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 rank one; an exact decomposition uses
equality in place of approximation.
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.
