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 , 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 k-mode product (k = 1, 2, 3) of by is denoted as with entries as
Altogether, the decomposition may also be written more directly 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
References
- ^ Ledyard R. Tucker (September 1966). "Some mathematical notes on three-mode factor analysis". Psychometrika. 31 (3): 279–311. doi:10.1007/BF02289464. PMID 5221127.
- ^ 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. Vol. 11. pp. 809–816.