Typical rank and indscal dimensionality for symmetric three-way arrays of order \(I\times 2\times 2\) or \(I\times 3\times 3\)

DOI10.1016/j.laa.2004.03.009zbMath1057.15002OpenAlexW2125518871MaRDI QIDQ1881083

Roberto Rocci, Nikolaos D. Sidiropoulos, Jos M. F. ten Berge

Publication date: 4 October 2004

Published in: Linear Algebra and its Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.laa.2004.03.009

zbMATH Keywords

CandecompParafacIndscalthree-mode component analysisthree-way ranktypical tensorial rankouter products of three vectorsperfect fitsymmetric arraysthree-way array

Mathematics Subject Classification ID

Factor analysis and principal components; correspondence analysis (62H25) Vector spaces, linear dependence, rank, lineability (15A03)

Related Items (13)

Symmetry transformations for square sliced three-way arrays, with applications to their typical rank ⋮ Symmetric tensor decomposition ⋮ A comparison of different notions of ranks of symmetric tensors ⋮ Simplicity and typical rank results for three-way arrays ⋮ The \(K\)-INDSCAL model for heterogeneous three-way dissimilarity data ⋮ Kruskal's condition for uniqueness in Candecomp/Parafac when ranks and \(k\)-ranks coincide ⋮ The Carroll and Chang conjecture of equal Indscal components when Candecomp/Parafac gives perfect fit ⋮ Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices ⋮ Subtracting a best rank-1 approximation may increase tensor rank ⋮ Simplicity transformations for three-way arrays with symmetric slices, and applications to Tucker-3 models with sparse core arrays ⋮ Degeneracy in Candecomp/Parafac and Indscal explained for several three-sliced arrays with a two-valued typical rank ⋮ On uniqueness conditions for Candecomp/Parafac and Indscal with full column rank in one mode ⋮ Generic and typical ranks of multi-way arrays

Cites Work

• On uniqueness in CANDECOMP/PARAFAC
• Simplicity of core arrays in three-way principal component analysis and the typical rank of \(p\times q\times 2\) arrays
• Some clarifications of the CANDECOMP algorithm applied to INDSCAL
• Kruskal's polynomial for \(2 \times{}2 \times{}2\) arrays and a generalization to \(2 \times{}n \times{}n\) arrays
• Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics
• Computational solutions for the problem of negative saliences and nonsymmetry in INDSCAL
• The typical rank of tall three-way arrays
• Analysis of individual differences in multidimensional scaling via an \(n\)-way generalization of ``Eckart-Young decomposition

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