Duality based interpretation of uniqueness in the trilinear decompositions

Abstract

Application of trilinearity constraint in curve resolution of three-way data sets can play an important role in the triadic decompositions. Matrix Augmented-MCR-ALS is a two-way analysis method while its incorporation with the trilinearity constraint can provide a triadic decomposition like PARAFAC model. Trilinearity is a very strong constraint which can lead unique decomposition under mild conditions. Additionally, duality concept represents a relation between column and row spaces of bilinear data matrices. Thus, in case of a unique solution or uniqueness condition, the duality concept is a general and powerful approach for visualization. Based on the duality concept, it is necessary to define a particular hyper-plane in the dual abstract space in order to have a unique profile. It is discussed and visualized that the trilinearity constraint with the idea of parallel proportional profiles can define a particular hyper-plane in the space causing/leading to uniqueness in the dual space. Several simulated and real data sets were exemplified to show the ability of the duality concept to study and visualize the uniqueness due to the application of trilinearity constraint.