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.
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 third-order tensor T (n1 x n2 x n3) , a core tensor R (r1 x r2 x r3) , and matrices A (n1 x r1) ;B (n2 x r2) ;C (n3 x r3) T ≈ R x1 A x2 B x3 C. Where x1 is mode-1 product and similarly x2 and x3. There are two special cases of Tucker decomposition (order three tensors):
Tucker1: B an C are identity, thus T ≈ R x1 A
Tucker2: C is identity, thus T ≈ R x1 A x2 B.
RESCAL decomposition  can be seen as a special case of Tucker where C is identity and A is equal to B.
- Ledyard R. Tucker (September 1966). "Some mathematical notes on three-mode factor analysis". Psychometrika. 31 (3): 279–311. doi:10.1007/BF02289464.
- 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.
- 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.
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