Uniqueness, rank, and simplicity of three-way arrays with symmetric slices
Faculty of Behavioural and Social Sciences
University of Groningen
Project executed at: University of Groningen
Project financed by: FCT (Fundaçāo para a Ciência e a Tecnologia, Portugal)
Project running from: 1 October 2006 – 1 October 2010
Promotores: Prof. dr J.M.F. Ten Berge, prof. dr H.A.L. Kiers
For the analysis of three-way data, various generalizations of Principal Components Analysis have been proposed. Tucker (1966) proposed 3-mode PCA, which decomposes a data array as a weighted sum of rank-one arrays (outer products of columns of matrices A, B and C), the weights for each outer product being an element of the so-called core array. Carroll and Chang (1970) and Harshman (1970) independently proposed a method which they christened Candecomp and Parafac, respectively. We refer to it as CP. It can be verified that CP is a special case of 3-mode PCA.
There is a great need to study more deeply several algebraic properties regarding three-way arrays, such as uniqueness, typical rank, and simplicity. These properties are crucial for a better understanding of the application of methods such as CP. In particular, attention will be devoted to three-way arrays with symmetric slices.
Date of defence: 28 October 2010
Title of thesis: Some mathematical results on three-way component analysis