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Title: Three-mode PCA for finding a solution intermediate between Tucker3 and Parafac Authors:  Aya Nakashima - Osaka University (Japan) [presenting]
Kohei Adachi - Osaka University (Japan)
Abstract: Three-mode PCA (3MPCA) refers to the modified PCA procedures specially designed for analyzing a three-mode data array. In 3MPCA, the data array is approximately decomposed into three loading matrices and a core array. This array describes the relationships among the components occupying the columns of the three loading matrices. Tucker3 and Parafac models are among popular 3MPCA ones. The Tucker3 model can be considered as too less restrictive, in that its core array is unconstrained. Thus, it is not easy to interpret the array. In contrast, the Parafac model is too restrictive, as the elements in its core array are forced to be zeros except the super-diagonal elements. Thus, Parafac likely provides the solutions with bad fit to data. Those discussions suggest that the model is useful which is intermediate between Tucker3 and Parafac, i.e., whose core array includes a suitable number of zero elements. For exploring such an intermediate model, we propose a new 3MPCA procedure. In this procedure, the Tucker3 loss function is minimized subject to the constraint that a specified number of core elements are exactly zeros, with which elements are zeros being unknown. Therefore, the optimal locations of the zero elements and nonzero parameter values are to be estimated simultaneously. For the estimation, we present an alternating least squares algorithm. Its behaviors are assessed in a simulation study and the proposed procedure is illustrated with real data examples.