The concept of tensor rank was introduced in the twenties. In the seventies, when methods of Component Analysis on arrays with more than two indices became popular, tensor rank became a much studied topic. The generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor. We explain in this paper how to obtain numerically in the complex field t...

Tensors are an integral part of many scientific disciplines [22], [16], [2], [12], [11]. At their most basic, tensors are simply a multidimensional collection of data (or a multidimensional array, as expressed in many programming languages). In other cases, tensors represent multidimensional transformations, extending the theory of vectors and matrices. The logic of handling, transforming, and ...

Operations with tensors, or multiway arrays, have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-1 outer products using either the CANDECOMP/PARAFAC (CP) or the Tucker models, or some variation thereof. Such decompositions are motivated by specific applications where the goal is to find an approximate such representation for ...

A novel algorithm is proposed for CANDECOMP/PARAFAC tensor decomposition to exploit best rank-1 tensor approximation. Different from the existing algorithms, our algorithm updates rank-1 tensors simultaneously in-parallel. In order to achieve this, we develop new all-at-once algorithms for best rank-1 tensor approximation based on the Levenberg-Marquardt method and the rotational update. We sho...