Tucker decomposition (original) (raw)
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 (HOSVD). It may be regarded as a more flexible PARAFAC (parallel factor analysis) model. In PARAFAC the core tensor is restricted to be "diagonal". Tucker1: if and are identity, then
| Property | Value |
|---|---|
| dbo:abstract | 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 (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 can be seen as a special case of Tucker where is identity and is equal to . (en) |
| dbo:wikiPageID | 18964319 (xsd:integer) |
| dbo:wikiPageLength | 4646 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1102167371 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Frank_Lauren_Hitchcock dbr:Frobenius_norm dbr:Factor_analysis dbr:Principal_component_analysis dbr:Tensor dbr:Higher-order_singular_value_decomposition dbc:Dimension_reduction dbr:Multilinear_principal_component_analysis dbr:PARAFAC dbr:Ledyard_R._Tucker |
| dbp:wikiPageUsesTemplate | dbt:Reflist dbt:Short_description dbt:Statistics-stub |
| dct:subject | dbc:Dimension_reduction |
| rdfs:comment | 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 (HOSVD). It may be regarded as a more flexible PARAFAC (parallel factor analysis) model. In PARAFAC the core tensor is restricted to be "diagonal". Tucker1: if and are identity, then (en) |
| rdfs:label | Tucker decomposition (en) |
| owl:sameAs | freebase:Tucker decomposition wikidata:Tucker decomposition https://global.dbpedia.org/id/4wf66 |
| prov:wasDerivedFrom | wikipedia-en:Tucker_decomposition?oldid=1102167371&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Tucker_decomposition |
| is dbo:knownFor of | dbr:Ledyard_Tucker |
| is dbo:wikiPageDisambiguates of | dbr:Tucker |
| is dbo:wikiPageRedirects of | dbr:Tucker3 |
| is dbo:wikiPageWikiLink of | dbr:Educational_Testing_Service dbr:Ledyard_Tucker dbr:Singular_value_decomposition dbr:Knowledge_graph_embedding dbr:Principal_component_analysis dbr:L1-norm_principal_component_analysis dbr:Higher-order_singular_value_decomposition dbr:Tucker dbr:Multilinear_principal_component_analysis dbr:Tensor_rank_decomposition dbr:List_of_statistics_articles dbr:Multilinear_subspace_learning dbr:Outline_of_machine_learning dbr:Tensor_decomposition dbr:Tucker3 |
| is dbp:knownFor of | dbr:Ledyard_Tucker |
| is foaf:primaryTopic of | wikipedia-en:Tucker_decomposition |