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In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tuckeralthough 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.
It may be regarded as a more flexible PARAFAC 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 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- 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 ∈ F n 1 × n 2 × n 3 {\displaystyle T\in F^{n_{1}\times n_{2}\times n_{3}}} , where F {\displaystyle F} is either R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } , Tucker Decomposition can be denoted as follows,