A new array decomposition method for multiway data analysis

In chemometrics, two-way singular value decomposition (SVD), CANDECOMP–PARAFAC decomposition (PARAFAC), and Tucker decomposition (TUKER) are three main array decomposition methods. There are disadvantages with the three methods. If multiway data are indeed multilinear, PARAFAC and TUCKER can provide more robust and interpretable models compared to two-way SVD. However, PARAFAC is sometimes numerically unstable, and TUCKER cannot guarantee the uniqueness of an approximate solution. This paper proposes a new array decomposition model with multiple bilinear structure. Then, utilizing this model, a new method, called multiple bilinear decomposition (MBD), is proposed as a generalization of two-way SVD. An algorithm is established to successively decompose an array without a full decomposition, which is not based on alternating least squares. Theoretically, the proposed method has an advantage over PARAFAC and TUCKER in its three important properties, including orthonormality of loading vectors, closed-form decomposition, and successive decomposition of variation. The simulation results based on orthogonal PARAFAC models show that the proposed method outperforms PARAFAC with respect to accuracy and robustness of loading estimate and data-fitting of model, even though the former does not use the priori information of multilinear structure. And, especially in the simulation under no noise, the equivalence of loading estimates indicates that as a successive decomposition, MBD is a superior alternative to PARAFAC.