Definitions

Given $T$ a tensor decomposition $\sum _{i=0}^r{T}_{i1}\otimes T_{i2}\otimes T_{i3}$ where each summand is a rank-one tensor:

• the integer $r$ is the tensor rank of tensor decomposition $T$;

• the ordered list ${\left[{\left(rank{T}_{ij}\right)}_{j=1\cdots 3}\right]}_{i=1\cdots r}$ is called the type of tensor decomposition (where $rankM$ being the classical rank of the matrix $M$).

  The type could be summarized as the multivariate polynomial  $\sum _{i=0}^r{X}^{rank{T}_{i1}}{Y}^{rank{T}_{i2}}{Z}^{rank{T}_{i3}}$.