Tucker decomposition

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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. 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 , where is either or , 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 , which are defined as the Frobenius norm of the 1-mode, 2-mode and 3-mode slices of tensor respectively. are unitary matrices in respectively. The j-mode product (j = 1, 2, 3) of by is denoted as with entries as

Taking for all is always sufficient to represent exactly, but often can be compressed or efficiently approximately by choosing . 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 and are identity, then

Tucker2: if is identity, then .

RESCAL decomposition [3] can be seen as a special case of Tucker where is identity and is equal to .

See also[edit]

References[edit]

  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. Vol. 11. pp. 809–816.