In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor.
It is named after Ledyard R. Tucker[1] although it goes back to Hitchcock in 1927.
[2] Initially described as a three-mode extension of factor analysis and principal component analysis it may actually be generalized to higher mode analysis, which is also called higher-order singular value decomposition (HOSVD).
It may be regarded as a more flexible PARAFAC (parallel factor analysis) model.
In PARAFAC the core tensor is restricted to be "diagonal".
In practice, Tucker decomposition is used as a modelling tool.
For instance, it is used to model three-way (or higher way) data by means of relatively small numbers of components for each of the three or more modes, and the components are linked to each other by a three- (or higher-) way core array.
The model parameters are estimated in such a way that, given fixed numbers of components, the modelled data optimally resemble the actual data in the least squares sense.
The model gives a summary of the information in the data, in the same way as principal components analysis does for two-way data.
For a 3rd-order tensor
, Tucker Decomposition can be denoted as follows,
is the core tensor, a 3rd-order tensor that contains the 1-mode, 2-mode and 3-mode singular values of
, which are defined as the Frobenius norm of the 1-mode, 2-mode and 3-mode slices of tensor
are unitary matrices in
The k-mode product (k = 1, 2, 3) of
is denoted as
with entries as Altogether, the decomposition may also be written more directly as Taking
is always sufficient to represent
can be compressed or efficiently approximately by choosing
A common choice is
= min (
, which can be effective when the difference in dimension sizes is large.
There are two special cases of Tucker decomposition: Tucker1: if
are identity, then
is identity, then
RESCAL decomposition [3] can be seen as a special case of Tucker where
is identity and
is equal to
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