chosen to do the comparison. Reformulation to their original forms shows these three methods can all be attributed to a generalized eigenvalue problem with different matrix pairs. With this general format, we can see that single-input multioutput (SIMO) methods can easily be extended to multi-input multioutput (MIMO) cases by taking advantage of a generalized Hankel matrix or a generalized Toeplitz matrix.

Model reduction of bilinear systems described by input–output difference equation

A class of single input single output bilinear systems described by their input–output difference equation is considered. A simple expression for the Volterra kernels of the system is derived in terms of the coefficients of difference equation. An algorithm, based on the singular value decomposition of a generalized Hankel matrix, is also developed. The algorithm is then used to find a reduced-order bilinear state-space model. The Hankel approach will be extensively studied under different data length cases and different orders of the state-space models. A numerical example is presented to illustrate the effectiveness of the proposed algorithm.

Matrix measures, moment spaces and Favard's theorem for the interval [0,1] and [0,∞)

In this paper we study the moment spaces corresponding to matrix measures on compact intervals and on the nonnegative line [0,∞). A representation for nonnegative definite matrix polynomials is obtained, which is used to characterize moment points by properties of generalized Hankel matrices. We also derive an explicit representation of the orthogonal polynomials with respect to a given matrix measure, which generalizes the classical determinant representation of the one-dimensional case. Moreover, the coefficients in the recurrence relations can be expressed explicitly in terms of the moments of the matrix measure. These results are finally used to prove a refinement of the well-known Favard theorem for matrix measures, which characterizes the domain of the underlying measure of orthogonality by properties of the coefficients in the recurrence relationships.

The initial algebra of maximal minors of a generalized hankel matrix

One deals with a certain generalization of the (generic) Hankel matrix. Fixing a field K, for a suitable diagonal term order on the polynomial ring B over K generated by the entries of this matrix, one considers the K-subalgebra A ⊂ B generated by the set of initial terms of the the maximal minors of the matrix. The algebra A detects the underlying combinatorics in terms of certain generalized standard tableaux and these give a way of showing that A is normal. A more involved technique also yields that the Rees algebra of the ideal of B generated by the generators of A is normal. In particular, both algebras are Cohen-Macaulay. We also present a Gröbner basis for the presentation ideal of A and compute some numerical invariants using the simplicial complex associated to A.

A fast algorithm for generalized Hankel matrices arising in finite-moment problems

Consider an n × n lower triangular matrix L whose ( i + 1) st row is defined by the coefficients of the real polynomial p i ( x) of degree i such that { p i ( x)} is a set of orthogonal polynomials satisfying a standard three-term recurrence relation. If H is an n × n real Hankel matrix with nonsingular leading principal submatrices, then H ̂ = LHL T will be referred to as a strongly nonsingular Hankel matrix with respect to the orthogonal polynomial basis { p i ( x)}. In this paper we develop an efficient O( n 2) algorithm for the solution of a system of linear equations with a real symmetric coefficient matrix H ̂ which is a Hankel matrix with respect to a suitable orthogonal polynomial basis. This leads to an efficient method for computing polynomial approximations of an unknown function given its modified moments.

A Fast Algorithm for Generalized Hankel Matrices Arising in Finite-Moment Problems

A Fast Algorithm for Generalized Hankel Matrices Arising in Finite-Moment Problems

Removing bias from discrete time models based on finite interval response data

Modern off-line system identification methods necessarily form a system realization using a finite amount of data. The models constructed are usually of an order smaller than the actual order of the physical system because it is not practical to model the relatively unimportant states. A method is presented to improve the selection of the important states and the model order in system identification methods based on generalized Hankel matrices. When forming a reduced-order model, it is critical to be able to accurately identify the important states and to select the appropriate model order. Some currently popular methods implicitly contain a frequency weighting that comes from the use of a finite tune interval of data in the identification. A method is presented to remove this implicit bias, and excellent results are obtained, even when the response data used span only a fraction of a cycle of the dominant frequency.

Recursive independent state-space and matrix fraction realizations

A recursive multivariate partial realization algorithm is proposed. The novel features are that it shows the explicit link between state-space and matrix fraction solutions, and that each solution is obtained not from the other but directly from data. This is achieved by means of a certain factorization of the generalized Hankel matrix, which yields the entries of both realizations, updated with a low computational effort.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A generalized bezoutian matrix

Bezoutian matrices play an important role in linear systems theory, providing iinks between such topics as controllability and observability, canonical forms, realization, root location for polynomials, and Hankel matrices. Like the usual bezoutian matrix, the proposed generalized bezoutian, termed the conjugate-bezoutian, is defined in terms of two polynomials. It is shown that some ol the properties and applications of the standard case extend to this new matrix. I hese include a generalization of a related Sylvester matrix, a companion matrix formula, and expression of the inverse of the conjugate bezoutian in a generalized Hankel matrix form.

Generalized Hankel matrices of Markov parameters and their applications to control problems

The concept of Hankel matrices of Markov parameters associated with two polynomials is generalized for matrices. The generalized Hankel matrices of Markov parameters are then used to develop methods for testing the relative primeness of two matrices A and B, for determining stability and inertia of a matrix, and for constructing a class of matrices C such that A + C has a desired spectrum. Neither the method of construction of the generalized Hankel matrices nor the methods developed using these matrices require explicit computation of the characteristic polynomial of A (or of B).

Two-Dimensional Discrete Space-Varying Systems: Identification, Balancing and Model Reduction

A recursive state-space model for discrete, two-dimensional, space-varying systems is proposed, and a class of uniform realizations is introduced in that context. We derive the conditions for existence of such realizations, and a model reduction scheme via balancing is described. Also, an identification procedure via generalized Hankel matrices is shown for finite support pulse responses. Both approaches yield the same realization.