Wikipedia has a perhaps more accessible description.
In multilinear algebra, there does not exist a general decomposition method for multi-way arrays (also known as N-arrays, higher-order arrays, or data-tensors) with all the properties of a matrix singular value decomposition (SVD).
A matrix SVD simultaneously computes
(a) a rank-R decomposition and
(b) the orthonormal row/column matrices.
These two properties can be
captured separately by two different decompositions for multi-way
arrays. Property (a) is extended to higher order by a class of closely
related constructions known collectively as CP decomposition (named
after the two most popular and general variants, CANDECOMP and
PARAFAC). Such decompositions represent a tensor as the sum of the
n-fold outter products of rank-1 tensors, where n is the dimension of
the tensor indices.
Property (b) is extended to higher order by a
class of methods known variably as Tucker3, N-mode SVD, and N-mode
principal component analysis (PCA). (This article will use the general
term "Tucker decomposition".) These methods compute the othonormal
spaces associated with the different axes (or modes) of a tensor. The
Tucker decomposition is also used in multilinear subspace learning as
multilinear principal component analysis. This terminology was coined
by P. Kroonenberg in the 1980s, but it was later called multilinear
SVD and HOSVD (higher-order SVD) by L. De Lathauwer. Historically,
much of the interest in higher-order SVDs was driven by the need to
analyze empirical data, especial in psychometrics and chemometrics. As
such, many of the methods have been independently invented several
times, often with subtle variations, leading to a confusing
literature. Abstract and general mathematical theorems are rare
(though see Kruskal1 with regard to the CP decomposition); instead,
the methods are often designed for analyzing specific data types. The
2008 review article by Kolda and Bader2 provides a compact summary
of the history of these decompositions, and many references for
further reading.
Also, see this package for MathTensor. I think it got updated for v8.
"The creation of MathTensor will open a new chapter in the development of all fields of science and engineering that use serious tensor analysis." - Stephan Wolfram
All this said, it's hard to answer such a general question. Try to provide a specific example and some code of what you've tried.
