We analyze the decomposition problem of multivariate polynomial-exponential functions from truncated series and present new algorithms to compute their decomposition. Using the duality between polynomials and formal power series, we first show how the elements in the dual of an Artinian algebra correspond to polynomial-exponential functions. They are also the solutions of systems of partial differential equations with constant coefficients. We relate their representation to the inverse system of the roots of the characteristic variety. Using the properties of Hankel operators, we establish a correspondence between polynomial exponential series and Artinian Gorenstein algebras. We generalize Kronecker theorem to the multivariate case, by showing that the symbol of a Hankel operator of finite rank is a polynomial-exponential series and by connecting the rank of the Hankel operator with the decomposition of the symbol. A generalization of Prony's approach to multivariate decomposition problems is presented , exploiting eigenvector methods for solving polynomial equations. We show how to compute the frequencies and weights of a minimal polynomial-exponential decomposition , using the first coefficients of the series. A key ingredient of the approach is the flat extension criteria, which leads to a multivariate generalization of a rank condition for a Carath 'e odory-Fej 'e r decomposition of multivariate Hankel matrices. A new algorithm is given to compute a basis of the Artinian Gorenstein algebra, based on a Gram-Schmidt orthogonalization process and to decompose polynomial-exponential series. A general framework for the applications of this approach is described and illustrated in different problems. We provide Kronecker-type theorems for convolution operators, showing that a convolution operator (or a cross-correlation operator) is of finite rank, if and only if, its symbol is a polynomial-exponential function, and we relate its rank to the decomposition of its symbol. We also present Kronecker-type theorems for the reconstruction of measures as weighted sums of Dirac measures from moments and for the decomposition of polynomial-exponential functions from values. Finally, we describe an application of this method for the sparse interpolation of polylog functions from values.
Hankel matrices are central in the theory of Pad 'e approximants for functions of one variable. Here also a large literature exists for univariate Pad 'e approximants: see e.g. @cite_29 for approximation properties, @cite_17 for numerical stability problems, @cite_52 , @cite_30 for algorithmic aspects. The extension to multivariate functions is much less developed @cite_47 , @cite_4 .
@cite_30 @cite_4 @cite_29 @cite_52 @cite_47 @cite_17
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The generalized eigenvalue problem @math with H a Hankel matrix and @math the corresponding shifted Hankel matrix occurs in number of applications such as the reconstruction of the shape of a polygon from its moments, the determination of abscissa of quadrature formulas, of poles of Pade approximants, or of the unknown powers of a sparse black box polynomial in computer algebra. In many of these applications, the entries of the Hankel matrix are only known up to a certain precision. We study the sensitivity of the nonlinear application mapping the vector of Hankel entries to its generalized eigenvalues. A basic tool in this study is a result on the condition number of Vandermonde matrices with not necessarily real abscissas which are possibly row-scaled. Recently, a uniform approach was given by B. Beckermann and G. Labahn [Numer. Algorithms, 3 (1992), pp. 45-54] for different concepts of matrix-type Pade approximants, such as descriptions of vector and matrix Pade approximants along with generalizations of simultaneous and Hermite Pade approximants. The considerations in this paper are based on this generalized form of the classical scalar Hermite Pade approximation problem, power Hermite Pade approximation. In particular, this paper studies the problem of computing these new approximants. A recurrence relation is presented for the computation of a basis for the corresponding linear solution space of these approximants. This recurrence also provides bases for particular subproblems. This generalizes previous work by Van Barel and Bultheel and, in a more general form, by Beckermann. The computation of the bases has complexity @math , where @math is the order of the desired approximant and requires no conditions on the input data. A second algorithm using the same recurrence relation along with divide-and-conquer methods is also presented. When the coefficient field allows for fast polynomial multiplication, this second algorithm computes a basis in the superfast complexity @math . In both cases the algorithms are reliable in exact arithmetic. That is, they never break down, and the complexity depends neither on any normality assumptions nor on the singular structure of the corresponding solution table. As a further application, these methods result in fast (and superfast) reliable algorithms for the inversion of striped Hankel, layered Hankel, and (rectangular) block-Hankel matrices. Let H0, …, H n be m x m matrices with entries in Q and Hankel structure, i.e. constant skew diagonals. We consider the linear Hankel matrix H(x) = H0+x1H_1+…+xnHn and the problem of computing sample points in each connected component of the real algebraic set defined by the rank constraint rank (H(x))≤ r, for a given integer r ≤ m-1. Computing sample points in real algebraic sets defined by rank defects in linear matrices is a general problem that finds applications in many areas such as control theory, computational geometry, optimization, etc. Moreover, Hankel matrices appear in many areas of engineering sciences. Also, since Hankel matrices are symmetric, any algorithmic development for this problem can be seen as a first step towards a dedicated exact algorithm for solving semi-definite programming problems, i.e. linear matrix inequalities. Under some genericity assumptions on the input (such as smoothness of an incidence variety), we design a probabilistic algorithm for tackling this problem. It is an adaptation of the so-called critical point method that takes advantage of the special structure of the problem. Its complexity reflects this: it is essentially quadratic in specific degree bounds on an incidence variety. We report on practical experiments and analyze how the algorithm takes advantage of this special structure. A first implementation outperforms existing implementations for computing sample points in general real algebraic sets: it tackles examples that are out of reach of the state-of-the-art. What we know about multivariate Pade approximation has been developed in the last 25 years. In the next sections we compare and discuss many of these results. It will become clear that simple properties or requirements, such as the uniqueness of the Pade approximant and consequently its consistency property, can play a crucial role in the development of the multivariate theory. A separate section is devoted to a discussion of the convergence properties. At the end we include an extensive reference list on the topic. Abstract We obtain a several variables generalization of Kronecker's well-known result on finite rank Hankel matrices. Given all moments of the marginals of a measure on Rn, one provides (a) explicit bounds on its support and (b), a numerical scheme to compute the smallest box that contains the support. It consists of solving a hierarchy of generalized eigenvalue problems associated with some Hankel matrices. The so-called Berlekamp -- Massey -- Sakata algorithm computes a Grobner basis of a @math -dimensional ideal of relations satisfied by an input table. It extends the Berlekamp -- Massey algorithm to @math -dimensional tables, for @math . We investigate this problem and design several algorithms for computing such a Grobner basis of an ideal of relations using linear algebra techniques. The first one performs a lot of table queries and is analogous to a change of variables on the ideal of relations. As each query to the table can be expensive, we design a second algorithm requiring fewer queries, in general. This -like algorithm allows us to compute the relations of the table by extracting a full rank submatrix of a matrix (a multivariate generalization of Hankel matrices). Under some additional assumptions, we make a third, adaptive, algorithm and reduce further the number of table queries. Then, we relate the number of queries of this third algorithm to the of the final staircase and we show that it is essentially linear in the size of the output when the staircase is convex. As a direct application to this, we decode @math -cyclic codes, a generalization in dimension @math of Reed Solomon codes. We show that the multi-Hankel matrices are heavily structured when using the ordering and that we can speed up the computations using fast algorithms for quasi-Hankel matrices. Finally, we design algorithms for computing the generating series of a linear recursive table. In this paper, we consider the approximation of dense block Toeplitz-plus-Hankel matrices by sums of Kronecker products of Toeplitz-plus-Hankel matrices. We present an algorithm for efficiently computing the matrix approximation that requires the factorization of matrices of much smaller dimension than that of the original. The main results are described for block Toeplitz matrices with Toeplitz-plus-Hankel blocks, but the algorithms can be readily adjusted for other related structures that arise in image processing applications, such as block Toeplitz with Toeplitz blocks and block Toeplitz-plus-Hankel with Toeplitz-plus-Hankel blocks. Our work extends the techniques in Kamm and Nagy (SIAM J. Matrix Anal. Appl. 2000; 22:155–172) and (SIAM J. Matrix Anal. Appl. 2004; 25:829–841), which consider similar matrices, but with the added restriction that the matrices have a banded block-banded structure. We illustrate the effectiveness of our algorithm by using the output of the algorithm to construct preconditioners for systems from two different applications: diffuse optical tomography and atmospheric image deblurring. Copyright © 2007 John Wiley & Sons, Ltd. We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category ( C ) , an equivariant indicator of an object in ( C ) is defined as a functional on the Grothendieck algebra of the quantum double ( Z( C ) ) via generalized Frobenius-Schur indicators. The set of all equivariant indicators admits a natural action of the modular group. Using the properties of equivariant indicators, we prove a congruence subgroup theorem for modular categories. As a consequence, all modular representations of a modular category have finite images, and they satisfy a conjecture of Eholzer. In addition, we obtain two formulae for the generalized indicators, one of them a generalization of Bantay’s second indicator formula for a rational conformal field theory. This formula implies a conjecture of Pradisi-Sagnotti-Stanev, as well as a conjecture of Borisov-Halpern-Schweigert. Suppose that @math is a Toeplitz matrix whose entries come from a sequence of independent but not necessarily identically distributed random variables with mean zero. Under some additional tail conditions, we show that the spectral norm of @math is of the order @math . The same result holds for random Hankel matrices as well as other variants of random Toeplitz matrices which have been studied in the literature. This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of application, ease of use, and strength of conclusion that have made the scalar inequalities so valuable. Abstract Elementary proofs are given for theorems of Bapat and Raghavan on the scaling of nonnegative multidimensional matrices. Theorems of Sinkhorn and of Brualdi, Parter, and Schneider are derived as corollaries. For positive two-dimensional matrices, Hilbert's projective metric and a theorem of G. Birkhoff are used to prove that Sinkhorn's original iterative procedure converges geometrically; the ratio of convergence is estimated from the given data. We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haar-distributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson–Zuber type. Abstract Let X 1 , … , X N ∈ R n be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability one has sup x ∈ S n − 1 | 1 N ∑ i = 1 N ( | 〈 X i , x 〉 | 2 − E | 〈 X i , x 〉 | 2 ) | ⩽ C n N , where C is an absolute positive constant. This result is valid in a more general framework when the linear forms ( 〈 X i , x 〉 ) i ⩽ N , x ∈ S n − 1 and the Euclidean norms ( | X i | n ) i ⩽ N exhibit uniformly a sub-exponential decay. As a consequence, if A denotes the random matrix with columns ( X i ) , then with overwhelming probability, the extremal singular values λ min and λ max of A A ⊤ satisfy the inequalities 1 − C n N ⩽ λ min N ⩽ λ max N ⩽ 1 + C n N which is a quantitative version of Bai–Yin theorem (Z.D. Bai, Y.Q. Yin, 1993 [4] ) known for random matrices with i.i.d. entries. Let A be a complex matrix with arbitrary Jordan structure and @math an eigenvalue of A whose largest Jordan block has size n. We review previous results due to Lidskii [U.S.S. R. Comput. Math. and Math. Phys., 1 (1965), pp. 73--85], showing that the splitting of @math under a small perturbation of A of order @math is, generically, of order @math . Explicit formulas for the leading coefficients are obtained, involving the perturbation matrix and the eigenvectors of A. We also present an alternative proof of Lidskii's main theorem, based on the use of the Newton diagram. This approach clarifies certain difficulties which arise in the nongeneric case and leads, in some situations, to the extension of Lidskii's results. These results suggest a new notion of Holder condition number for multiple eigenvalues, depending only on the associated left and right eigenvectors, appropriately normalized, not on the Jordan vectors. We present an efficient and practical algorithm for the online prediction of discrete-time linear dynamical systems with a symmetric transition matrix. We circumvent the non-convex optimization problem using improper learning: carefully overparameterize the class of LDSs by a polylogarithmic factor, in exchange for convexity of the loss functions. From this arises a polynomial-time algorithm with a near-optimal regret guarantee, with an analogous sample complexity bound for agnostic learning. Our algorithm is based on a novel filtering technique, which may be of independent interest: we convolve the time series with the eigenvectors of a certain Hankel matrix. The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling technique, based on tropical algebra, which applies in particular to this companion form. This scaling, which is inspired by an earlier work of Akian, Bapat, and Gaubert, relies on the computation of “tropical roots”. We give explicit bounds, in a typical case, indicating that these roots provide accurate estimates of the order of magnitude of the different eigenvalues, and we show by experiments that this scaling improves the accuracy (measured by normwise backward error) of the computations, particularly in situations in which the data have various orders of magnitude. In the case of quadratic polynomial matrices, we recover in this way a scaling due to Fan, Lin, and Van Dooren, which coincides with the tropical scaling when the two tropical roots are equal. If not, the eigenvalues generally split in two groups, and the tropical method leads to making one specific scaling for each of the groups. We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs @math converging locally to a Galton--Watson tree @math (GWT), we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function @math of @math . In the first part, we show that the adjacency operator associated with @math is always self-adjoint; we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we establish a sufficient condition on @math for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of @math . Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory and statistical physics. The capacity of symmetric instance of the multiple unicast index coding problem with neighboring antidotes (side-information) with number of messages equal to the number of receivers was given by Maleki In this paper, we construct matrices of size @math over @math such that any @math adjacent rows of the matrix are linearly independent. By using such matrices, we give an optimal scalar linear index codes over @math for the symmetric one-sided antidote problems considered by Maleki for any given number of messages and one-sided antidotes. The constructed codes are independent of field size and hence works over every field. Given an n x n complex matrix A, let mu(A)(x, y) := 1 n vertical bar 1 <= i <= n, Re lambda(i) <= x, Im lambda(i) <= y vertical bar be the empirical spectral distribution (ESD) of its eigenvalues lambda(i) is an element of C, i = l, ... , n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD mu(1 root n An) of a random matrix A(n) = (a(ij))(1 <= i, j <= n), where the random variables a(ij) - E(a(ij)) are i.i.d. copies of a fixed random variable x with unit variance. We prove a universality principle for such ensembles, namely, that the limit distribution in question is independent of the actual choice of x. In particular, in order to compute this distribution, one can assume that x is real or complex Gaussian. As a related result, we show how laws for this ESD follow from laws for the singular value distribution of 1 root n A(n) - zI for complex z. As a corollary, we establish the circular law conjecture (both almost surely and in probability), which asserts that mu(1 root n An) converges to the uniform measure on the unit disc when the a(ij) have zero mean.