The tensor decomposition rank, also called CP rank, of a tensor 
is the smallest number of decomposable summands in an exact tensor
decomposition (Kolda and Bader 2009). For 
, it is the least 
such that
 |
where 
and 
denotes the tensor direct product. The zero
tensor has decomposition rank 0, and a nonzero decomposable
tensor has decomposition rank 1. For order 2, this quantity
is the matrix rank.
This meaning of rank differs from tensor rank as the number of indices, also called tensor order. For a symmetric tensor, restricting each summand to repeated factors gives symmetric tensor rank. The Comon conjecture concerns the relation between these two decomposition ranks.
