Multilinear Algebra

Multilinear Algebra

The multilinear algebra is a very powerful tool, which is applied in several science fields, such as psychometrics, telecommunications, chemistry, and medical imaging. By using tensor algebra, we obtain several advantages, which are not possible by using matrices. One first advantage is the identifiability which means that the tensor rank can largely exceed its dimensions. By using tensors, we have the uniqueness of the data, which means that the decomposed data is unique, except by permutation and scaling ambiguities. Finally, we can apply to the tensors the multilinear rank reduction, which gives us a significant improvement of the accuracy of parameters.

Our objective is to propose new operators, decompositions and tensor based techniques by using concepts of multilinear algebra.

Figure 1. Sum of three rank of tensors.

In Fig. 1, we consider an example of a 3-D data composed by the sum of three rank one tensors. Usually, depending on the field, each rank one tensor has a different interpretation. For instance, in the telecommunications, each rank one tensor is related to one path of the MIMO channel.

One example of multilinear operator is the n-mode product, which is exemplified below.

Figure 2. n-mode product

With the n-mode product operator, it is possible to make a product between a tensor and a matrix