This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or N -way array. Decompositions of higher-order tensors (i.e., N -way arrays with N ≥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, etc. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decompo- sition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal components analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox and Tensor Toolbox, both for MATLAB, and the Multilinear Engine are examples of software packages for working with tensors.

L’évaluation numérique à grande précision de constantes comme π, e, γ, ln 2, etc., de fonctions élémentaires comme exp et arctan, puis de fonctions spéciales comme Γ ou ζ est un problème classique. D’un point de vue informatique, « grande précision » s’oppose à précision fixe, mais sous-entend aussi que l’on cherche des algorithmes asymptotiquement efficaces quand le nombre de chiffres demandés grandit. Le développement d’algorithmes de complexité quasilinéaire en le nombre de chiffres du résultat remonte aux années 1970, avec par exemple les travaux de Richard Brent, Eugene Salamin ou R. William Gosper. Les fonctions holonomes sont les solutions d’équations différentielles linéaires à coefficients polynomiaux. Leurs propriétés élémentaires sont bien connues depuis le dix-neuvième siècle, mais elles ont pris une place importante en combinatoire (comme séries génératrices) et en calcul formel (en tant que classe de fonctions bénéficiant de propriétés algorithmiques agréables, tant du point de vue de la calculabilité que de celui de la complexité) depuis les années 1980. Parmi les responsables de ce regain d’intérêt, on peut citer Richard Stanley, Leonard Lipshitz et Doron Zeilberger. Mon travail de stage s’incrit dans une démarche générale du projet Algo de développer pour toute la classe des fonctions holonomes une algorithmique efficace utilisant

We consider the multiplication of a sparse N × N matrix A with a dense N × N matrix B in the I/O model. We determine the worst-case non-uniform complexity of this task up to a constant factor for all meaningful choices of the parameters N (dimension of the matrices), k (average number of non-zero entries per column or row in A, i.e., there are in total kN non-zero entries), M (main memory size), and B (block size), as long as M ≥ B 2 (tall cache assumption). For large and small k, the structure of the algorithm does not need to depend on the structure of the sparse matrix A, whereas for intermediate densities it is possible and necessary to find submatrices that fit in memory and are slightly denser than on average. The focus of this work is asymptotic worst-case complexity, i.e., the existence of matrices that require a certain number of I/Os and the existence of algorithms (sometimes depending on the shape of the sparse matrix) that use only a constant factor more I/Os. 1

Abstract. We investigate differences between X-rank and X-border rank, focusing on the cases of tensors and partially symmetric tensors. As an aid to our study, and as an object of interest in its own right, we define notions of X-rank and border rank for a linear subspace. Results include determining and bounding the maximum X-rank of points in several cases of interest. 1.

sous la direction de Bruno SALVY Génération automatique de procédures numériques pour les fonctions D-finies Rapport de stage de Master 2

Abstract. We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring’s problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester’s approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasi-Hankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding

Abstract. The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.

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Algebraic Geometry is a deep and well-established field within pure mathematics that is increasingly finding applications outside of mathematics. These applications in turn are the source of new questions and challenges for the subject. Many applications flow from and contribute to the more combinatorial and computational parts of algebraic geometry, and this often involves real-number or positivity questions. The scientific development of this area devoted to applications of algebraic geometry is facilitated by the sociological development of administrative structures and meetings, and by the development of human resources through the training and education of younger researchers. One goal of this project is to deepen the dialog between algebraic geometry and its applications. This will be accomplished by supporting the research of Sottile in applications of algebraic geometry and in its application-friendly areas of combinatorial and computational algebraic geometry. It will be accomplished in a completely different way by supporting Sottile’s activities as an officer within SIAM and as an organizer of scientific meetings. Yet a third way to accomplish this goal will be through Sottile’s training and mentoring of graduate students, postdocs, and junior collaborators.