Polynomial–Exponential Decomposition From Moments

Abstract

We analyze the decomposition problem of multivariate polynomial-exponential functions from their 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 isolated points 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 Carathéodory-Fejér 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.

Algorithms for computing cubatures based on moment theory

“…(theorem 4.5) We give a proof of this result by exhibiting the left eigenvectors of the matrix of . This is a hint that duality can shed further light on the structure of in general Theorem Let I be a zero‐dimensional ideal in and assume that is a basis of .…”

“…Under some natural assumptions, solutions have been proposed for the univariate case, for the multivariate case with a univariate resolution (projection method) and with a multivariate approach …”

“…The connection between Hankel operators and multiplication maps, which allows one to compute the cubature nodes as an eigenvalue problem, was, for instance, used in Refs. and in the context of symmetric tensor decomposition and multivariate Prony's method.…”

Decomposition of Low Rank Multi-symmetric Tensor

“…The proposed method extends the techniques of [16] to more general tensors and to tensors of higher rank. It is closely connected to the multivariate Prony method investigated in [14] and to the structured low rank decomposition of Hankel matrix [7].…”

“…We slice variables into bunches of sub-variables and we adapt the description of Artinian Gorenstein Algebra to this case. We adapt the method of decomposition of Hankel matrices of low rank described in [7] to a decomposition of multi linear tensors method which is based on the decomposition of a formal power series as a weighted sum of exponential described in [14]. The computation of multiplication matrices depend on the dimension of tensor, and the number of given moments or coefficients.…”