Tucker Decomposition

In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker
although it goes back to Hitchcock in 1927.
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) or the M-mode SVD. The algorithm to which the literature typically refers when discussing the Tucker decomposition or the HOSVD is the M-mode SVD algorithm introduced by Vasilescu and Terzopoulos, but misattributed to Tucker or De Lathauwer etal.

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 $T \in F^$, where $F$ is either $\mathbb$ or $\mathbb$, Tucker Decomposition can be denoted as follows,
$T = \mathcal \times_ U^ \times_ U^ \times_ U^$
where $\mathcal \in F^$ is the ''core tensor'', a 3rd-order tensor that contains the 1-mode, 2-mode and 3-mode singular values of $T$, which are defined as the ''Frobenius norm'' of the 1-mode, 2-mode and 3-mode slices of tensor $\mathcal$ respectively. $U^, U^, U^$ are unitary matrices in $F^, F^, F^$ respectively. The ''k''-mode product (''k'' = 1, 2, 3) of $\mathcal$ by $U^$ is denoted as $\mathcal \times U^$ with entries as
:$\begin (\mathcal \times_ U^)(i_, j_, j_) &= \sum_^ \mathcal(j_, j_, j_)U^(j_, i_) \\ (\mathcal \times_ U^)(j_, i_, j_) &= \sum_^ \mathcal(j_, j_, j_)U^(j_, i_) \\ (\mathcal \times_ U^)(j_, j_, i_) &= \sum_^ \mathcal(j_, j_, j_)U^(j_, i_) \end$

Altogether, the decomposition may also be written more directly as
:$T(i_, i_, i_) = \sum_^ \sum_^ \sum_^ \mathcal(j_, j_, j_) U^(j_, i_) U^(j_, i_) U^(j_, i_)$

Taking $d_i = n_i$ for all $i$ is always sufficient to represent $T$ exactly, but often $T$ can be compressed or efficiently approximately by choosing $d_i < n_i$. A common choice is $d_1 = d_2 = d_3 = \min(n_1, n_2, n_3)$, which can be effective when the difference in dimension sizes is large.

There are two special cases of Tucker decomposition:

Tucker1: if $U^$ and $U^$ are identity, then $T = \mathcal \times_ U^$

Tucker2: if $U^$ is identity, then $T = \mathcal \times_ U^ \times_ U^$ .

RESCAL decomposition can be seen as a special case of Tucker where $U^$ is identity and $U^$ is equal to $U^$ .

See also

* Higher-order singular value decomposition
* Multilinear principal component analysis

References

Dimension reduction
{{statistics-stub