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, where F is either \mathbb{R} or \mathbb{C}, Tucker Decomposition can be denoted as follows, where 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. are unitary matrices in respectively. The k-mode product (k = 1, 2, 3) of \mathcal{T} by U^{(k)} is denoted as with entries as Altogether, the decomposition may also be written more directly as 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, 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 Tucker2: if U^{(3)} is identity, then. RESCAL decomposition can be seen as a special case of Tucker where U^{(3)} is identity and U^{(1)} is equal to U^{(2)}.