Analysis of Interactions Among Hidden Components for Tucker Model

Tensor representation and tensor decompositions are natural approaches to deal with large amounts of data with multiple aspects and high dimensionality in modern applications, such as environmental analysis, chemometrices, pharmaceutical analysis, spectral analysis, neuroscience. The two most popular decomposition/factorization models for N -th order tensors are the Tucker model and the more restricted PARAFAC model. The Tucker decomposition allows for the extraction of different numbers of factors in each of the modes, and permits interactions within each modality while PARAFAC does not. This advantage, however, is also one of the weakness of this decomposition. The difficult problem is to identify the dominant relationships between components, and to establish unique representation. In this paper, we will introduce a new measure index which is called the Joint Rate (JR) index, in order to evaluate interactions among various components in the general Tucker decomposition. The Hinton diagram is also extended to 3-D visualization. The use of the JR index will be illustrated with the analysis of EEG data for classification and BCI applications. I. TENSOR DECOMPOSITIONS AND INTERACTIVE RELATION AMONG THEIR HIDDEN COMPONENTS Standard matrix factorizations, such as PCA/SVD, ICA, NMF, and their variants, are invaluable tools for feature selection, dimensionality reduction, noise reduction, and data mining [1]. However, they have only two modes or 2-way representations, and their use is therefore limited. In many applications such as studies in neuroscience, the data structures often contain higher-order ways (modes) such as trials, task conditions, subjects, and groups together with the intrinsic dimensions of space, time, and frequency. If the data for every subject were analyzed separately by extracting a matrix or slice from a data block, we would lose the covariance information among subjects. To discover hidden components within the data and retain the integrative information, the analysis tools should reflect the multi-dimensional structure of the data [2], [3]. This way all dimensions or modes are retained by virtue of multi-linear models which often produce unique and physically meaningful components. The two most popular decompositions/factorizations for N th order tensors are the Tucker model and the more restricted PARAFAC model. The Tucker decomposition is described as a “decomposition of a given N -th order tensor Y ∈ R I1×I2···×IN into an unknown core tensor G ∈ R12N multiplied by a set of N unknown component matrices, A (n) = [a (n) 1 ,a (n) 2 , . . . ,a (n) Jn ] ∈ Rnn (n = 1, 2, . . . , N ), = Y G E + ( ) I J 3 3