The independent component analysis (ICA) of a random vector consists of searching for a linear transformation that minimizes the statistical dependence between its components. In order to define suitable search criteria, the expansion of mutual information is utilized as a function of cumulants of increasing orders. An efficient algorithm is proposed, which allows the computation of the ICA of a data matrix within a polynomial time. The concept oflCA may actually be seen as an extension of the principal component analysis (PCA), which can only impose independence up to the second order and, consequently, defines directions that are orthogonal. Potential applications of ICA include data analysis and compression, Bayesian detection, localization of sources, and blind identification and deconvolution. Zusammenfassung Die Analyse unabhfingiger Komponenten (ICA) eines Vektors beruht auf der Suche nach einer linearen Transformation, die die statistische Abh~ingigkeit zwischen den Komponenten minimiert. Zur Definition geeigneter Such-Kriterien wird die Entwicklung gemeinsamer Information als Funktion von Kumulanten steigender Ordnung genutzt. Es wird ein effizienter Algorithmus vorgeschlagen, der die Berechnung der ICA ffir Datenmatrizen innerhalb einer polynomischen Zeit erlaubt. Das Konzept der ICA kann eigentlich als Erweiterung der 'Principal Component Analysis' (PCA) betrachtet werden, die nur die Unabh~ingigkeit bis zur zweiten Ordnung erzwingen kann und deshalb Richtungen definiert, die orthogonal sind. Potentielle Anwendungen der ICA beinhalten Daten-Analyse und Kompression, Bayes-Detektion, Quellenlokalisierung und blinde Identifikation und Entfaltung. R~sum~ L'Analyse en Composantes Ind6pendantes (ICA) d'un vecteur al6atoire consiste en la recherche d'une transformation lin6aire qui minimise la d6pendance statistique entre ses composantes. Afin de d6finir des crit6res d'optimisation appropribs, on utilise un d6veloppement en s6rie de l'information mutuelle en fonction de cumulants d'ordre croissant. On propose ensuite un algorithme pratique permettant le calcul de I'ICA d'une matrice de donn6es en un temps polynomial. Le concept d'ICA peut 6tre vu en r~alitb comme une extension de l'Analyse en Composantes Principales (PCA) qui, elle, ne peut imposer l'ind6pendance qu'au second ordre et d6finit par cons6quent des directions orthogonales. Les applications potentielles de I'ICA incluent l'analyse et la compression de donn6es, la d&ection bayesienne, la localisation de sources, et l'identification et la d6convolution aveugles.

A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank-1 order-k tensor is the outer product of k non-zero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1.

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aThis work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple 'bottlenecks', and on 'swamps'. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the enhanced line search (ELS) enhancement in these algorithms is discussed. Computer simulations feed this discussion.

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.