Rotational freedom and uniqueness in three-way component models

2 Rotational Freedom and Uniqueness in Three-way Component Models 

Uniqueness of models is an important issue. A unique model is a model where all parameters are identified under the said premises, e.g. structure and additional constraints such as orthogonality. Requiring a model of a matrix to be bilinear is not sufficient for identifying the parameters of the model uniquely. However, if additional constraints are used, the model can become unique. In principal component analysis it is additionally required that the components are orthogonal and extracted such that the first component explains as much variation as possible the second explains as much variation of the yet unexplained part, etc. Applying these additional constraints makes the bilinear model unique (apart from sign indeterminacies). 

In order to discuss uniqueness and rotational freedom, a clear definition of uniqueness is necessary. There are different levels of uniqueness. Consider, e.g., aPCA model of X (/ x J) 

If the matrix S is an orthogonal matrix (S S = SS = I), then the transformation is called an orthogonal transformation. An orthogonal transformation is a reflection and/or a rotation. The property, as exemplified in and below Equation (5.15), is called rotational freedom and although this term is rather sloppy, it is commonly used. 

Regardless of whether the parameters of the Tucker3 model are uniquely estimated, it can be shown that, like PCA, the Tucker3 model provides unique subspaces [Kroonenberg etal. 1989, Lathauwer 1997], 

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