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, $$T={𝒯}{\times }_1{U}^{(1)}{\times }_2{U}^{(2)}{\times }_3{U}^{(3)}$$ where ${𝒯}\in F^{d_1\times d_2\times d_3}$ 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 ${𝒯}$ 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 ${𝒯}$ by $U^{(k)}$ is denoted as ${𝒯}\times U^{(k)}$ with entries as

$\begin{array}{rl}({𝒯}{\times }_1{U}^{(1)})(i_1,j_2,j_3)& =\sum _{j_1=1}^{d_1}{𝒯}(j_1,j_2,j_3)U^{(1)}(j_1,i_1)\\ ({𝒯}{\times }_2{U}^{(2)})(j_1,i_2,j_3)& =\sum _{j_2=1}^{d_2}{𝒯}(j_1,j_2,j_3)U^{(2)}(j_2,i_2)\\ ({𝒯}{\times }_3{U}^{(3)})(j_1,j_2,i_3)& =\sum _{j_3=1}^{d_3}{𝒯}(j_1,j_2,j_3)U^{(3)}(j_3,i_3)\end{array}$

Altogether, the decomposition may also be written more directly as

$T(i_1,i_2,i_3)=\sum _{j_1=1}^{d_1}\sum _{j_2=1}^{d_2}\sum _{j_3=1}^{d_3}{𝒯}(j_1,j_2,j_3)U^{(1)}(j_1,i_1)U^{(2)}(j_2,i_2)U^{(3)}(j_3,i_3)$

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 $T={𝒯}{\times }_1{U}^{(1)}$ Tucker2: if $U^{(3)}$ is identity, then $T={𝒯}{\times }_1{U}^{(1)}{\times }_2{U}^{(2)}$ . 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