We present a method to decompose a set of multivariate real polynomials into linear combinations of univariate polynomials in linear forms of the input variables. The method proceeds by collecting the first-order information of the polynomials in a set of operating points, which is captured by the Jacobian matrix evaluated at the operating points. The polyadic canonical decomposition of the three-way tensor of Jacobian matrices directly returns the unknown linear relations, as well as the necessary information to reconstruct the univariate polynomials. The conditions under which this decoupling procedure works are discussed, and the method is illustrated on several numerical examples.1 The total degree is defined as the maximal sum of the exponents of the variables in a term.

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Abstract. The problem discussed in this paper is the symmetric best low multilinear rank approximation of third-order symmetric tensors. We propose an algorithm based on Jacobi rotations, for which symmetry is preserved at each iteration. Two numerical examples are provided indicating the need for such algorithms. An important part of the paper consists of proving that our algorithm converges to stationary points of the objective function. This can be considered an advantage of the proposed algorithm over existing symmetry-preserving algorithms in the literature.Key words. multilinear algebra, higher-order tensor, rank reduction, singular value decomposition, Jacobi rotation AMS subject classifications. 15A69, 65F99 DOI. 10.1137/11085743X1. Introduction. Higher-order tensors (three-way arrays) have been used as a tool in higher-order statistics [39,36,47,38] and independent component analysis (ICA) [13,14,19,9] for several decades. Other application areas include chemometrics, scientific computing, biomedical signal processing, image processing, and telecommunications. For an exhaustive list and references we refer to [46,34,32,8,12].Let us first consider the general low multilinear rank approximation of thirdorder tensors. The problem consists of finding the best approximation of a given tensor A ∈ R I1×I2×I3 , subject to a constraint on the multilinear rank of the approximation. The concept of multilinear rank was first introduced in [25,26] and is simply a generalization of the row and column rank of matrices to higher-order tensors. We define the problem more precisely in the next section.A closed-form solution of the best low multilinear rank approximation problem is not known. A generalization of the singular value decomposition (SVD) [23, sect. 2.5] called higher-order SVD (HOSVD) has been studied in [15]. A variation of this decomposition is know as the Tucker decomposition [49,50]. In general, truncation of the HOSVD leads to a good but not necessarily to the best low multilinear rank approximation. Recent iterative algorithms solving the problem include geometric Newton [21,29]

We consider the problem of approximating an affinely structured matrix, for example a Hankel matrix, by a low-rank matrix with the same structure. This problem occurs in system identification, signal processing and computer algebra, among others. We impose the low-rank by modeling the approximation as a product of two factors with reduced dimension. The structure of the low-rank model is enforced by introducing a penalty term in the objective function. The proposed local optimization algorithm is able to solve the weighted structured low-rank approximation problem, as well as to deal with the cases of missing or fixed elements. In contrast to approaches based on kernel representations (in linear algebraic sense), the proposed algorithm is designed to address the case of small targeted rank. We compare it to existing approaches on numerical examples of system identification, approximate greatest common divisor problem, and symmetric tensor decomposition and demonstrate its consistently good performance.

An increasing number of applications are based on the manipulation of higher-order tensors. In this paper, we derive a differential-geometric Newton method for computing the best rank-(R 1 , R 2 , R 3 ) approximation of a third-order tensor. The generalization to tensors of order higher than three is straightforward. We illustrate the fast quadratic convergence of the algorithm in a neighborhood of the solution and compare it with the known higher-order