Direct transformations yielding the knight's move pattern in 3 脳 3 脳 3 arrays
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- 作者：Jorge N. Tendeiro ; Jos M.F. Ten Berge ; Vartan Choulakian
- 关键词：Three-mode component analysis ; Tucker transformations ; Simplicity ; Grö ; bner basis
- 刊名：Chemometrics and Intelligent Laboratory Systems
- 出版年：15 November, 2013
- 年：2013
- 卷：129
- 期：Complete
- 页码：10-14
- 全文大小：901 K
文摘
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 脳 3 脳 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 鈥渒night's move pattern鈥?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 Gr枚bner basis. This approach seems to work almost surely for randomly generated arrays, whether symmetric or nonsymmetric.