Tensor decomposition

Remarks

The smallest length of a CP decomposition of $T$ is the tensor rank of $T$. A CP decomposition is a Tucker decomposition whose core is diagonal, with $r_1 = \cdots = r_d = r$. Example: the $2 \times 2 \times 2$ array with $t_{111} = t_{222} = 1$ and all other entries $0$ has the CP decomposition $e_1 \otimes e_1 \otimes e_1 + e_2 \otimes e_2 \otimes e_2$ of length $2$. Check: the first term has only the entry $(1,1,1)$ equal to $1$, and the second only the entry $(2,2,2)$. Non-example: $e_1 \otimes e_1 \otimes e_1$ alone is not a CP decomposition of this array, since it misses the entry $t_{222} = 1$. For $d = 2$, a CP decomposition is a factorization $T = A B^T$ with $r$ columns in $A$ and $B$, and the singular value decomposition gives one of minimal length. For $d = 3$, Kruskal's theorem gives conditions under which a CP decomposition of minimal length is essentially unique. Here essentially unique means unique up to the order of the terms and the scaling of the factors. This has no analogue for matrices: a minimal factorization $T = A B^T$ of a matrix of rank at least $2$ is never unique in this sense.