Tensor decompositions in Engineering

Description

In Engineering, the identification of a linear statistical model is omnipresent (antenna array processing, factor analysis, etc). This problem has been extensively addressed since the fifties, refer e.g. to early contributions of Darmois and Skitovich. It consists of estimating a mixing matrix from observed realizations, under the assumption that the unobserved source random variables exciting the model are statistically independent. However, existence and uniqueness theorems were generally not constructive, in the sense that they did not yield practical algorithms. It is shown how such problems can be stated in terms of the decomposition of symmetric tensors into a sum of rank-1 terms. For instance, Cumulant tensors have been used in different Engineering problems, and are symmetric as high-order derivatives of a multivariate scalar function. Various numerical algorithms are surveyed, and actually attempt to find a compromise between sub-optimality, numerical complexity, and simplicity. Principles underlying these algorithms are outlined, depending on the type of mixing matrix in the model: orthogonal, square invertible, or rectangular with full row rank. The case of square mixtures leads to approximate decompositions, whereas rectangular mixtures allow exact decompositions of so-called Under-determined Mixtures (UDM). It is pointed out during the course that some problems remain open. In some cases, the observation model itself is multi-linear, which may avoid to compute tensors (e.g. cumulants) from data. In that case, solutions become deterministic, and tensors to be decomposed generally do not enjoy symmetries.

Abstract

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