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).

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 $T\in F^{d_{1}\times d_{2}\times d_{3}}$, where $F$ is either $\mathbb {R}$ or $\mathbb {C}$, Tucker Decomposition can be denoted as follows,

$$T={\mathcal {T}}\times _{1}U^{(1)}\times _{2}U^{(2)}\times _{3}U^{(3)}$$

where ${\mathcal {T}}\in F^{d_{1}\times d_{2}\times d_{3}}$ 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 ${\mathcal {T}}$ respectively. $U^{(1)},U^{(2)},U^{(3)}$ are unitary matrices in $F^{d_{1}\times d_{1}},F^{d_{2}\times d_{2}},F^{d_{3}\times d_{3}}$ respectively. The j-mode product (j = 1, 2, 3) of ${\mathcal {T}}$ by $U^{(j)}$ is denoted as ${\mathcal {T}}\times U^{(j)}$ with entries as

${\begin{aligned}({\mathcal {T}}\times _{1}U^{(1)})(d_{1},d_{2},d_{3})&=\sum _{i_{1}=1}^{d_{1}}{\mathcal {T}}(i_{1},d_{2},d_{3})U^{(1)}(d_{1},i_{1})\\({\mathcal {T}}\times _{2}U^{(2)})(d_{1},d_{2},d_{3})&=\sum _{i_{2}=1}^{d_{2}}{\mathcal {T}}(d_{1},i_{2},d_{3})U^{(2)}(d_{2},i_{2})\\({\mathcal {T}}\times _{3}U^{(3)})(d_{1},d_{2},d_{3})&=\sum _{i_{3}=1}^{d_{3}}{\mathcal {T}}(d_{1},d_{2},i_{3})U^{(3)}(d_{3},i_{3})\end{aligned}}$

There are two special cases of Tucker decomposition:

Tucker1: if $U^{(2)}$ and $U^{(3)}$ are identity, then $T={\mathcal {T}}\times _{1}U^{(1)}$

Tucker2: if $U^{(3)}$ is identity, then $T={\mathcal {T}}\times _{1}U^{(1)}\times _{2}U^{(2)}$ .

RESCAL decomposition can be seen as a special case of Tucker where $U^{(3)}$ is identity and $U^{(1)}$ is equal to $U^{(2)}$ .

See also

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