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

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

Altogether, the decomposition may also be written more directly as

$T(i_{1},i_{2},i_{3})=\sum _{j_{1}=1}^{d_{1}}\sum _{j_{2}=1}^{d_{2}}\sum _{j_{3}=1}^{d_{3}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(1)}(j_{1},i_{1})U^{(2)}(j_{2},i_{2})U^{(3)}(j_{3},i_{3})$

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 $U^{(2)}$ 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