Tensor ranks and some properties of tensor spaces
In: 2nd Workshop on Tensor Decompositions and Applications, 29 Aug - 02 Sept 2005, Luminiy, France.
The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the High-Order SVD, and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the Tensor Rank. Important basic questions can be raised, such as the maximal achievable rank of a tensor of given dimensions, or the computation of a CanD. Some questions are answered, and it turns out that the answers depend on the choice of the underlying field, and on tensor symmetry structure, which outlines major differences compared to matrices.