A tensor decomposition expresses a tensor as a combination of simpler tensors. For a multidimensional array 
, the canonical polyadic (CP) decomposition,
also called the CANDECOMP/PARAFAC decomposition, has the form
 |
where 
denotes the tensor direct product. Each summand
is an 
-fold
tensor direct product of vectors
and has tensor decomposition rank one;
an exact decomposition uses equality in place of approximation. The minimum number
of summands in an exact CP decomposition is the tensor
decomposition rank. For a symmetric tensor,
a decomposition using repeated factors defines symmetric
tensor rank. The Comon conjecture asserted
equality of these two ranks.
The Tucker decomposition represents the tensor by a smaller core tensor 
and one factor matrix for each
mode,
 |
where 
denotes multiplication along the 
th index. CP and Tucker decompositions generalize aspects of
the matrix singular value decomposition;
the Tucker decomposition is also a higher-dimensional analog of principal
component analysis. Tensor decompositions are used for compression, dimensional
reduction, and the analysis of multiway data.
