The Tucker decomposition expresses a tensor by a core tensor and a factor matrix
along each index (Kolda and Bader 2009). For an order-
tensor, it has the form
 |
where 
denotes multiplication along index 
. An exact decomposition uses equality. If 
has size 
, the core 
has size 
and 
has size 
.
Taking 
gives compression and dimensionality reduction.
The factor matrices are often chosen to have columns that
form an orthonormal set. This makes the decomposition
a higher-order analog of principal component
analysis. Unlike the CANDECOMP/PARAFAC
decomposition, the core allows interactions among distinct components along different
indices.
