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

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

where ${\displaystyle {\mathcal {T}}\in F^{d_{1}\times d_{2}\times d_{3}}}$ is the core tensor, a 3rd-order tensor that contains the 1-mode, 2-mode and 3-mode singular values of ${\displaystyle T}$, which are defined as the Frobenius norm of the 1-mode, 2-mode and 3-mode slices of tensor ${\displaystyle {\mathcal {T}}}$ respectively. ${\displaystyle U^{(1)},U^{(2)},U^{(3)}}$ are unitary matrices in ${\displaystyle F^{d_{1}\times d_{1}},F^{d_{2}\times d_{2}},F^{d_{3}\times d_{3}}}$ respectively. The j-mode product (j = 1, 2, 3) of ${\displaystyle {\mathcal {T}}}$ by ${\displaystyle U^{(j)}}$ is denoted as ${\displaystyle {\mathcal {T}}\times U^{(j)}}$ with entries as

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

There are two special cases of Tucker decomposition:

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

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

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

L1-Tucker tensor decomposition is the L1-norm-based, corruption-resistant variant of Tucker.[4][5][6]

See also

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

References

1. Ledyard R. Tucker (September 1966). "Some mathematical notes on three-mode factor analysis". Psychometrika. 31 (3): 279–311. doi:10.1007/BF02289464. PMID 5221127.
2. F. L. Hitchcock (1927). "The expression of a tensor or a polyadic as a sum of products". Journal of Mathematics and Physics. 6: 164–189.
3. Nickel, Maximilian; Tresp, Volker; Kriegel, Hans-Peter (28 June 2011). A Three-Way Model for Collective Learning on Multi-Relational Data. ICML. 11. pp. 809–816.
4. Chachlakis, Dimitris G.; Prater-Bennette, Ashley; Markopoulos, Panos P. (22 November 2019). "L1-norm Tucker Tensor Decomposition". IEEE Access. 7: 178454–178465. doi:10.1109/ACCESS.2019.2955134.
5. Markopoulos, Panos P.; Chachlakis, Dimitris G.; Prater-Bennette, Ashley (21 February 2019). "L1-norm Higher-Order Singular-Value Decomposition". IEEE Proc. 2018 IEEE Global Conference on Signal and Information Processing. doi:10.1109/GlobalSIP.2018.8646385.
6. Markopoulos, Panos P.; Chachlakis, Dimitris G.; Papalexakis, Evangelos (April 2018). "The Exact Solution to Rank-1 L1-Norm TUCKER2 Decomposition". IEEE Signal Processing Letters. 25 (4). arXiv:1710.11306. doi:10.1109/LSP.2018.2790901.