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Direct transformations yielding the knight's move pattern in 3x3x3 arrays

Tendeiro, J., ten Berge, J. M. F. & Choulakian, V., 15-Nov-2013, In : Chemometrics and Intelligent Laboratory Systems. 129, p. 10-14 5 p.

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Three-way arrays (or tensors) can be regarded as extensions of the traditional two-way data matrices that have a third dimension. Studying algebraic properties of arrays is relevant, for example, for the Tucker three-way PCA method, which generalizes principal component analysis to three-way data. One important algebraic property of arrays is concerned with the possibility of transformations to simplicity. An array is said to be transformed to a simple form when it can be manipulated by a sequence of invertible operations such that a vast majority of its entries become zero. This paper shows how 3 × 3 × 3 arrays, whether symmetric or nonsymmetric, can be transformed to a simple form with 18 out of its 27 entries equal to zero. We call this simple form the “knight's move pattern” due to a loose resemblance to the moves of a knight in a game of chess. The pattern was examined by Kiers, Ten Berge, and Rocci. It will be shown how the knight's move pattern can be found by means of a numeric–algebraic procedure based on the Gröbner basis. This approach seems to work almost surely for randomly generated arrays, whether symmetric or nonsymmetric.
Original languageEnglish
Pages (from-to)10-14
Number of pages5
JournalChemometrics and Intelligent Laboratory Systems
Volume129
Publication statusPublished - 15-Nov-2013

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